The diffusion coefficient is evaluated for some dipole molecules of fission products in water vapor taking account of the dipole-dipole interaction. Results are ...
Atomic Energy, Vol. 111, No. 3, January, 2012 (Russian Original Vol. 111, No. 3, September, 2011)
DIFFUSION OF FISSION-PRODUCT DIPOLE MOLECULES IN WATER VAPOR
A. A. Sorokin, V. M. Alipchenkov, A. E. Kiselev, and V. F. Strizhov
UDC 621.039.58
The diffusion coefficient is evaluated for some dipole molecules of fission products in water vapor taking account of the dipole-dipole interaction. Results are presented for CsI, CsOH, HI, and Cs2MoO4, which are the main components of the fission products in transport of radioactive iodine and cesium in the first loop of a reactor. It is shown that this contribution decreases the diffusion coefficient of the dipole molecules in water vapor as compared with the coefficient computed with the standard Lennard–Jones potential. The decrease of the diffusion coefficient can be substantial (several-fold) for molecules with a large dipole moment.
Modeling of the transport of radioactive fission products in the first loop of a reactor during a serious accident is a necessary condition for evaluating the consequences of radiological contamination of the environment. If insufficient heat is removed, the fuel elements heat up strongly and the coolant in the core becomes overheated (water in VVER is pressurized) and coolant vaporizes. In this case, the diffusion of fission products in overheated water vapor will be the main mechanism determining their settling and condensation on the surface of the process channels and thereby the strength of the possible source of radioactive fission products with a depressurized first loop. However, the particulars of the diffusion of dipole molecules in a polar gas, which water vapor is, are neglected in the models used in the modern computer codes ASTEC (France) and MELCOR (USA). Likewise, there are no experimental data on the diffusion of most dipole molecules of fission products in water vapor for conditions corresponding to the parameters of the first loop during an accident. It is important to note that some fission product molecules have a quite large dipole moment. Correspondingly, because of the dipole-dipole interaction such molecules have an additional attraction to polar water molecules. In turn, this increases the cross section of their collisions with molecules of the gaseous medium. As a result, the diffusion coefficient of dipole molecules of fission products in the polar medium of the coolant gas is lower than the diffusion coefficient in the absence of this effect. In consequence, their rate of settling on the surface of the process channel decreases, which ultimately increases the amount of radioactive vapor and aerosols flowing into the room of the protective shell in the case of an accident in which the first loop becomes unsealed. This qualitative analysis shows the necessity and importance of additional study of this effect. In this article, we present a theoretical evaluation of binary diffusion coefficients of dipole molecules of fission products in water vapor taking account of the dipole interaction. Specifically, results are presented for CSI, CsOH, HI, and Cs2MoO4, which are the main components of the fission products participating in the transport of radioactive iodine and cesium in the first loop. In the Chapman–Enskog approximation, the expression for the binary diffusion coefficient DAB of the minority impurity A (component of the fission products) in the carrier gas B (superheated water vapor) has the form [1, 2] D AB =
3 (2 πkT / M AB )1 / 2 , 16 Nπσ 2 Ω (1,1) AB
(1)
Institute of Problems in the Safe Development of Nuclear Energy, Russian Academy of Sciences (IBRAE RAN), Vol. 111, No. 3, pp. 154–158, September, 2011. Original article submitted May 18, 2011. 1063-4258/12/11103-0203 ©2012 Springer Science+Business Media, Inc.
203
where k is Boltzmann’s constant; T is the temperature of the gas; MAB = mM/(m + M); m and M are, respectively, the mass of the impurity and carrier gas molecules; N is the concentration of molecules in the mixture; σAB is the characteristic interaction length (collision diameter) of the impurity and carrier gas molecules; Ω(1,1) is the reduced collision integral of the molecules, which depends on the reduced temperature T* = kT/εAB; and εAB is the characteristic interaction energy of the molecules. In what follows, for notational convenience, the superscripts in the collision integral of the molecule will be omitted. Ordinarily, the collision diameter σAB and the interaction energy εAB of the molecules are determined using combination relations for the parameters of the interaction potentials of single-component (pure) substances, for example, σA = σAA and εA = εAA. The most widely used potential for pure substances is the Lennard–Jones pair interaction potential, for which there is a quite extensive and experimentally verified database [1, 2]. Simple combination relations which are often used in calculations are [2] σAB = 0.5(σA + σB); εAB = (σAεB)1/2. (2) The tabulated values of the collision integral Ω for the Lennard–Jones interparticle interaction potential as a function of temperature are indicated in [1, 3]. Analytical approximations are also available for the function Ω(T*), mainly for the Lennard–Jones potential [4, 5]. For example, the dependence in [4] has the form Ω=
A * B
(T )
+
C *
exp ( DT )
+
E *
+
exp ( FT )
G exp ( HT * )
,
(3)
where T* = kT/εAB, A = 1.06036, B = 0.15610, C = 0.19300, D = 0.47635, E = 1.03587, F = 1.52996, G = 1.76474, and H = = 3.89411. It is important to note that this dependence is applicable only for models with the Lennard–Jones interaction potential. For dipole molecules interacting with one another, the computational error in the diffusion coefficient according to models (1)–(3) can be quite large. A direct calculation of the collision integral for modeling the transport of fission products in the first loop is inefficient because of the large amounts of computer time required. For this reason, in integral codes such as MELCOR and SOPHAEROS/ASTEC, where model (1) is used with the parameters in the intermolecular interaction potential approximated according to combination relations (2), either tabulated values of the collision integral as a function of the reduced temperature or their analytical approximation in the (3) form are used. However, no modern computer code takes account of the effect of the dipole moment of the molecules of fission products on the nature of their diffusion in a polar gas of the carrying medium, which water vapor is. In the case where polar molecules interact with one another, the effective potential (Stockmeier potential) contains an additional term in the equation with the Lennard–Jones potential that takes account of the interaction of two dipoles and depends on the relative orientation of the molecules. In the approximation of equally probably orientations, the approximate expression for the collision integral averaged over the orientation of the molecules has the form [1, 3]: Ω(T * , δ , ω ) = Ω LJ (T * ) + A1
δ 2AB T*
,
(4)
where ω is the vector of angles determining the relative orientation of the dipoles [3], ΩLJ is the collision integral for the Lennard–Jones potential (3), and the constant A1 = 0.19. The reduced dipole parameter δAB depends on the dipole moment of the molecules [2]: μ2 (5) δ AB = (δ Aδ B )1 / 2 ; δ = . 2 εσ 3 where μ is the dipole moment of a molecule, C·m. Thus, ⎡ Ω(T * , δ , ω ) − Ω (T * ) ⎤ ∝ μ μ . LJ A B ⎢⎣ ⎥⎦ 204
TABLE 1. Parameters of the Interaction Potential of Dipole Molecules [6–8] Molecule
M, g/mole
σ, nm
ε, K
µ, 3.34·10–30 C·m
H2O
18
0.248
707
1.85
NH3
17
0.353
163
1.4
CsI
259.8
0.403
685
11.7
CsOH
149.9
0.39
692
7.1
Cs2MoO4
425.7
0.242
2931
3
HI
127.9
0.369
703
0.5
I2
253.8
0.374
801
1.3
Te
127.6
0.335
1850
0
Therefore, taking account of the dipole-dipole interaction of polar molecules of the fission products and water molecules in addition to the Lennard–Jones interaction potential increases the collision integral, and therefore decreases the diffusion coefficient of the polar molecules of fission products in water vapor and their rate of settling on a surface. This means that the concentration of radioactive fission products in the coolant volume and the magnitude of the outflow of radioactive fission products in the room of the protective shell increase in an accident where first-loop seal is depressurized. For this reason, it is obviously of interest to obtain a quantitative evaluation of the effect of a decrease of the diffusion coefficient of the dipole molecules of the fission products in water vapor taking account of the dipole-dipole interaction with water molecules in addition to the Lennard–Jones potential. As an example, we shall examine the calculation of the diffusion coefficient for single-component polar gases NH3 and H2O and compare them with measurements performed at different temperatures. The computational results for the binary diffusion coefficients of several fission products molecules in superheated water vapor, specifically, CsI, CsOH, Cs2MoO4, I2, and HI, are examined as an application of this model. It is important to note that these components are of the main interest for modeling the transport of radioactive cesium and iodine in the first loop during a serious accident with radioactive fission products flowing into the coolant. The interaction potential parameters required for the calculations are presented in Table 1. The computational results for the diffusion coefficients in single-component polar gases NH3 and H2O for different temperatures are presented in Table 2. The computational and experimental data are compared in this table also. The computational results were obtained using two models: the standard Lennard–Jones potential (LJ model – modeling of the collision integral using expression (3)) and the Lennard–Jones potential taking account of the dipole-dipole interaction of the molecules (LD + dip model – using expressions (3)–(5) to model the collision integral). Analysis shows that the computational results obtained taking account of the dipole-dipole interaction of polar molecules in addition to the standard Lennar–Jones interaction potential agree much better with the experimental data. This is especially true for the calculation of the diffusion coefficient for water vapor. Of course, this result is not new and is presented here only from the methodological standpoint. The modeling of transport coefficients in polar gases is examined in greater detail in [6, 10]. In summary, for molecules with a large dipole moment the dipole-dipole interaction with the polar molecules of the carrier gas can appreciably decrease the diffusion coefficient because the collision integral of the molecules increases. For this reason, it is important to examine the influence of this effect on the diffusion of the dipole molecules of fission products in the polar gas of the carrier medium, which superheated water vapor in the first loop of a reactor is at the stage where a serious accident is unfolding. The modeling results for the binary diffusion coefficient for several dipole molecules of fission products in water vapor are presented in Table 3. The data for the diffusion coefficient in single-component water vapor are presented for comparison. Since the dipole moment of Cs2MoO4 is unknown to the authors of the present article it was taken to be 10–29 C·m 205
TABLE 2. Diffusion Coefficient of Single-Component Mixtures of the Molecules NH3 and H2O for Different Temperatures and Pressures 105 Pa, cm2/sec T, K
Experiment
LJ model
LJ + dip model
NH3 [9] 273
0.155
0.201
0.152
373
0.307
0.358
0.301
473
0.504
0.549
0.487
573
0.733
0.769
0.705
673
0.993
1.016
0.951
H2O [6] 300
0.17
0.23
0.14
400
0.32
0.42
0.28
500
0.52
0.65
0.48
600
0.77
0.95
0.74
700
1.08
1.3
1.05
800
1.42
1.68
1.42
900
1.82
2.11
1.83
1000
2.26
2.6
2.29
1200
3.3
3.67
3.34
1400
4.4
4.9
4.5
1600
5.7
6.3
5.9
1800
7.2
7.73
7.36
2000
8.8
9.33
8.96
by analogy with structurally similar molecules, for example, H2SO4. In the absence of a dipole moment in this molecule the computational results correspond to the standard Lennard–Jones potential model. Analysis of Table 3 shows that the diffusion coefficient for different fission products molecules in water vapor differs appreciably from the diffusion coefficient for the water molecule. This means that using one diffusion coefficient for all components of the fission products in the case of a serious accident can result in large errors in calculations of their transport in the first loop of a reactor. However, the main computational result is that the effect of the dipole interaction on the diffusion coefficient of fission products dipole molecules in water vapor is large. For example, the diffusion coefficient for CsI and CsOH molecules in water vapor decreases by approximately a factor of 2–5 as compared with the value obtained neglecting this effect. Conversely, for fission products molecules with a large dipole moment, for example, I2 or HI and fission product components in the atomic state, for example, Te, model (3) with the parameters of the standard Lennard–Jones potential can be used to calculate the diffusion coefficient in water vapor. In summary, the calculations show that quite simple relations can be used to calculate the diffusion coefficients of different components of fission products in superheated water vapor. In so doing, the contribution of the dipole-dipole interaction with polar water molecules must be taken into account for dipole molecules of fission products in addition to the standard 206
TABLE 3. Binary Diffusion Coefficient of Fission Product Polar Molecules in a Mixture with Water Vapor for Different Temperatures and Pressures 105 Pa, cm2/sec T, K
LJ model + dip
LJ model
LJ model
LJ model + dip
LJ model
H2O
Te
300
0.14
0.1
0.103
0.012
0.106
0.026
500
0.48
0.28
0.28
0.062
0.3
0.123
700
1.05
0.55
0.551
0.175
0.59
0.31
900
1.83
0.91
0.9
0.37
0.96
0.61
1100
2.80
1.37
1.33
0.64
1.41
0.99
1300
3.93
1.9
1.82
1
1.93
1.46
1500
5.20
2.52
2.37
1.44
2.52
2
1700
6.61
3.21
2.97
1.96
3.16
2.62
1900
8.15
3.96
3.63
2.54
3.86
3.29
2100
9.8
4.78
4.34
3.19
4.61
4.03
CsI
I2
LJ model + dip
CsOH
Cs2MoO4
HI
300
0.105
0.095
0.114
0.112
0.13
0.07
500
0.295
0.281
0.32
0.318
0.34
0.25
700
0.58
0.56
0.63
0.63
0.66
0.54
900
0.96
0.94
1.03
1.03
1.1
0.95
1100
1.41
1.39
1.52
1.52
1.65
1.48
1300
1.94
1.91
2.08
2.08
2.3
2.12
1500
2.53
2.51
2.71
2.71
3.06
2.86
1700
3.18
3.16
3.41
3.41
3.92
3.71
1900
3.89
3.87
4.16
4.16
4.87
4.65
2100
4.65
4.63
4.97
4.97
5.91
5.67
Lennard–Jones interaction potential. Taking account of this contribution increases the integral cross section for collisions with water molecules and, in consequence, decreases the diffusion coefficient in water vapor. The decrease of the diffusion coefficient can be substantial (several-fold) for molecules with a large dipole moment. Corrections taking account of the particulars of the diffusion of polar molecules in a polar gas are now being incorporated in the PROFIT module (fission products and aerosols) in the SOKRAT integral code for modeling the transport of vapors of radioactive fission products in water vapor in the first loop of a reactor during accidents. This study was supported by the Russian Foundation for Basic Research (Project No. 11-08-00410-a).
REFERENCES 1.
J. Hirschfelder, C. Curtiss, and R. Bird, Molecular Theory of Gases and Liquids [Russian translation], Izd. Inostr. Lit., Moscow (1961). 207
2. 3. 4. 5. 6. 7. 8. 9. 10.
208
J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases [Russian translation], Mir, Moscow (1976). L. Monchick and E. Mason, “Transport properties of polar gases,” J. Chem. Phys., 35, 1676–1697 (1961). P. Neufeld, A. Jansen, and R. Aziz, “Empirical equations to calculate 16 of the transport collision integrals Ω(l,s) for the L-J(12-6) potential,” J. Chem. Phys., 57, 1100 (1972). L. P. Fokin, V. N. Popov, and A. N. Kalashnikov, “Analytical representation of collision integrals for the m-6 Lennard–Jones potentials in the EPIDIF database,” Teplofiz. Vys. Temp., 37, No. 1, 49–55 (1999). L. P. Fokin and A. N. Kalashnikov, “Viscosity and self-diffusion of rarefied water vapor: refinement of handbook data,” Teplofiz. Vys. Temp., 46, No. 5, 674–679 (2008). O. A. Osipov and V. I. Minkin, Handbook of Dipole Moments, Vysshaya Shkola, Moscow (1965). J. Henshaw, M. Miganelli, M. Newland, and A. Yates, Status of Diffusion and Thermochemical Data for Fission Product Species, AEA-TSD-0296, Harwell, UK (1994). Y. Singh and A. Gupta, “Transport and equilibrium properties of polar gases,” Chem. Phys., 52, 3064–3067 (1970). L. R. Fokin and A. N. Kalashnikov, “Handbook data in the viscosity of rarefied water vapor at temperatures to 2000–2500 K,” Teplofiz. Vys. Temp., 38, No. 2, 242–248 (2000).
