The property values of therapeutic gas mixtures are important in designing devices, defining delivery parameters, and in understanding the therapeutic effects. In the medical related literature the vast majority of articles related to gas mixtures report property values only for the pure substances or estimates based on concentration weighted averages. However, if the molecular size or structures of the component gases are very different a more accurate estimate should be considered.

Findings

In this paper estimates based on kinetic theory are provided of density, viscosity, mean free path, thermal conductivity, specific heat at constant pressure, and diffusivity over a range of concentrations of He-O_{2}, Xe-O_{2}, N_{2}O-O_{2} and N_{2}-O_{2} mixtures at room (or normal) and body temperature, 20 and 37°C, respectively and at atmospheric pressure.

Conclusions

Property value estimations have been provided for therapeutic gas mixtures and compared to experimental values obtained from the literature where possible.

Introduction

Inhaled therapeutic gases in use today include helium (He) for respiratory treatments, and xenon (Xe) and nitrous oxide (N_{2}O) for anesthesia. For clinical applications these gases are used in the form of mixtures with oxygen in a range of concentrations (typically starting from 20% oxygen (O_{2}) concentration by volume, which is equivalent to a mole fraction of 0.20) so as to maintain adequate oxygenation. Other gases, such as nitric oxide (NO) for pulmonary vascular dilation, are used only in trace amounts.

The property values of therapeutic gas mixtures are important in designing devices, defining delivery parameters, and in understanding the therapeutic effects. Properties of interest include density, viscosity, mean free path, thermal conductivity, specific heat, and diffusivity. In the medical literature the vast majority of articles related to gas mixtures report property values only for the pure substances or estimates based on (volume or molar) concentration weighted averages [1–7]. However, if the molecular size or structures of the component gases are very different a more accurate estimate could be considered [8–10]. For this reason property values of helium and xenon mixtures should be considered for more accurate estimation.

Starting with kinetic theory for molecules treated as hard spheres as a basis, a rich literature has developed regarding the modeling of property values based on first principles and increasing complexity of the molecular interactions; in particular, the attraction and repulsion of molecules as first formulated by Chapman and Enskog [8, 9]. The empirically determined Lennard-Jones potential energy function has proved to be a good model for many applications. Extensive measurements of the viscosity of gases using oscillating-disk viscometry have primarily been published by Kestin and his colleagues [11–16]. Other equilibrium and transport properties have been extrapolated from the viscosity measurements using the models described above [8, 9]. There also exists limited thermal conductivity data measured using a hot wire method [17].

The objective of this short communication is to give a straightforward reference to the applied scientist, engineer, and medical personnel who perform research with therapeutic gas mixtures. We anticipate that this information will assist both in the design and interpretation of experiments. Estimates of density, viscosity, mean free path, thermal conductivity, specific heat at constant pressure, and diffusivity are provided over a range of concentrations of He-O_{2}, Xe-O_{2}, and N_{2}O-O_{2} mixtures at room (or normal) and body temperature, 20 and 37°C, respectively and at atmospheric pressure; based on kinetic theory and compared to experimental values obtained from the literature where it is possible. For further comparison N_{2}-O_{2} mixtures will be included because this mixture makes up the composition of medical air.

Methods

Density

All of the mixtures can be evaluated as ideal gases under the conditions considered. As such the density is based on the state equation as,

${\rho}_{mix}=\frac{p}{{R}_{mix}T}$

(1)

where ρ_{mix} is the mixture density, p is the pressure, T is the absolute temperature and R_{mix} is the gas constant defined for the mixture as

In Equation (2) R_{univ} is the universal gas constant, X_{i} is the mole fraction of the pure gas component, and MW_{i} is the molecular weight of the pure gas component (32 is the molecular weight for oxygen). The units of R_{mix} depends on the value chosen for R_{univ} (e.g., 8314 N-m/kgmol-K).

Viscosity

For viscosity we use a semi-empirical method by Wilke [8] that extends the model for collisions between hard spheres to mixtures.

μ_{
i
} and ${\mu}_{{O}_{2}}$are the viscosities of the pure gas component and oxygen, respectively. The pure gas viscosity estimates are based on the Lennard-Jones empirical function for the potential:

where r is the distance between the molecules, ε is a characteristic energy of the interaction between molecules and σ is a characteristic diameter, or collision diameter. Equation (5) is a viscosity formula based on the Lennard-Jones parameters in units of kg/s-m derived for monatomic gases that has also been shown to work well for polyatomic gases [8],

where Ω_{
μ
} is a function of ε. Lennard-Jones parameters are tabulated for common gases [8, 9] and for the gases herein in Table 1.

Table 1

Molecular parameters and Lennard-Jones potential parameters for the pure gas components [9].

Gas

MW

R (J/kg-K)

σ (Å)

ε/κ (°K)

Ωμ

at 20°C

Ωμ

at 37°C

Atomic Diffusion Volume

(Σv)

He

4.003

2076.9

2.551

10.22

0.7061

0.7004

2.67

Xe

131.3

63.3

4.047

231.

1.4140

1.3798

32.7

N_{
2
}O

44.02

188.9

3.828

232.4

1.4190

1.3846

35.9

N_{
2
}

28.02

296.7

3.798

71.4

0.9697

0.9535

18.5

O_{
2
}

32.00

259.8

3.467

106.7

1.0635

1.047

16.3

Values for Ω have been interpolated from Table B-2 in Bird et al. [8]. κ is the Boltzmann constant.

Mean Free Path

The estimation of mean free path is based on the Chapman-Enskog formulation for hard spheres [18], where the mixture viscosity and density account for the interactions of the different molecules:

The specific heat at constant pressure (on a per unit mass basis) for all of the mixtures can be evaluated assuming ideal gas behavior and therefore the specific heat curve is a linear function of the mass fraction, though nonlinear in terms of the mole fraction

where ${{c}_{p}}_{mix}$ and ${{c}_{p}}_{i}$ are the specific heats of the mixture and of the pure gas component, respectively. The pure gas values for the monatomic gases are based on the theoretical value ${c}_{{p}_{i}}=\frac{2.5{R}_{univ}}{M{W}_{i}}$ The polyatomic estimates are based on empirically derived 4^{th} order polynomials in temperature found in Poling et al. [9].

Thermal Conductivity

Thermal conductivity is treated in an analogous manner to viscosity, where Equation (8a) is equivalent to Equation (3a) and the coefficients are exactly the same based on the pure gas viscosity values.

The self diffusivity for a binary system D_{ij}, represents the movement of species i relative to the mixture, where D_{ij} = D_{ji}. The presentation here is based on the method of Fuller et al. given in Poling et al [9], which uses empirically obtained atomic diffusion volumes (Σv).

In Equation (10) j always represents oxygen, the diffusivity is in m^{2}/s, T is the temperature in degrees Kelvin, p is the pressure in bar and the atomic diffusion volumes are given in Table 1 for each gas. ${D}_{i{O}_{2}}$is almost independent of composition at low pressures so only a single value will be calculated for each binary gas pair [8].

Of much practical interest is the diffusivity of water vapor or carbon dioxide through the gas mixtures. Values are calculated for these mixtures based on Blanc's law [9].

Where m represents the therapeutic gas mixture considered, j represents the specific therapeutic gas, and k corresponds to H_{2}O or CO_{2}. The diffusion constants in Equation 11 of H_{2}O or CO_{2} through the therapeutic gas or oxygen are calculated using Equation 10 with atomic diffusion volumes of 13.1 and 26.9 for H_{2}O or CO_{2}, respectively.

Results

The molecular weights, gas constants, Lennard-Jones parameters, and atomic diffusion volumes for the pure gases are given in Table 1. The mixture results are given in tabular and graphical forms. Tables 2, 3, 4, and 5 give the property values for He, Xe, N_{2}O, and N_{2} with O_{2} mixtures, as a function of mole fraction at 20°C. Tables 6, 7, 8, and 9 are the analogous tables for 37°C. Table 10 gives binary diffusivities for the gas mixtures. Figures 1, 2, 3, 4, and 5 are plots of the 20°C data of density, viscosity, mean free path, thermal conductivity, and specific heat, respectively.

In this paper thermophysical property values have been presented for inhaled therapeutic binary gas mixtures. Pure substance values at 20°C and 37°C and mixing formulas based on kinetic theory were used to estimate the mixture values. The approach was to use relatively simple estimates for nonpolar gases [8]. That is, more complex intermolecular interactions that occur, for example, at high pressure, were not included.

Whereas many therapeutic gases (e.g.; CO and NO) are used at trace concentrations such that property values of the bulk mixture are essentially equivalent to those of air, mixtures considered herein have significantly different properties than air which change as a function of component concentration. Mechanical property values of density and viscosity are fundamental to the understanding of gas transport and airway resistance. The thermal properties of conductivity and capacity are necessary to accurately predict how gas treatments will affect the temperature and humidity of the respiratory tract. They also will influence the thermodynamic interaction of inhaled aerosols with the gas, and thus the deposition distribution which is particularly relevant for helium-oxygen mixtures. Diffusion is a key mode of gas transport deep in the lung potentially affecting exchange with the blood.

Bird et al. [8] note that the concept of the mean free path is applicable only if there are no long range forces associated with the hard sphere kinetic theory models. For this reason it is not typically an element of modern kinetic theory. Nevertheless, it is an important parameter in modeling the interaction of aerosols and gases [19], and thus for combination therapies involving aerosols and gas mixtures. In contrast to the scheme employed by Loeb [20], the estimation method employed here does not directly take into account the molecular collisions. However, Equation (6) for the mean free path does account for the collisions of different molecules through the mixture viscosity. As the utility of this parameter in aerosol mechanics is to estimate a reduced drag on small particles where their size is comparable to the mean free path, this approach would appear to be self consistent.

A comparison of estimated data based on Equation (3) to experimental data for the viscosity at 20°C of helium-oxygen mixtures [14] is shown in Figure 6, along with the linear curve representing the concentration weighted average. The maximum relative difference of 0.9% between the theory and experiment occurs at X_{He} = 0.82. For the concentration weighted average value the maximum relative error of 7.9% occurs at X_{He} = 0.67.

Figure 7 shows comparisons of experimental thermal conductivity values [17] for helium-oxygen and xenon-oxygen mixtures at 30°C compared to theoretical values calculated using Equation (8). The maximum relative differences between the theory and experiment are 4.2% at X_{He} = 0.68 and 4.7% at X_{Xe} = 0.27, respectively.

Table 11 shows a good agreement between experimental data for binary diffusivity of He-O_{2} and Xe-O_{2} [14, 21] with theoretical data calculated using Equation (10). For the diffusivity of water vapor or carbon dioxide, the simplifying assumption leading to Blanc's law is for a trace component diffusing into a homogeneous, binary mixture. A quantitative definition of trace for the applicability of this assumption was not found. However, experiments testing diffusion of He, CO and SF_{6} through gas mixtures similar to alveolar gas (14% O_{2}, 6% CO_{2} and 80% N_{2}) did not show significant departures from values predicted on the basis of binary diffusion coefficient values weighted according to fractional concentrations [22] in agreement with Blanc's law. These experiments were performed with test gas concentrations varying from 0 to 10% suggesting Blanc's law would be appropriate for typical applications of the gases considered herein.

Table 11

Comparison of experimental and theoretical binary diffusivities based on Equation (10).

In conclusion, the methods presented above allow accurate estimation of thermophysical property values for inhaled therapeutic binary gas mixtures, including He-O_{2}, Xe-O_{2}, and N_{2}O-O_{2}, over a range of concentrations.

Declarations

Acknowledgements

We thank Paul Finlay for performing some of the calculations.

Authors’ Affiliations

(1)

Medical Gases Group, Air Liquide Santé International, Centre de Recherche Claude-Delorme

(2)

Department of Mechanical Engineering, Lafayette College

(3)

Scientific Direction, Air Liquide Research and Development, Centre de Recherche Claude-Delorme

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