Article pubs.acs.org/jced
Ab Initio Values of the Gas Transport Properties of Hydrogen Isotopologues and Helium−Hydrogen Mixtures at Low Density Bo Song, Kai Kang, Zhuo Zhang, Xiaopo Wang,* and Zhigang Liu Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China S Supporting Information *
ABSTRACT: We employed the classical kinetic theory to calculate transport properties as a function of temperature for hydrogen isotopologues and helium−hydrogen mixtures. The viscosity, thermal conductivity, diffusion coefficient, and thermal diffusion factor were computed at low density and over a wide temperature range from 298 to 2000 K. The calculation utilized the spherically symmetric versions of the latest ab initio potentials for the interactions of the like and unlike gas molecules. The computed values are generally consistent with the limited experimental data for HD, 4He−H2, 4He− D2, 4He−HD, 3He−H2, 3He−D2, H2−HD, and D2−HD. Our results provide reliable transport property data for the considered species at temperatures where experimental information is absent.
1. INTRODUCTION Modern ab initio methods can describe the potentials for the interactions between small molecules to high accuracy, allowing the quantitative predictions of thermophysical properties with uncertainties as small as the best experimental measurements. For the He−He pair and H2−H2 pair, the intermolecular forces have been extensively studied because of their simple molecular structures. Recently, Przybytek and Patkowski along with other collaborators developed high-level potentials for the two systems, respectively (He−He in ref 1 and H2−H2 in ref 2). They determined interaction energies on very large basis sets with corrections for a number of important physical effects so that the new pair potentials are substantially better than any published to date. Garberoglio et al.3,4 and Song et al.5,6 respectively reported accurate values of the second virial coefficient and transport properties computed from these potentials not only for pure gases 4He, 3He, H2, D2, T2, HD, HT, and DT but also for their mixtures. The wide-ranging results are of interest in many applications related to helium and hydrogen. With properties now well-understood for pure helium and pure hydrogen, it is natural to consider helium−hydrogen mixtures. More recently, Bakr et al.7 constructed an interaction potential for the He−H2 complex using a variety of state-of-theart electronic structure methods. Their uncertainty in the van der Waals well region is an order of magnitude smaller than that of the best previous potential. Helium-4 and hydrogen (H2) are widely used in scientific investigations and engineering applications. The other isotopologues are interesting mainly from a theoretical point of view, but their transport properties are also important for the nuclear energy industry and analytical chemistry. Experimental data on both these pure isotopologues © XXXX American Chemical Society
and their mixtures are scarce. In this article, our work would be extended to calculate the low-density transport properties of hydrogen isotopologues and helium−hydrogen mixtures from the spherical versions of the new potentials for the interactions between like and unlike molecules. Calculations were performed for viscosity, thermal conductivity, diffusion coefficient, and thermal diffusion factor at temperatures 298− 2000 K. The systems include four pure gases: T2, HD, HT, and DT and 26 binary mixtures: 4He−H2, 4He−D2, 4He−T2, 4He− HD, 4He−HT, 4He−DT, 3He−H2, 3He−D2, 3He−T2, 3He− HD, 3He−HT, 3He−DT, H2−T2, H2−HD, H2−HT, H2−DT, D2−T2, D2−HD, D2−HT, D2−DT, T2−HD, T2−HT, T2−DT, HD−HT, HD−DT, HT−DT. In Section 2, we introduce our theoretical calculation. In Section 3, we describe the available experimental data for transport properties and compare it with our results, and we provide our conclusions in Section 4.
2. THEORETICAL EVALUATION OF TRANSPORT PROPERTIES We utilized the classical kinetic theory to compute transport properties, where the isotropic potentials of like and unlike interactions are necessary. The detailed information on the He2 and (H2)2 potentials were given in our previous publications5,6 so that they are not repeated here. For the He−H2 system, the 2D potential function of ref 7 can be simplified to a onedimensional form by including only the terms with a variable subscript of 0, Received: January 26, 2016 Accepted: April 25, 2016
A
DOI: 10.1021/acs.jced.6b00076 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
Article
V (R ) = (A 000 + A 001R + A 002 R2 + A 003R3)e−α0R C − ∑ fn (dR ) n00 Rn n = 6,8,10
We utilized a standard integration routine, the Clenshaw− Curtis quadrature, to evaluate the integrals in eqs 3−6. This procedure is designed for semi-infinite intervals and automatically uses nonlinear transformation and extrapolation to achieve user-specified relative tolerances for a user-specified function. The relative error was set to 2 × 10−4 in this study. Finally, for a pure gas with the molecular mass m, the viscosity η, the thermal conductivity λ, the self-diffusion coefficient D,, and the thermal diffusion factor αT are given by9
(1)
where V is the interaction energy as a function of the intermolecular distance R. The famous Tang−Toennies damping function f n is defined as n
fn (dR ) = 1 − e−dR
∑ m=0
(dR )m m!
(2)
All of the fit parameters in eqs 1 and 2 can be found directly in Table 4 of ref 7, but these parameters as well as the potential parameters ε/k and σ in atomic units are also listed for convenience in Table 1 of this work. Here ε/k and σ denote the well depth and the location of the zero of the potential energy function, respectively. Table 1. Parameters of the 1D Potential Energy Function V(R) of the He−H2 Dimer parameter
value
unit
α0 A000 A001 A002 A003 d C600 C800 C1000 ε/k σ
2.01155883 4.36363025454046 6.2752828628237 −8.441037182029 × 10−1 1.86986347364 × 10−2 3.16011458 3.97449506 6.87606662 × 101 2.67151004 × 102 4.665270 × 10−5 5.630480
bohr−1 hartree hartree·bohr−1 hartree·bohr−2 hartree·bohr−3 bohr−1 hartree·bohr6 hartree·bohr8 hartree·bohr10 hartree bohr
⎡ 1 + ( −1) ⎥ Q(l)(E) = 2π ⎢1 − 2(1 + l) ⎦ ⎣
∫0
∞
∫R
0
dR /R2 (4)
where R0 represents the closest distance during a molecular collision, 1 − b2 /R 02 − V (R 0)/E = 0
∫0
∞
(8)
D=
fD 3 (πk3T 3/m)1/2 8 P Ω(1,1)
(9)
αT =
15 (6C* − 5)(2A* + 5) (1 + κ0) 2 A*(16A* − 12B* + 55)
(10)
1 (6C* − 5)2 (2A* + 5)−1 8
(11)
⎧ ⎧ 1 2A* ⎨H * (7 − 8E*)⎨ 9 ⎩ 35/4 + 7A* + 4F * ⎩
+
[A*(7 − 8E*) − 7(6C* − 5)](35/8 + 28A* − 6F *) ⎫ ⎬ 42A*(2A* + 5) ⎭
−
⎤⎫ 7(6C* − 5) 5⎡ 3 − (7 − 8E*)⎥⎬ ⎢H * + ⎦⎭ 7⎣ 5(2A* + 5) 10
(12)
A* = Ω(2,2)/Ω(1,1)
(13)
B* = (5Ω(1,2) − 4Ω(1,3))/Ω(1,1)
(14)
C* = Ω(1,2)/Ω(1,1)
(15)
E* = Ω(2,3)/Ω(2,2)
(16)
F * = Ω(3,3)/Ω(1,1)
(17)
H * = (3B* + 6C* − 35/4)/(6C* − 5)
(18)
For a mixture of gases 1 and 2, the viscosity ηmix is computed according to,
(5)
Next, the collision integral Ω(l,s) is derived by integrating the cross section over the collision energy, Ω(l , s)(T ) = [(s + 1) ! (kT )s + 2 ]−1
fλ 75 (πk3T /m)1/2 (2,2) 64 Ω
A*−H* are derived from the combination of collision integrals Ω(l,s),
(1 − cosl θ )b db
1 − b2 /R2 − V (R )/E
λ=
κ0 =
where E denotes the collision energy, l a positive integer, and b the impact parameter. The deflection angle θ is evaluated as follows, ∞
(7)
fD = 1 +
(3)
θ(E , b) = π − 2b
fη 5 (πmkT )1/2 (2,2) 16 Ω
here P is the pressure of 1 atm (101.3 kPa). The formulations of fη and fλ can be found in the Appendix of Viehland et al.10 They are expressed by the ratios of 25 element determinants so they are not listed here. f D and κ0 are the corrections to the selfdiffusion coefficient and thermal diffusion factor, respectively,
In the classical kinetics theory of dilute gases, transport properties are evaluated by the following steps. First, the cross sections Q(l), depending on molecular collisions, are computed by,8 l ⎤−1
η=
ηmix =
Q(l)(E)e−E / kT E s + 1dE
1 + Zη X η + Yη
(19)
where
(6)
Xη =
where T is the temperature, s a positive integer, and k the Boltzmann constant (1.380658 × 10−23 J·K−1). B
x12 2x x x2 + 1 2 + 2 η1 η12 η2
(20) DOI: 10.1021/acs.jced.6b00076 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data Yη =
Article
2 2 2x x (m + m2)2 η12 x2 m ⎞ 3 * ⎛ x1 m1 + 1 2 1 + 2 2 ⎟⎟ A12 ⎜⎜ η12 η2 m1 ⎠ 5 ⎝ η1 m2 4m1m2 η1η2
(21)
Zη =
* − 5)2 ax1/(1 + bx1) Δ = 1.3(6C12
(22)
1 + Zλ Xλ + Yλ
x12 2x x x2 + 1 2 + 2 λ1 λ12 λ2
(26)
+
⎞m 4 * 1 ⎛ 12 * 1 (m1 − m2) ⎜ A12 − B12 + 1⎟ 1 + ⎠ m2 15 12 ⎝ 5 2 m1m2
⎛ 2m2 ⎞ 2 ⎟ ⎜ m2(m1 + m2) ⎝ m1 + m2 ⎠
(2,2) Ω11 (1,1) Ω12
(37)
(2,2) ⎛ Ω(2,2) 8(m1 + m2) Ω11 12 * ⎞ 22 × ⎜11 − B12 ⎟ + 1/2 (1,1) (1,1) ⎝ ⎠ 5 5(m1m2) Ω12 Ω12
(28)
(38)
2 2 ⎞ 4 * (m1 + m2) λ12 1 ⎛ 12 * Uy = A12 − ⎜ B12 + 1⎟ ⎠ 15 4m1m2 λ1λ 2 2⎝ 5
The expressions of S2 and Q2 can be derived from those of S1 and Q1 by simply exchanging subscripts for corresponding parameters. Since our calculations were performed classically using spherically averaged potentials at temperatures 298−2000 K, several factors would influence the accuracy of transport properties computed in this way. One possible source is the uncertainty of interaction potentials themselves. In the example of pure noble gases,11−13 two lower and upper bound potentials V−(R) = V(R) − U(R) and V+(R) = V(R) + U(R) were constructed, where U denotes the potential energy uncertainty. The uncertainty in a given property was evaluated as half the differences of values calculated with V−(R) and V+(R), respectively. For the He−H 2 and (H 2 ) 2 dimers, the corresponding lower- and upper-limit potentials are not available in the literature. On the other hand, the hydrogen potential used here has a relative uncertainty of 0.3% at its well depth,2 which is the same as that of argon potential.14 Therefore, we chose argon as a reference to assess the effect in an analogy manner. For the considered temperatures in this work, the calculations of argon12 indicates that the propagated
(29)
2 ⎤ λ ⎞ 4 * ⎡ (m1 + m2) ⎛ λ12 A12 ⎢ + 12 ⎟ − 1⎥ ⎜ ⎥⎦ 15 ⎢⎣ 4m1m2 ⎝ λ1 λ2 ⎠
⎞ 1 ⎛ 12 * ⎜ B12 + 1⎟ ⎠ 12 ⎝ 5
(36)
⎛ m − m 2 ⎞2 ⎛ 5 * 4m1m2A12 6 *⎞ ⎟ + Q 12 = 15⎜ 1 ⎟ ⎜ − B12 5 ⎠ (m1 + m2)2 ⎝ m1 + m2 ⎠ ⎝ 2
2 ⎞m 4 * 1 ⎛ 12 * 1 (m2 − m1) ⎜ A12 − B12 + 1⎟ 2 + ⎠ m1 15 12 ⎝ 5 2 m1m2
−
2(m1 + m2)2
⎡⎛ 5 ⎤ 6 *⎞ 2 8 2 *⎥ ⎟m + 3m + m1m2A12 × ⎢⎜ − B12 1 2 ⎣⎝ 2 ⎦ 5 ⎠ 5
(27)
Uz =
15m2(m1 − m2)
1/2
Q1 =
2
⎞ (m − m2)2 5 ⎛ 12 * ⎜ B12 − 5⎟ 1 ⎠ m1m2 *⎝ 5 32A12
(35)
where
(24)
Zλ = x12U1 + 2x1x 2UZ + x 22U2
−
(34)
1/2 (2,2) * m1 ⎛ 2m2 ⎞ Ω11 4m1m2A12 S1 = − ⎟ ⎜ (1,1) m2 ⎝ m1 + m2 ⎠ Ω12 (m1 + m2)2
(25)
U2 =
b = 10a(1 + 1.8c + 3c 2) − 1
⎛ ⎞ x1S1 − x 2S2 * − 5)⎜⎜ ⎟⎟ αT = (6C12 2 2 ⎝ x1 Q 1 + x1x 2Q 12 + x 2 Q 2 ⎠
(23)
x2 2x x x2 Yλ = 1 U1 + 1 2 UY + 2 U2 λ1 λ12 λ2
U1 =
(33)
and c = m2/m1. The thermal diffusion factor αT of a mixture is determined by,
where, Xλ =
(32)
(1,1) Ω12 21/2 8(1 + 1.8c)2 Ω(2,2) 22
a=
here x is the mole fraction. The subscripts 1 and 2 represent the values for the gases 1 and 2 respectively, whereas 12 denotes the quantities from the interaction potential between unlike molecules. The thermal conductivity λmix for a mixture is evaluated as follows,
λmix =
(31)
where
⎧ ⎡ (m + m )2 ⎛ η η ⎞ 3 * ⎪ 2 m1 2 ⎜⎜ 12 + 12 ⎟⎟ + 2x1x 2⎢ 1 A12 ⎨x1 η2 ⎠ 5 ⎪ ⎢⎣ 4m1m2 ⎝ η1 ⎩ m2 ⎫ ⎤ m ⎪ − 1⎥ + x 22 2 ⎬ ⎪ m1 ⎭ ⎥⎦
1/2 3 3 3 ⎡ πk T (m1 + m2) ⎤ 1 + Δ ⎥ ⎢ (1,1) 8⎣ 2m1m2 ⎦ P Ω12
D12 =
(30)
The internal degrees of freedom of diatomic molecules have a non-negligible effect on the final values of thermal conductivity. Equations 8 and 23 need to be modified to include this contribution when evaluating the thermal conductivity of gases containing hydrogen isotopologues. We refer the reader to ref 6 and references therein for the detailed formulations. The binary diffusion coefficient D12 between gases 1 and 2 can be obtained by, C
DOI: 10.1021/acs.jced.6b00076 J. Chem. Eng. Data XXXX, XXX, XXX−XXX
Journal of Chemical & Engineering Data
Article
uncertainty of transport properties reaches the maximum value of 0.05% at T = 1000 K. Here we employed isotropic potentials as the input data. The calculation in this manner also introduces additional error to the derived transport properties. In principle, transport properties of polyatomic gases are evaluated from full nonspherical potentials to account for the translational and the internal rotational motion of energy. However, for the systems involving hydrogen isotopologues, the interaction potentials are nearly isotropic, and the effects of the anisotropy on transport properties are small. The computation by Köhler and Schaefer15 and the measurement by Hermans et al.16 on hydrogen can yield some insight: The fractional difference (ηsph − η)/η is 0.06% and 0.08% at 200 and 293 K, respectively, and the corresponding values are 0.5% for λ in both papers. The deviations for thermal conductivity are significantly larger than those of viscosity, which is further validated by the recent work of Schaefer.17 Their precise calculation shows that the rotational inelastic contributions to the viscosity η are found of the order of
