A Deeper Look Into the Thermodynamic Perfection of the Debye Equation of State for Helium-3 Yonghua Huanga*

Available online at www.sciencedirect.com ScienceDirect

Physics Procedia 67 ( 2015 ) 582 – 590

25th International Cryogenic Engineering Conference and the International Cryogenic Materials Conference in 2014, ICEC 25–ICMC 2014 A deeper look into the thermodynamic perfection of the Debye equation of state for helium-3 Yonghua Huanga*

a Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University, Shanghai 20020, China

Abstract A new form of a state equation for helium-3 in wide range of temperature and pressure, based on a conceptual extrapolation from the Debye equation for specific heat of solid materials was previously developed. A deeper look into the performance of the state equation was recently considered necessary and valuable, due to some feedback from cryogenic applications. The evaluation of has been conducted with the Grüneisen parameter along isobar and isotherms, cubic sound velocity – pressure linearity, enthalpy and entropy at low pressures, and the virial coefficients as the benchmarks. Some similar analysis was applied to helium-4 for the sake of analogizing the common behavior of these isotopes, particularly in the critical region. The results confirmed the good thermodynamic perfection of the Debye state equation for helium-3 in most areas on the phase diagram as declared before.

©© 2015 2014 The The Authors. Authors. Published Published by Elsevierby Elsevier B.V. B.V. This is an open access article under the CC BY-NC-ND license (Peer-reviewhttp://creativecommons.org/licenses/by-nc-nd/4.0/ under responsibility of the organizing). committee of ICEC 25-ICMC 2014. Peer-review under responsibility of the organizing committee of ICEC 25-ICMC 2014 Keywords: helium-3; equation of state; thermodynamic properties; evaluation

Nomenclature T temperature ȡ density B(T) second virial coefficient h enthalpy C(T) third virial coefficient s entropy c sound velocity ī Grüneisen parameter cv specific heat Ĭ Debye temperature p pressure į reduced density v specific volume Ĳ reduced temperature

* Corresponding author. Tel.: +86-21-34206295; fax: +86-21-34206295. E-mail address: [email protected]

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICEC 25-ICMC 2014 doi: 10.1016/j.phpro.2015.06.079 Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590 583

1. Introduction

Helium-3 (3He) is one of the two isotopes of helium that shows attractive potential for as a cryogen to achieve lower temperatures down to mili-Kelvin or greater cooling capacity below about 10 K. The normal boiling temperature of helium-3 is 3.197 K, and it turns to superfluid below 0.0026 K, in marked contrast to 4He (the ordinary helium) with the normal boiling temperature of 4.23 K and which transitions to a superfluid phase below 2.1768 K. Without doubt, 3He is the last cryogenic fluid without a state equation covering the phase region from compressed liquid to superheated gas. Compared to 4He, the state equation for 3He in wide range of temperature and pressure was available much later until 2006 by Huang et al (2006). The Debye specific heat model for crystal was successfully applied/extended to a cryogenic fluid for the first time. Meanwhile, equilibrium equations over wide ranges with high precision along the vapor-liquid by Huang et al (2005), Huang and Chen (2006) and liquid-solid lines by Huang and Chen (2005) were also proposed. Based on these state equations and classical thermodynamics, a computer program named “He3Pak” was developed to simplify the calculation of 3He properties over a wide range of temperature from 0.003 K to 1500 K and pressures up to 20 MPa. Previously, the state equations were mostly evaluated by comparing the predicted p-v-T data and specific heat data with available measurements published in the history. This paper will focus on evaluating the equation of state from some new perspective such as the Grüneisen parameter, the sound velocity, enthalpy and entropy, and the virial coefficient, which will help to deeper examine the thermodynamic accuracy of the equation.

2. Debye equation of state for fluid 3He

The modified Debye equation of state developed for fluid 3He covering the normal liquid and gas regions using the Helmholtz potential energy (A) function was written as,

432i 24 i 11GW()CC25ii 26 W i ªºG AH 4014216217 CiiG () CC i Ci12(1)WG¦¦¦C (1) «»C3 (1)G C4 (1)W 11eeii 111 i ¬¼ 1 e 1 e (1) C 2 1 e 5W 4 i32 ªº()(CCCC5WGGW3ii 19 3 20 3 i 21 ) CC34 35 C6 (1/G 1) ¦ ¬¼ 1 e i 1 where the Debye temperature Ĭ is a function only of density for fluid rather than a constant as for solids. The details of the other variables and parameters can be referred to Huang et al (2006). With this state equation, all other state properties such as p-v-T relations, specific heats, thermal expansion, and sound velocity for 3He could be determined by standard thermodynamics derivation. Fig. 1 shows the agreement between the calculations by equation (1) and the gaseous p-v-T experimental data on isotherms up to 60 K by Karnus and Rukenko (1974), which is a Russian article found after the development of the state equation. It was expected to see an excellent agreement between the calculations and measurements at each points. 140 14.13 K 32.92 K 120 16.95 K 37.94 K 20.38 K 100 23.73 K 28.06 K ]

3 80 60 , [kg/m U 40 42.90 K 48.26 K 20 53.29 K 0 59.85 K 024681012 p, [MPa] Fig. 1. Comparison of calculated p-ȡ-T (lines) with measurements from Karnus et al. (1974) (symbols)

584 Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590

3. Further evaluation of the Debye state equation

3.1. From the perspective of specific heat

Since the Debye state equation for helium-3 was started from the behavior of specific heat at constant volume, it is necessary to check the agreement of the predicted specific heat curves with the corresponding experimental data in the low temperature range. As shown in Fig. 2, the state equation precisely predicted the cv data in the liquid phase in wide temperature range from 0.005 K to 3 K with reference to the measurements by Greywall (1983) who claimed the uncertainty of data is within ±1%. When we change the temperature coordinates to logarithmic, it clearly shows the intersection of the cv curves between 0.1 and 0.2 K, which was an important observation in Greywall’s experiment. The cv curves below 0.2 K well agree with the theoretical predictions by the Fermi theory. It is interesting to find that the overall specific heat behavior of helium-3 is significantly different from that of its isotope helium-4, as shown in Fig. 3 with curves calculated by the latest NIST database by Lemmon et al (2013). At

the lower end of the temperature range from the superfluid transition point TȜ to the critical point Tc, the specific heat of helium-3 drops rapidly with the decreasing of temperature. However, the specific heat of helium-4 remains fairly constant at high densities, while at lower densities it turns out to ascend quickly instead due to the lambda transition mechanism. It could also be expected, although not studied here, for helium-3 a monotone increasing specific heat behavior when approaching its extremely low lambda transition temperature 0.0026 K. At the higher temperature end close to the critical point, the helium-4 curves behave much “flatter” than those of helium-3 at densities greater ~1.5ȡc .

3.2. Linearity of cubic sound velocity vs. pressure

Maris (1991) used to make an important observation of the very precise sound velocity (c, in m/s) measurements by Abraham et al (1971) that c3 appears to be linear in pressure (p) over all the measured range. This characteristics was also well predicted by our Debye state equation for helium-3 and verified by experimental results by Vignos and Fairbank (1966), as shown in Fig. 4. The results could be of interest to us because this trend was not predicted by any theory. Since the velocity of sound shows the compressibility of the Fermi-Dirac ground state and 2nd derivative of the state equation, further theoretical study would be necessary to understand the underlying mechanism. Whatever, this trend could be considered as an assistant proof of the correctness of the equation at temperatures in the liquid phase.

(a) (b)

3.0 3 3.0 e n i . m l g/ k d lines: state equation i

He-3 3

1 u . .9 e scatters: exp. data by Greywall (1983) q 1 i 2.5 n 2.5 l i 8 3

d d m 3 i / c g e

u t k q /m 7 li 3 g 3 a

5 5 k r d . 7 c 81.913 kg/m e 2 4 u 2.0 t 9 2 2.0 t a 9. 7 3 r 9 3 92.553 kg/m a tu s K] a /m K] s kg 3 98 99.247 kg/m . .8 . 1.5 O 08 1.5 O 3 7 1 7 108.898 kg/m

1.0 melting line 1.0 , [kJ/kg , [kJ/kg v v line c c ing melt 0.5 lines: state equation 0.5 scatters: exp. data by Greywall (1983) He-3 0.0 0.0 01234 0.001 0.01 0.1 1 4 T, [K] T, [K] Fig. 2. Helium-3 specific heat vs. temperature along isochors (scatters are measurements) (a) linear scale; (b)logarithm scale

Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590 585

5 120

3 He-4: from Refprop 9.1 100 He liquid 4 Melting pressure

] .

3 80 s . O

K] 3

7 c 7 60 3 3 120kg/m , [m ne 3 -6 id li m d liqu 140kg/

e 10 40

, [kJ/kg t ura 3 v at s /m u State equation [T=1.0K] c 60kg 3 1 3 2 c /m Exp. data by Abraham (a) 180kg 20 3 State equation [T=2.5K] /m 210kg Exp. data by Atkins melting line 1 0 23456 0123456 T, [K] p, [MPa]

Fig.3. Calculated cv value from Refprop 9.1 for helium-4 Fig. 4. Linearity of the cubic velocity of sound vs. pressure

3.3. Virial coefficients

One of the most commonly used equations of state for gas relating pressure p, density ȡ, and temperature T is the virial equation, which is competitive for high speed calculation in practical.

2 p UUURTªº¬1()() B T C T L ¼ (2) where B(T) and C(T) are the second and third virial coefficients, respectively. This equation is useful theoretically in that the coefficients can be related to particular molecular interactions, the second coefficient to pair interactions, the third to three-body interactions, and so on. Therefore, these coefficients are either theoretically or semi-theoretically determined in origin, especially for cases where as yet no experimental data are available, or empirically determined by fitting the temperature statistical equations (with quantum corrections) to experimental data. To my knowledge, there were very few measurements (derived from p-v-T data) 2nd and 3rd virial coefficients of helium-3. Checking the consistency of the B(T) and C(T) with the theoretical derived values by Hurly and Moldover (2000) via ab-initio method and experimental data, will be valuable to evaluate the performance of the helium-3 state equation. Meanwhile, it will also be valuable to further studies of the interatomic potentials theory (hard-core square-well potential, Lennard-Jones 12-6 potential) for this fluid. Fig. 5 shows comparisons of calculated virial coefficients for dilute 3He gas with experimental values. The predicted second virial coefficient by the state equation shows good consistency with either the ab-initio values or the measurements by Matacotta et al (1987), Cameron and Seidel (1985) in wide temperature range. However, the calculated third virial coefficient is found to be overestimated at temperatures below ~4 K. Notice that the magnitude of the calculated quadratic term must not differ from zero by a statistically significant amount. Nor on theoretical grounds would the data at these low densities (never greater than 2×10~4 mol·cm-3 in these isotherm measurements) be expected to exhibit a contribution from the third virial coefficient. With C(T) estimated to be the order of 3×103 cm6·mol-2 in the temperature range of these experiments, C(T)ȡ2 would be at maximum only 10-4. On the other hand, the only available experimental data by McConville and Hurly (1991) were highly scattered. It is hard to tell the accuracy of the third virial coefficient below around 4 K, although McConville and Hurly presented an fitted equation.

3.4. Enthalpy and entropy at low pressures

The enthalpy properties of 3He are of great interest for cryogenic applications like regenerative cryocoolers, usually at operating pressure from 0.5 ~2.0 MPa. However, for pre-cooled J-T coolers by going down to less than 1 K, the working pressure of the helium-3 gas might be several orders of magnitude lower. Hence, enthalpy data at low 586 Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590

(a) Second virial coefficient (b) Third virial coefficient 0.02 16 He3 EoS in this work 0.00 McConville (1991) fitted equation 12 McConville (1991) Exp. data ] 2

-0.04 /kg) 8 3 /kg] 3 4 5

, [(m C(T)=93+2971/T-5.47u10 /T , [m Hurly (2000) 4 3 2

B orinal in units: (cm /mol) -0.08 He3 EoS calculation (this work) u

Exp. Matacotta (1987) C Exp. Cameron (1985) 0 -0.12 1 10 100 1000 1 10 100 1000 T, [K] T, [K] Fig. 5. Comparison of the calculated virial coefficients with experimental values

pressures should also be reliable although it is usually neglected by most applications. Generally, for developing a multi-parameter equation of state, the validity to large spans of both pressure and temperature across several orders of magnitude of the absolute values is always a big challenge. The ratio of T0/Ttri (room temperature over triple point 4 temperature) for any other fluids (except He) is less than 4. Although T0/Ttri for helium-4 is about 140, the value for helium-3 reaches 105. It should be pointed out that triple point for helium-3 and helium-4 do not exist but their superfluid transition point/line are generally treated as the pseudo-triple point when it has to be one for parallel theoretical calculations. Therefore, an evaluation of the state equation for pressures down to 0.001 – 10 kPa either in liquid or vapor phase will be important. Fig. 6 and Fig. 7. present the calculated T-s and T-h relationship respectively as well as available measurements along with the saturation line. It can be seen that the state equation predictions agree satisfyingly with the experimental data, except those from Singwi (1952), which were taken in the earliest year and diverged from the others at the lower temperatures. The reference point for zero enthalpy of Rauch’s data (1961) was set to the saturated vapor at 0 K, which means enthalpy for the saturated liquid has negative values. In order to keep all the data consistent, Rauch’s data have been shifted a value of 21.08736 JÂmol-1, which equals the latent heat of evaporation of 3He at 0 K. After processing, Rauch’s data agree well with the others, as well as the solid line representing the smooth calculations by the equation of state in this work. Here the early enthalpy-pressure data from Kraus et al (1974) was also checked. Since different reference state for enthalpy was adopted by Kraus, a shifts similar as the above was first conducted. As shown in Fig. 8, the differences 300 60 Singwi Brewer (a) Rauch Brewer (b) 100 Roberts (a) Colyer ]

-1 Roberts (b) Abraham (b) 40 K Saturated vapor Rauch -1 ]

-1 Colyer Tc

mole Betts mol State eq. in this work , [J 10 [J 20 s Saturated liquid

114.6kPa , h 114.6kPa Lee 1.16kPa Strongin 3 10-5kPa -5 u 3u10 kPa State eq. in this work 0 1.16kPa 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 T, [K] T, [K] 3 Fig. 6. Entropy of saturated 3He (low pressure) Fig. 7. Saturated enthalpy of He

Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590 587

35 1.8 a 4.17 K P a 1.6 0 4.00 K P 30 5 1 7 7 3.80 K = = 3.50 K p 25 p 3.00 K 1.2 h=15.0 kJ/kg 2.50 K 20 2.00 K h=12.5 kJ/kg

t 0.8

15 a , [K]

s h=11.4 kJ/kg T , [kJ/kg] sa h=10.7 kJ/kg h tur 10 at ion 0.4 lin 5 e 0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10 15 20 25 30 35 40 45 p, [MPa] s, [kJ/kgK] Fig. 8. Comparing enthalpy- pressure data by Kraus and this work Fig. 9. Low pressure curves on T-s diagram between the curves representing the equation of state and the scatters by Kraus were acceptable for general cryogenic applications, in particular at pressures less than 0.3 MPa. Relatively larger deviation was observed above 0.4 MPa and at temperatures 3.3 K. Since these data are located in the gas region (more precisely, supercritical region), the lines by the equation of state are believed to be more reliable. During the calculations for a 3He Joule-Thomson cryocooler, Dr. Takeshi Shimazaki from AIST Japan reported that 1) the inflection points of the isobar lines (less than 1 kPa) and isenthalpic lines expected to be on the saturation line but the figure seems differently; 2) unexpected constant temperature region are observed on the isenthalpic lines. Finally the problem was located to an mathematical iteration routine for (T, p) input-pair flash calculations, while the state equation (1) is expressed as a function of T and density ȡ. After fixing the code, correct curves were obtained as shown in Fig. 9, in which the constant enthalpy curves smoothly connect the saturated vapor line at one end and turn to horizontal lines as expected for any ideal gas. The trend and position of very low pressure isobars are also thermodynamically reasonable. Excess wet regions on the isobar lines are no longer observed. These information helps to confirm the correctness of the state equation itself.

3.5. From the perspective of Grüneisen parameter

The Grüneisen parameters ( ī ) has long been used to describe the effect that changing the volume of a crystal lattice has on its vibrational properties. As a consequence, the effect that changing temperature has on the size or dynamics of the lattice, in other words, represents the thermal pressure from a collection of vibrating atoms. However,

0.8

Nitrogen K 00 2 K 0K 60 30 1 0K 0.6 14

0K 13 , [-]

* 0.4 3 = c 0.2 U 69.641 kg/m 0 50 100 150 200 250 300 350 400 3 U, [kg/m ] Fig. 10. Grüneisen parameter of nitrogen

588 Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590

ī in fluids has rarely been recognized or studied by Arp et al (1984). Because of the equivalences between many properties and derivatives within thermodynamics, there are many formulations of the Grüneisen parameter which

are equally valid. The one convenient for fluids is chosen to be * (/vcvv )(w p / w T ). As we know, properties like specific heat, always behaves large/sharp variations in the vicinity of the critical point, which absolute value is hard to assess. It was found that the Grüneisen parameters exhibit much less variation over the near critical range. This characteristics over wide ranges of fluid suggests that ī might be rather simply parameterized with in the context of ordinary fluid state models and useful for examining the state equation as a benchmark. Arp et al (1984) concluded that the Grüneisen parameter in fluids (except water) usually is close to or within the decade range from 0.2 to 2, which could be confirmed by Fig.10 for nitrogen by Lemmon et al (2013) as an example. The Grüneisen parameter has never been evaluated for helium-3. It is interesting to show how it occurs in the critical point problem in compressible fluid hydrodynamics on helium-3, dropping the considerations of liquid structure. Fig. 11(a) gives ī curves with respect to reduced temperature (Tr = T/Tc) at several isobars (also in reduced form, pr = p/pc). The reduce form will be more comparable for models of different fluids. I could be seen that the value ī of helium-3 is normally be below 1.5, which follows the conclusion by Arp et al (1984). 3 í3 Keeping in mind that ȡc and Tc for He are 3.3157 K and 41.191 kg·m respectively (or Tr = 1 and pr = 1), these curves show relatively little influence of the critical point. It should be mentioned that the isobars across each other around Tr = 0.6, similar to the characteristics observed in the cv curves as shown in Fig.2(b). However, their crossing points do not occur at the same temperature. Physics explanations related to this phenomenon should be addressed in next studies. At reduced temperatures below 0.6, ī drops quickly.

(a) (b) 1.50 1.8

1.6 Gruniesen parameter 1.25 8p He4 by Refprop 9.1 8 p r r 1.4 1 2 4 p p p 1.00 r r r 4p

0.5 1.2 r

n i l p , [-] , [-] r n * * 1.0 o i 2p 0.75 t r

0.8 a 1p r saturation line s 0.50 0.5p 0.6 r sat line by Hepak 0.25 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T , [-] T , [-] r r (c) (d) 8 2.5

K a 0 0 K P 3 K 2 00 M 0 3 0 0 a 6 K 2.0 3 K 1 . 0 0 P 1 5 4

3 M

3 475.59MPa

K 6 K 0 . 00 1 8 2 8 0K 0K 4 1.5 10 30 55.66MPa 0K

=69.641 kg/m 5 , [-] c , [-] U * * = 41.191 kg/m c 2 U 1.0

Helium-3 Helium-4 ideal gas value =2/3 0 0.5 0 100 200 300 400 450 0 50 100 150 200 250 300 350 400 3 3 U, [kg/m ] U, [kg/m ] Fig. 11. Grüneisen parameters of helium isotopes (a) 3He ī-T; (b) 4He ī-T ; (c) 3He ī-ȡ; (d) 4He ī-ȡ Yonghua Huang / Physics Procedia 67 ( 2015 ) 582 – 590 589

For comparison, the ī curves for helium-4 (calculated by Refprop, Lemmon et al (2013)) was plotted as well in Fig. 11(b). We can see the general similarities between these two isotopes. At low pressures including the saturation line, the curves for helium-4 look strange with sharp negative peaks, which was not expected/acceptable by any theory. To confirm that, Grüneisen parameter along the saturation line was also calculated by using Hepak, Arp et al (2005), one of most accepted computer program for helium-4 properties. The saturation live by Hepak agrees with that by Refprop at the liquid side but not around the critical point and on the vapor side. Whatever, the helium-3 curve behaves closer to the Hepak prediction, which is smooth around the critical point. The intersection phenomenon of the ī curves was not observed. Grüneisen parameter versus density curves are also plotted for helium-3 and helium-4 as shown in Fig. 11( c) and (d). The data for helium-4 were generated by Hepak. It would be reasonable to conclude that ī(ȡ) is nearly independent of temperature from 50 to 300 K for either helium-3 or helium-4, if we make some allowance for possible systematic error in the calculation of ī(ȡ)-T from p-v data. Especially for helium-3, this independency is reliable up to 3ȡc even for temperatures down to 10 K. Besides, at zero density limit, the Grüneisen parameter goes to it’s value corresponding ideal gas, 2/3 for helium isotopes. The diversity difference between 3He and 4He might 3 be due to significant specific heat difference as shown in Fig. 2, where the cv-T curves of He are nearly independent of density at low temperatures. Generally, the 3He Grüneisen curves shown in Fig.11(a) and (c) hint a correct expression of the state equation. It should be pointed the big difference between helium isotopes and other cryogenic fluids such as nitrogen (with relatively higher boiling point temperatures). By comparing Fig. 10 and Fig.11 (c)(d), we find that for fluids like nitrogen, at any specified density, higher temperature leads to higher value of Grüneisen parameter. However, for the quantum fluids helium-3 and helium-4, they behaves the opposite way.

4. Conclusion

The Debye equation of state for helum-3 was evaluated from perspectives other than it was developed. p-v-T surface was not the only concerned aspect. An deeper look into the specific heat reveals the accuracy of the state equation for all compressed normal liquid helium-3 in a wide span of temperature (down to at least 5 mK). It precisely describes the intersection of the isochoric specific heat. This detail in fact confirms the applicability of the Debye model to quantum fluid. The cubic speed of sound to pressure relationship demonstrated the same relationship observed for helium-4, which reflect an hidden thermodynamic theory for helium physics. The enthalpy and entropy properties at extremely low pressures show expected behaviors both on the saturation line and the superheated vapor region and compressed liquid region. The Grüneisen parameter for helium-3 reaches the ideal gas value when density goes to zero and shows week variation around the critical point as expected. All the above examination show contributive proofs of the thermodynamic reliability of the state equation, which is satisfying to the requirement of engineering applications.

Acknowledgements

This work was financially support from the National Natural Science Foundation of China (No.50806047, No.50376055 and No.51176112). The author thanks many experts for continuous help and stimulating discussions, especially Dr. Vincent Arp from NIST (retired).

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