He4: Reference manual

Introduction

The code of this library is stored in the file He4.f08 in the form of Fortran 2008 source. This file can be compiled using GFortran free compiler or other alternatives. For Linux operating systems, it is usually simpler to obtain the compiler from the official distribution repository. The code has been designed to not require further dependency and can be compiled using the command line reported in the source code.

Alternatively, the library is distributed in the form of compiled shared library He4.dll and He4.so for Windows™ 64 bit and Linux 64 bit operative systems. A 32 bit version is available only for Windows™.

The name of the functions, exposed by the library, are reported in the List of function session. It has been chosen to use the "C" format for the compiled code so that the functions can be called by almost all the development environment and software for numerical analysis.

All the quantities are represented using double precision floating point with a length of 8 bytes while integer indexes, when necessary, are implemented using signed integer of 4 bytes. Arrays are pointers to double.

All values are passed by reference in the form of pointers.

For a sake of testing, a LabView™ 16 bit library has been implemented to link He4 functions in a form of a set of virtual instruments. LabView™ Library is contained in the He4.llb file. A Maple™ 16 bit version is already available if requested.

Thermal properties are calculated using a virial expansion for the pressure truncated including the fourth virial D(T).

Installation

Please check the terms of use described in the license before installing this software.

He4 library is not provided with an automatic installed because its installation depends on the choices of the end-user. For LabView, the shared library, being He4.dll or He4.so, should be saved in the same folder of the He4.llb file. Other development platforms and software for numerical analysis might request to save libraries in determined places. Please refer to the manual of the software connecting to the He4 libraries to get the necessary information.

For 32 bits platforms (only Windows), it is possible use He4_32bit.dll. To this end, rename the file to He4.dll before using.

List of the functions

he4_rho(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	kg/m³	Density of the helium

This function returns the density of the helium calculated as a function of the temperature and pressure. The closed form for the density is obtained by the Newton algorithm applied symbolically to the pressure virial expansion. The initial guess value for the density is obtained using the ideal gas expression while the expression, explicit in the pressure, is the second symbolic iterate of the Newton algorithm. The obtained value is accurate at least at the 10th-figure.

Reference:

W. Cencek, M. Przybytek, J.Komasa, J. B. Mehl, B. Jeziorski and K.Szalewicz, Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium, J. Chem. Phys. 136, 224303 (2012); https://doi.org/10.1063/1.4712218
M. Przybytek, W. Cencek, B. Jeziorski and K. Szalewicz, Pair Potential with Submillikelvin Uncertainties and Nonadiabatic Treatment of the Halo State of the Helium Dimer, PRL 119, 123401 (2017); https://doi.org/10.1103/PhysRevLett.119.123401
G. Garberoglio, M. Moldover, A. H. Harvey, Improved First-Principles Calculation of the Third Virial Coefficient of Helium, J. Res. Natl. Inst. Stand. Technol. 116, 729-742 (2011); http://www.nist.gov/jres
G. Garberoglio, A. H. Harvey, Path-integral calculation of the fourth virial coefficient of helium isotopes, J. Chem. Phys. 154 (10) 2012; https://doi.org/10.1063/5.0043446

he4_vol(T, p)

T	Real(8)	K	Temperature of helium
p	Real(8)	kPa	Pressure of helium
Return	Real(8)	m³/mol	Molar volume of helium

he4_w(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	m/s	Speed of sound of helium

Acoustic virial coefficients are obtained from pressure virial coefficients and ideal specific heat capacity using Gillis et al. relations.

Reference:

K. A. Gillis and M. R. Moldover, Practical Determination of Gas Densities from the Speed of Sound Using Square-Well Potentials, Int. J. Thermophys. 17 (6) 1996;

he4_w_gk(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	m/s	Speed of sound of helium calculated using Gokul formulation

Acoustic virial coefficients are those publisched by Gokul et al.

Reference:

N. Gokul, A. Shultz, D. A. Kofke, Speed of sound in helium 4 from ab initio acoustic virial coefficients, J. Chem. Eng. Data 66 (8) 2021; https://doi.org/10.1021/acs.jced.1c00328

he4_cp(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	J/(kg K)	Constant pressure specific heat capacity of helium

Reference:

N. Gokul, A. Shultz, D. A. Kofke, Speed of sound in helium 4 from ab initio acoustic virial coefficients, J. Chem. Eng. Data 66 (8) 2021; https://doi.org/10.1021/acs.jced.1c00328

he4_cv(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	J/(kg K)	Constant volume specific heat capacity of helium

Reference:

N. Gokul, A. Shultz, D. A. Kofke, Speed of sound in helium 4 from ab initio acoustic virial coefficients, J. Chem. Eng. Data 66 (8) 2021; https://doi.org/10.1021/acs.jced.1c00328

he4_gamma(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	none	Heat capacities ratio of helium cp/cv

To determine gamma, cp(T,p) and cv(T,p) are obtained and then their ratio is calculated.

he4_nu(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	Pa s	Viscosity of helium

The viscosity of helium is calculated fitting the tabulated points of Cencek et al. provided in the limit of zero pressure. The pressure dependency is obtained using a virial expansion, in term of pressure, where the coefficients, depending only from temperature, have been calculated using the behaviour provided by Refprop 10.

Reference:

W. Cencek, M. Przybytek, J.Komasa, J. B. Mehl, B. Jeziorski and K.Szalewicz, Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium, J. Chem. Phys. 136, 224303 (2012); supplementary material https://aip.scitation.org/doi/suppl/10.1063/1.4712218/suppl_file/s4_he4prop.txt

he4_k(T, p)

T	Real(8)	K	Temperature of the helium
p	Real(8)	kPa	Pressure of the helium
Return	Real(8)	W/(m K)	Thermal conductivity of helium

The thermal conductivity of helium is calculated fitting the tabulated points of Cencek et al. provided in the limit of zero pressure. The pressure dependency is obtained using a virial expansion, in term of pressure, where the coefficients, depending only from temperature, have been calculated using the behaviour provided by Refprop 10.

Reference:

W. Cencek, M. Przybytek, J.Komasa, J. B. Mehl, B. Jeziorski and K.Szalewicz, Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium, J. Chem. Phys. 136, 224303 (2012); supplementary material https://aip.scitation.org/doi/suppl/10.1063/1.4712218/suppl_file/s4_he4prop.txt

temp_K(W2_ratio, P, Tref, Pref)

W2_ratio	Real(8)	none	Ratio W²exp/W²ref
P	Real(8)	kPa	Pressure of W²exp measurement
Tref	Real(8)	K	Temperature of W²ref
Pref	Real(8)	kPa	Pressure of W²ref
Return	Real(8)	K	Temperature of the helium according Gokul’s formulation

This function is used to determine the thermodynamic temperature knowing the square of the speed of sound at a fixed point W² ref with known reference temperature and pressure, T ref and p ref. The calculated temperature is corrected for the effect of pressure P that it usually not the same as p ref.

Reference:

N. Gokul, A. Shultz, D. A. Kofke, Speed of sound in helium 4 from ab initio acoustic virial coefficients, J. Chem. Eng. Data 66 (8) 2021; https://doi.org/10.1021/acs.jced.1c00328

Acknowledgement

This project (18SIB02-RMG1) has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.

License

MIT-like license with commercial restrictions.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute and sublicense copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
The Software, or part of it, is not included or used in commercial applications without prior written agreement from Le-Cnam/INRiM.
Except as contained in this notice, the name of Le-Cnam/INRiM shall not be used in advertising or otherwise to promote the sale, use or other dealings in this Software without prior written authorization from Le-Cnam/INRiM.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.