Property value estimation for inhaled therapeutic binary gas mixtures: He, Xe, N2O, and N2 with O2

  • Ira Katz1, 2Email author,

    Affiliated with

    • Georges Caillibotte1,

      Affiliated with

      • Andrew R Martin1 and

        Affiliated with

        • Philippe Arpentinier3

          Affiliated with

          Medical Gas Research20111:28

          DOI: 10.1186/2045-9912-1-28

          Received: 8 July 2011

          Accepted: 6 December 2011

          Published: 6 December 2011

          Abstract

          Background

          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-O2, Xe-O2, N2O-O2 and N2-O2 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 (N2O) 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 (O2) 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 [17]. However, if the molecular size or structures of the component gases are very different a more accurate estimate could be considered [810]. 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 [1116]. 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-O2, Xe-O2, and N2O-O2 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 N2-O2 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,
          ρ m i x = p R m i x T http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ1_HTML.gif
          (1)
          where ρmix is the mixture density, p is the pressure, T is the absolute temperature and Rmix is the gas constant defined for the mixture as
          R m i x = R u n i v X i M W i + ( 1 - X i ) 32 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ2_HTML.gif
          (2)

          In Equation (2) Runiv is the universal gas constant, Xi is the mole fraction of the pure gas component, and MWi is the molecular weight of the pure gas component (32 is the molecular weight for oxygen). The units of Rmix depends on the value chosen for Runiv (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 m i x = X i μ i X i + ( 1 - X i ) ϕ i - O 2 + ( 1 - X i ) μ O 2 X i ϕ O 2 - i + ( 1 - X i ) http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ3_HTML.gif
          (3a)
          ϕ i - O 2 = 1 + μ i μ O 2 32 M W i 1 4 2 8 1 + M W i 32 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ4_HTML.gif
          (3b)
          ϕ O 2 - i = 1 + μ O 2 μ i M W i 32 1 4 2 8 1 + 32 M W i http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ5_HTML.gif
          (3c)
          μ i and μ O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq1_HTML.gifare 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:
          ϕ ( r ) = 4 ε σ r 12 - σ r 6 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ6_HTML.gif
          (4)
          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],
          μ i = 0 . 26 69 3 x 1 0 - 5 M W i T σ 2 Ω μ http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ7_HTML.gif
          (5)
          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:
          λ m i x = 16 μ m i x 5 ρ m i x 2 π R m i x T http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ8_HTML.gif
          (6)

          The input values are obtained from Equations 1-3.

          Specific Heat at Constant Pressure

          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
          c p m i x = X i ρ i ρ m i x c p i + 1 - X i ρ O 2 ρ m i x c p O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ9_HTML.gif
          (7)

          where c p m i x http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq2_HTML.gif and c p i http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq3_HTML.gif 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 = 2 . 5 R u n i v M W i http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq4_HTML.gif The polyatomic estimates are based on empirically derived 4th 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.
          μ m i x = X i k i X i + ( 1 - X i ) ϕ i - O 2 + ( 1 - X i ) k O 2 X i ϕ O 2 - i + ( 1 - X i ) http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ10_HTML.gif
          (8a)
          ϕ i - O 2 = 1 + μ i μ O 2 32 M W i 1 4 2 8 1 + M W i 32 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ11_HTML.gif
          (8b)
          ϕ O 2 - i = 1 + μ O 2 μ i M W i 32 1 4 2 8 1 + 32 M W i http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ12_HTML.gif
          (8c)
          The pure gas conductivity estimates are based on a modified Eucken approximation found in Poling et al. [9].
          k i = μ i R i c p i R i - 1 1 . 15 + 2 . 03 c p i R i - 1 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ13_HTML.gif
          (9)

          Diffusivity

          The self diffusivity for a binary system Dij, represents the movement of species i relative to the mixture, where Dij = Dji. 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).
          D i O 2 = 1 . 43 x 10 - 7 T 1 . 75 2 p 1 M W i + 1 32 - 1 2 Σ v i 1 3 + 16 . 3 1 3 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ14_HTML.gif
          (10)

          In Equation (10) j always represents oxygen, the diffusivity is in m2/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 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq5_HTML.gifis 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].
          D m k = X j D j k + X O 2 D O 2 k - 1 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Equ15_HTML.gif
          (11)

          Where m represents the therapeutic gas mixture considered, j represents the specific therapeutic gas, and k corresponds to H2O or CO2. The diffusion constants in Equation 11 of H2O or CO2 through the therapeutic gas or oxygen are calculated using Equation 10 with atomic diffusion volumes of 13.1 and 26.9 for H2O or CO2, 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, N2O, and N2 with O2 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.
          Table 2

          He-O2 property values at 20°C and 1 atm.

          He Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (n m)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.330

          2.029

          70.561

          0.026

          917.5

          2.551

          1.573

          0.05

          1.272

          2.040

          72.547

          0.029

          945.5

          2.641

          1.632

          0.10

          1.214

          2.051

          74.673

          0.032

          976.1

          2.739

          1.695

          0.15

          1.156

          2.063

          76.954

          0.035

          1009.8

          2.844

          1.764

          0.20

          1.098

          2.074

          79.409

          0.039

          1047.1

          2.957

          1.838

          0.25

          1.039

          2.086

          82.057

          0.043

          1088.6

          3.080

          1.919

          0.30

          0.981

          2.097

          84.924

          0.047

          1135.0

          3.214

          2.007

          0.35

          0.923

          2.109

          88.038

          0.051

          1187.3

          3.359

          2.104

          0.40

          0.865

          2.120

          91.432

          0.055

          1246.6

          3.519

          2.210

          0.45

          0.807

          2.131

          95.148

          0.060

          1314.4

          3.694

          2.328

          0.50

          0.748

          2.141

          99.235

          0.066

          1392.8

          3.888

          2.459

          0.55

          0.690

          2.149

          103.751

          0.071

          1484.4

          4.103

          2.606

          0.60

          0.632

          2.156

          108.773

          0.077

          1592.9

          4.343

          2.772

          0.65

          0.574

          2.161

          114.393

          0.084

          1723.4

          4.613

          2.960

          0.70

          0.516

          2.162

          120.735

          0.091

          1883.3

          4.919

          3.175

          0.75

          0.457

          2.158

          127.959

          0.099

          2084.0

          5.268

          3.424

          0.78

          0.422

          2.152

          132.807

          0.104

          2230.9

          5.503

          3.593

          0.79

          0.411

          2.150

          134.522

          0.106

          2285.5

          5.585

          3.653

          0.80

          0.399

          2.147

          136.291

          0.108

          2343.2

          5.671

          3.715

          0.85

          0.341

          2.127

          146.059

          0.117

          2690.8

          6.140

          4.060

          0.90

          0.283

          2.092

          157.788

          0.128

          3181.5

          6.694

          4.477

          0.95

          0.225

          2.037

          172.409

          0.139

          3926.4

          7.359

          4.988

          1.0

          0.166

          1.952

          191.912

          0.152

          5192.4

          8.169

          5.632

          Table 3

          Xe-O2 property values at 20°C and 1 atm.

          Xe Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.330

          2.029

          70.561

          0.026

          917.5

          2.551

          1.573

          0.05

          1.537

          2.084

          67.417

          0.024

          782.7

          2.487

          1.522

          0.10

          1.743

          2.128

          64.637

          0.023

          679.8

          2.427

          1.474

          0.15

          1.950

          2.163

          62.138

          0.021

          598.6

          2.369

          1.429

          0.20

          2.156

          2.192

          59.866

          0.020

          533.1

          2.314

          1.387

          0.25

          2.362

          2.215

          57.783

          0.019

          478.9

          2.262

          1.347

          0.30

          2.569

          2.232

          55.863

          0.017

          433.5

          2.211

          1.309

          0.35

          2.775

          2.247

          54.083

          0.016

          394.9

          2.164

          1.273

          0.40

          2.982

          2.257

          52.428

          0.015

          361.5

          2.118

          1.240

          0.45

          3.188

          2.265

          50.883

          0.014

          332.5

          2.074

          1.208

          0.50

          3.394

          2.271

          49.437

          0.013

          307.1

          2.031

          1.177

          0.55

          3.601

          2.275

          48.080

          0.012

          284.5

          1.991

          1.148

          0.60

          3.807

          2.277

          46.804

          0.011

          264.4

          1.952

          1.121

          0.65

          4.014

          2.278

          45.602

          0.010

          246.4

          1.915

          1.095

          0.70

          4.220

          2.278

          44.467

          0.010

          230.1

          1.878

          1.070

          0.75

          4.427

          2.276

          43.395

          0.009

          215.3

          1.844

          1.046

          0.80

          4.633

          2.274

          42.379

          0.008

          201.9

          1.810

          1.023

          0.85

          4.839

          2.272

          41.415

          0.007

          189.6

          1.778

          1.001

          0.90

          5.046

          2.268

          40.500

          0.007

          178.3

          1.747

          0.980

          0.95

          5.252

          2.265

          39.630

          0.006

          167.9

          1.717

          0.960

          1.0

          5.459

          2.260

          38.801

          0.005

          158.3

          1.688

          0.940

          Table 4

          N2O-O2 property values at 20°C and 1 atm.

          N2O Mole Fraction

          ρ (kg/m3)

          μx105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.330

          2.029

          70.561

          0.026

          917.5

          2.551

          1.573

          0.05

          1.355

          1.956

          67.394

          0.025

          914.3

          2.500

          1.542

          0.10

          1.380

          1.892

          64.577

          0.025

          911.1

          2.451

          1.511

          0.15

          1.405

          1.835

          62.065

          0.024

          908.1

          2.404

          1.482

          0.20

          1.430

          1.784

          59.820

          0.024

          905.2

          2.358

          1.454

          0.25

          1.455

          1.739

          57.810

          0.023

          902.4

          2.315

          1.426

          0.30

          1.480

          1.699

          56.005

          0.023

          899.7

          2.273

          1.400

          0.35

          1.505

          1.664

          54.383

          0.022

          897.1

          2.232

          1.375

          0.40

          1.530

          1.632

          52.923

          0.022

          894.6

          2.193

          1.351

          0.45

          1.555

          1.605

          51.605

          0.021

          892.1

          2.156

          1.327

          0.50

          1.580

          1.580

          50.414

          0.021

          889.7

          2.119

          1.305

          0.55

          1.605

          1.559

          49.337

          0.020

          887.4

          2.084

          1.283

          0.60

          1.630

          1.540

          48.361

          0.020

          885.2

          2.050

          1.262

          0.65

          1.655

          1.523

          47.475

          0.019

          883.0

          2.017

          1.241

          0.70

          1.680

          1.508

          46.670

          0.019

          880.9

          1.985

          1.221

          0.75

          1.705

          1.496

          45.938

          0.019

          878.9

          1.954

          1.202

          0.80

          1.730

          1.485

          45.271

          0.018

          876.9

          1.924

          1.184

          0.85

          1.755

          1.475

          44.663

          0.018

          875.0

          1.895

          1.166

          0.90

          1.780

          1.467

          44.108

          0.018

          873.1

          1.866

          1.148

          0.95

          1.805

          1.461

          43.601

          0.017

          871.3

          1.839

          1.131

          1.0

          1.830

          1.455

          43.137

          0.017

          869.6

          1.812

          1.115

          Table 5

          N2-O2 property values at 20°C and 1 atm.

          N2 Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.330

          2.029

          70.561

          0.026

          917.5

          2.551

          1.573

          0.05

          1.322

          2.015

          70.289

          0.026

          922.8

          2.548

          1.573

          0.10

          1.314

          2.001

          70.016

          0.026

          928.2

          2.546

          1.573

          0.15

          1.306

          1.987

          69.743

          0.026

          933.7

          2.543

          1.573

          0.20

          1.297

          1.973

          69.468

          0.026

          939.3

          2.541

          1.573

          0.25

          1.289

          1.959

          69.192

          0.026

          944.9

          2.538

          1.573

          0.30

          1.281

          1.945

          68.915

          0.026

          950.6

          2.536

          1.573

          0.35

          1.272

          1.931

          68.637

          0.026

          956.3

          2.533

          1.573

          0.40

          1.264

          1.916

          68.358

          0.026

          962.2

          2.531

          1.574

          0.45

          1.256

          1.902

          68.077

          0.026

          968.1

          2.529

          1.574

          0.50

          1.248

          1.888

          67.796

          0.026

          974.1

          2.526

          1.574

          0.55

          1.239

          1.874

          67.513

          0.026

          980.2

          2.524

          1.574

          0.60

          1.231

          1.860

          67.230

          0.026

          986.3

          2.521

          1.574

          0.65

          1.223

          1.846

          66.945

          0.026

          992.6

          2.519

          1.574

          0.70

          1.215

          1.832

          66.659

          0.026

          998.9

          2.516

          1.574

          0.75

          1.206

          1.818

          66.371

          0.026

          1005.3

          2.514

          1.574

          0.78

          1.201

          1.809

          66.198

          0.026

          1009.2

          2.513

          1.574

          0.79

          1.200

          1.806

          66.141

          0.026

          1010.5

          2.512

          1.574

          0.80

          1.198

          1.803

          66.083

          0.026

          1011.8

          2.512

          1.574

          0.85

          1.190

          1.789

          65.793

          0.026

          1018.4

          2.509

          1.574

          0.90

          1.181

          1.775

          65.502

          0.025

          1025.1

          2.507

          1.574

          0.95

          1.173

          1.761

          65.210

          0.025

          1031.9

          2.504

          1.574

          1.0

          1.165

          1.747

          64.916

          0.025

          1038.7

          2.502

          1.574

          Table 6

          He-O2 property values at 37°C and 1 atm.

          He Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.257

          2.113

          75.572

          0.027

          920.7

          2.815

          1.736

          0.05

          1.202

          2.125

          77.716

          0.030

          948.7

          2.915

          1.801

          0.10

          1.147

          2.137

          80.012

          0.034

          979.3

          3.023

          1.871

          0.15

          1.092

          2.149

          82.477

          0.037

          1013.0

          3.139

          1.947

          0.20

          1.037

          2.162

          85.131

          0.041

          1050.3

          3.264

          2.029

          0.25

          0.982

          2.175

          87.996

          0.045

          1091.7

          3.400

          2.118

          0.30

          0.927

          2.188

          91.101

          0.049

          1138.1

          3.547

          2.215

          0.35

          0.872

          2.200

          94.475

          0.053

          1190.3

          3.708

          2.322

          0.40

          0.817

          2.213

          98.157

          0.058

          1249.5

          3.883

          2.440

          0.45

          0.762

          2.225

          102.191

          0.063

          1317.3

          4.077

          2.570

          0.50

          0.707

          2.236

          106.633

          0.069

          1395.7

          4.291

          2.714

          0.55

          0.652

          2.246

          111.548

          0.075

          1487.2

          4.528

          2.876

          0.60

          0.597

          2.255

          117.019

          0.081

          1595.6

          4.793

          3.059

          0.65

          0.542

          2.261

          123.153

          0.088

          1726.0

          5.091

          3.266

          0.70

          0.487

          2.264

          130.085

          0.096

          1885.8

          5.429

          3.504

          0.75

          0.432

          2.262

          137.997

          0.104

          2086.3

          5.814

          3.779

          0.78

          0.399

          2.258

          143.316

          0.110

          2233.2

          6.073

          3.965

          0.79

          0.388

          2.256

          145.199

          0.112

          2287.7

          6.165

          4.032

          0.80

          0.377

          2.254

          147.142

          0.114

          2345.3

          6.259

          4.100

          0.85

          0.322

          2.235

          157.891

          0.124

          2692.7

          6.777

          4.481

          0.90

          0.267

          2.202

          170.832

          0.135

          3183.0

          7.389

          4.941

          0.95

          0.212

          2.149

          187.014

          0.147

          3927.4

          8.122

          5.505

          1.0

          0.157

          2.064

          208.666

          0.161

          5192.4

          9.016

          6.216

          Table 7

          Xe-O2 property values at 37°C and 1 atm.

          Xe Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.257

          2.113

          75.572

          0.027

          920.7

          2.815

          1.736

          0.05

          1.453

          2.173

          72.306

          0.025

          785.3

          2.745

          1.680

          0.10

          1.648

          2.221

          69.409

          0.024

          682.0

          2.678

          1.627

          0.15

          1.843

          2.261

          66.798

          0.022

          600.5

          2.615

          1.577

          0.20

          2.038

          2.293

          64.418

          0.021

          534.7

          2.554

          1.530

          0.25

          2.233

          2.319

          62.231

          0.020

          480.3

          2.496

          1.486

          0.30

          2.428

          2.339

          60.210

          0.018

          434.7

          2.441

          1.445

          0.35

          2.623

          2.356

          58.334

          0.017

          395.9

          2.388

          1.405

          0.40

          2.818

          2.369

          56.586

          0.016

          362.4

          2.337

          1.368

          0.45

          3.013

          2.378

          54.951

          0.015

          333.3

          2.289

          1.333

          0.50

          3.208

          2.386

          53.419

          0.014

          307.7

          2.242

          1.299

          0.55

          3.404

          2.391

          51.980

          0.013

          285.1

          2.197

          1.267

          0.60

          3.599

          2.395

          50.625

          0.012

          264.9

          2.154

          1.237

          0.65

          3.794

          2.397

          49.346

          0.011

          246.7

          2.113

          1.208

          0.70

          3.989

          2.397

          48.138

          0.010

          230.4

          2.073

          1.180

          0.75

          4.184

          2.397

          46.995

          0.009

          215.6

          2.035

          1.154

          0.80

          4.379

          2.396

          45.911

          0.008

          202.1

          1.998

          1.129

          0.85

          4.574

          2.393

          44.882

          0.008

          189.7

          1.962

          1.105

          0.90

          4.769

          2.391

          43.904

          0.007

          178.4

          1.928

          1.081

          0.95

          4.964

          2.387

          42.973

          0.006

          168.0

          1.895

          1.059

          1.0

          5.159

          2.384

          42.086

          0.006

          158.3

          1.863

          1.038

          Table 8

          N2O-O2 property values at 37°C and 1 atm.

          N2O Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.257

          2.113

          75.572

          0.027

          920.7

          2.815

          1.736

          0.05

          1.281

          2.039

          72.254

          0.027

          918.4

          2.759

          1.701

          0.10

          1.305

          1.974

          69.300

          0.026

          916.2

          2.705

          1.668

          0.15

          1.328

          1.916

          66.666

          0.025

          914.0

          2.653

          1.635

          0.20

          1.352

          1.864

          64.312

          0.025

          911.9

          2.603

          1.604

          0.25

          1.376

          1.819

          62.203

          0.024

          909.9

          2.555

          1.574

          0.30

          1.399

          1.779

          60.311

          0.024

          908.0

          2.509

          1.546

          0.35

          1.423

          1.743

          58.611

          0.023

          906.1

          2.464

          1.518

          0.40

          1.446

          1.712

          57.080

          0.023

          904.3

          2.421

          1.491

          0.45

          1.470

          1.684

          55.700

          0.022

          902.5

          2.379

          1.465

          0.50

          1.494

          1.659

          54.454

          0.022

          900.8

          2.339

          1.440

          0.55

          1.517

          1.638

          53.327

          0.021

          899.1

          2.300

          1.416

          0.60

          1.541

          1.619

          52.307

          0.021

          897.5

          2.262

          1.393

          0.65

          1.564

          1.602

          51.382

          0.021

          896.0

          2.226

          1.370

          0.70

          1.588

          1.588

          50.543

          0.020

          894.5

          2.190

          1.348

          0.75

          1.612

          1.576

          49.780

          0.020

          893.0

          2.156

          1.327

          0.80

          1.635

          1.565

          49.087

          0.020

          891.6

          2.123

          1.306

          0.85

          1.659

          1.556

          48.455

          0.019

          890.2

          2.091

          1.286

          0.90

          1.683

          1.549

          47.880

          0.019

          888.9

          2.060

          1.267

          0.95

          1.706

          1.542

          47.355

          0.019

          887.6

          2.030

          1.248

          1.0

          1.730

          1.537

          46.876

          0.018

          886.3

          2.000

          1.230

          Table 9

          N2-O2 property values at 37°C and 1 atm.

          N2 Mole Fraction

          ρ (kg/m3)

          μ × 105 (kg/s-m)

          λ (nm)

          k (W/m-K)

          cp (J/kg-K)

          D H 2 O http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq6_HTML.gif× 105 (m2/s)

          D C O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq7_HTML.gif× 105 (m2/s)

          0

          1.257

          2.113

          75.572

          0.027

          920.7

          2.815

          1.736

          0.05

          1.250

          2.098

          75.262

          0.027

          925.9

          2.812

          1.736

          0.10

          1.242

          2.083

          74.950

          0.027

          931.2

          2.810

          1.736

          0.15

          1.234

          2.067

          74.638

          0.027

          936.6

          2.807

          1.736

          0.20

          1.226

          2.052

          74.324

          0.027

          942.0

          2.804

          1.737

          0.25

          1.218

          2.037

          74.010

          0.027

          947.5

          2.802

          1.737

          0.30

          1.211

          2.022

          73.694

          0.027

          953.1

          2.799

          1.737

          0.35

          1.203

          2.006

          73.377

          0.027

          958.7

          2.796

          1.737

          0.40

          1.195

          1.991

          73.059

          0.027

          964.4

          2.793

          1.737

          0.45

          1.187

          1.976

          72.740

          0.027

          970.2

          2.791

          1.737

          0.50

          1.179

          1.961

          72.420

          0.027

          976.1

          2.788

          1.737

          0.55

          1.171

          1.946

          72.098

          0.027

          982.0

          2.785

          1.737

          0.60

          1.164

          1.931

          71.776

          0.027

          988.0

          2.783

          1.737

          0.65

          1.156

          1.915

          71.452

          0.027

          994.1

          2.780

          1.737

          0.70

          1.148

          1.900

          71.127

          0.027

          1000.3

          2.777

          1.737

          0.75

          1.140

          1.885

          70.801

          0.027

          1006.6

          2.775

          1.737

          0.78

          1.135

          1.876

          70.605

          0.027

          1010.4

          2.773

          1.737

          0.79

          1.134

          1.873

          70.539

          0.027

          1011.6

          2.773

          1.737

          0.80

          1.132

          1.870

          70.474

          0.027

          1012.9

          2.772

          1.737

          0.85

          1.124

          1.855

          70.145

          0.026

          1019.4

          2.769

          1.737

          0.90

          1.117

          1.840

          69.815

          0.026

          1025.9

          2.767

          1.737

          0.95

          1.109

          1.824

          69.484

          0.026

          1032.5

          2.764

          1.737

          1.0

          1.101

          1.809

          69.152

          0.026

          1039.2

          2.761

          1.737

          Table 10

          Binary diffusivities at 1 atm.

          Gas

          D i O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq8_HTML.gif× 105 (m2/s)

           

          20°C

          37°C

          He-O 2

          7.142

          7.883

          Xe-O 2

          1.243

          1.372

          N 2 O-O 2

          1.415

          1.561

          N 2 -O 2

          1.999

          2.206

          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig1_HTML.jpg
          Figure 1

          Density of gas mixtures at 20°C and 1 atm.

          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig2_HTML.jpg
          Figure 2

          Viscosity of gas mixtures at 20°C and 1 atm.

          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig3_HTML.jpg
          Figure 3

          Mean free path of gas mixtures at 20°C and 1 atm.

          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig4_HTML.jpg
          Figure 4

          Thermal conductivity of gas mixtures at 20°C and 1 atm.

          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig5_HTML.jpg
          Figure 5

          Specific heat of gas mixtures at 20°C and 1 atm.

          Discussion

          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 XHe = 0.82. For the concentration weighted average value the maximum relative error of 7.9% occurs at XHe = 0.67.
          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig6_HTML.jpg
          Figure 6

          Viscosity of He-O 2 mixtures using Equation (3), based on a weighted average of the molar fractions and from experimental measurements [14].

          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 XHe = 0.68 and 4.7% at XXe = 0.27, respectively.
          http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_Fig7_HTML.jpg
          Figure 7

          Thermal conductivity at 30°C for He-O 2 and Xe-O 2 mixtures using Equation (8), based on a weighted average of the molar fractions and from experimental measurements [17].

          Table 11 shows a good agreement between experimental data for binary diffusivity of He-O2 and Xe-O2 [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 SF6 through gas mixtures similar to alveolar gas (14% O2, 6% CO2 and 80% N2) 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).

           

          D i - O 2 http://static-content.springer.com/image/art%3A10.1186%2F2045-9912-1-28/MediaObjects/13618_2011_Article_31_IEq9_HTML.gif× 105 (m2/s)

           

          T (K)

          Experimental

          Theoretical

          Percent Difference

          He-O 2

             

          298 [14]

          7.06

          7.357

          4.21

          300 [21]

          7.441

          7.437

          0.05

          Xe-O 2

             

          280 [21]

          1.147

          1.128

          1.68

          290 [21]

          1.220

          1.202

          1.50

          300 [21]

          1.295

          1.279

          1.25

          310 [21]

          1.371

          1.357

          1.03

          320 [21]

          1.449

          1.438

          0.76

          In conclusion, the methods presented above allow accurate estimation of thermophysical property values for inhaled therapeutic binary gas mixtures, including He-O2, Xe-O2, and N2O-O2, 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

          References

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          This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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