research article thermodynamic properties of vapors from...

Research ArticleThermodynamic Properties of Vapors from Speed of Sound

Muhamed BijediT and Sabina BegiT

Faculty of Technology University of Tuzla 75000 Tuzla Bosnia and Herzegovina

Correspondence should be addressed to Muhamed Bijedic muhamedbijedicuntzba

Received 5 May 2014 Accepted 9 July 2014 Published 21 July 2014

Academic Editor Angelo Lucia

Copyright copy 2014 M Bijedic and S Begic This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A numerical procedure for deriving the thermodynamic properties (119885 119888V and 119888119901) of the vapor phase in the subcritical temperaturerange from the speed of sound is presentedThe set of differential equations connecting these properties with the speed of sound issolved as the initial-value problem in119879-120601 domain (120601 = 120588120588sat)The initial values of119885 and (120597119885120597119879)120588 are specified along the isotherm1198790 with the highest temperature at a several values of 120601 [01 10] The values of 119885 are generated by the reference equation of statewhile the values of (120597119885120597119879)120588 are derived from the speed of sound by solving another set of differential equations in 119879-120588 domainin the transcritical temperature range This set of equations is solved as the initial-boundary-value problem The initial values of119885 and 119888V are specified along the isochore in the limit of the ideal gas at several isotherms distributed according to the Chebyshevpoints of the second kind The boundary values of 119885 are specified along the same isotherm 1198790 and along another isotherm with ahigher temperature at several values of 120588 The procedure is tested on Ar N2 CH4 and CO2 with the mean AADs for 119885 119888V and 119888119901at 00003 00046 and 00061 respectively (00007 00130 and 00189 along the saturation line)

1 Introduction

The speed of sound is the property of a fluid which ismeasuredwith an exceptional accuracy (Ewing andGoodwin[1] Estrada-Alexanders and Trusler [2] Costa Gomes andTrusler [3] Trusler and Zarari [4] and Estrada-Alexandersand Trusler [5]) While the speed of sound itself is a pieceof useful information about a fluid other thermodynamicproperties like the density and the heat capacity which maybe derived from itmake the speed of sound evenmore usefulHowever the speed of sound in a fluid is connected to theseproperties through the set of nonlinear partial differentialequations of the second orderThe general solution to this setof equations has not been found yet except for the gaseousphase under low pressures (Trusler et al [6] and Estela-Uribe and Trusler [7]) At moderate and higher pressures aprocedure based on the numerical integration may be usedfor obtaining the particular solution (Estrada-Alexanderset al [8] and Bijedic and Neimarlija [9ndash11]) The maindisadvantage of this approach is the need for the initial andorthe boundary values of the quantities being calculated (the

Dirichlet conditions) and their derivatives with respect to anindependent variable (the Neumann conditions)

In the previous paper (Bijedic and Neimarlija [11]) it wasshown that the compressibility factor and the heat capacity ofa gas may be derived from the speed of sound in the super-critical temperature range only with the initialboundaryconditions of the Dirichlet type In this paper that procedureis applied to the transcritical temperature range in order togenerate the initial conditions of the Neumann type whichare then used for deriving the compressibility factor andthe heat capacity of a vapor in the subcritical temperaturerange In this way the main disadvantage of the approachbased on the numerical integration is minimized becauseonly a few data points of the compressibility factor from othersources are needed to cover the whole gaseous phase exceptthe region around the critical point The boundary values ofthe compressibility factor are specified along two isothermsseveral degrees below the critical point The maximumdensity is chosen so as to correspond to that of the saturatedvapor at the initial temperature In this way the saturationline becomes a boundary of the domain of integration and

Hindawi Publishing CorporationJournal of ermodynamicsVolume 2014 Article ID 231296 5 pageshttpdxdoiorg1011552014231296

2 Journal of Thermodynamics

the thermodynamic properties of the saturated vapor arederived as well While the derived properties of a gas in thetranscritical temperature range have a very good agreementwith the corresponding reference data they are not discussedin this paper (see Table 7)

2 Theory

The thermodynamic speed of sound (the mechanical distur-bance of small amplitude and low frequency) is connectedto othermacroscopic thermodynamic properties through thefollowing fundamental relation (Trusler [12])

1199062= (120597119901

120597120588)119904

(1)

where 119906 is the speed of sound 119901 is the pressure 120588 is thedensity and 119904 is the specific entropy However since 119904 is not ameasurable property it is convenient to avoid it This can beaccomplished with the help of the following thermodynamicrelation (Moran and Shapiro [13])

(120597119901

120597120588)119904

=119888119901

119888V(120597119901

120597120588)119879

(2)

where 119879 is the thermodynamic temperature 119888119901 is the specificheat capacity at constant pressure and 119888V is the specific heatcapacity at constant volume Now (1) takes the followingform

1199062=119888119901

119888V(120597119901

120597120588)119879

(3)

The isobaric heat capacity may be eliminated from (3) onaccount of the following thermodynamic relation (Moranand Shapiro [13])

119888119901 minus 119888V = minus119879(120597V120597119879)

2

119901

(120597119901

120597V)119879

(4)

where V is the specific volume If the specific volume isreplaced with the density (4) becomes

119888119901 minus 119888V =119879

1205882(120597120588

120597119879)

2

119901

(120597119901

120597120588)119879

(5)

According to the chain rule from the calculus one may write

(120597119901

120597120588)119879

(120597120588

120597119879)119901

(120597119879

120597119901)120588

= minus1 (6)

From (6) it follows that

(120597120588

120597119879)119901

= minus(120597120588

120597119901)119879

(120597119901

120597119879)120588

(7)

If the right hand side of (7) is introduced into (5) one obtains

119888119901 = 119888V +119879

1205882(120597120588

120597119901)119879

(120597119901

120597119879)

2

120588

(8)

Introducing the right hand side of (8) into (3) one finallyobtains

1199062= (120597119901

120597120588)119879

+119879

1205882119888V(120597119901

120597119879)

2

120588

(9)

In (9) not only the density but also the isochoric heat capacityis unknown Therefore another equation which connectsthe heat capacity with the thermal properties is necessaryFor this purpose the following thermodynamic relation issuitable

(120597119888V

120597120588)119879

= minus119879

1205882(1205972119901

1205971198792)120588

(10)

If the density is replaced with a less varying compressibilityfactor

119885 =119872119901

120588119877119879 (11)

where 119885 is the compressibility factor 119872 is the molar massand 119877 is the universal gas constant (9) and (10) afterrearrangement take the following forms respectively

(120597119885

120597120588)119879

=1

1205881198721199062

119877119879minus 119885 minus119877

119872119888V[119885 + 119879(

120597119885

120597119879)120588

]

2

(12)

(120597119888V

120597120588)119879

= minus119877

119872120588[2119879(120597119885

120597119879)120588

+ 1198792(1205972119885

1205971198792)120588

] (13)

Equations (12) and (13) may be solved simultaneously for 119885and 119888V in the range of 119901 and 119879 in which accurate values ofthe speed of sound are available (Bijedic and Neimarlija [11])This set of equations is designed to be solved in rectangular119879-120588 domain

If the subcritical vapor phase is considered it is con-venient to introduce new independent variable instead ofthe density For this purpose the ratio of the density andits maximum value at the observed temperature (at thesaturation) is suitable that is

120601 =120588

120588119904 (14)

where 120601 is the new independent variable and 120588119904 is the densityof the saturated vapor at the observed temperature In such away new rectangular 119879-120601 domain is obtained which allowsone to cover the temperature range from the critical pointdown to the triple point and the density range from the idealgas limit up to the saturation line

According to the calculus one may write

(120597119885

120597119879)120601

= (120597119885

120597119879)120588

+ (120597119885

120597120588)119879

(120597120588

120597119879)120601

(15)

If new variable 119886 is defined as

119886 = (120597119885

120597119879)120588

(16)

Journal of Thermodynamics 3

Table 1 120588-119879-119901 ranges covered for the subcritical domain

Substance 120588 [kg timesmminus3] 119879 [K] 119901 [MPa]min max min max min max

Ar 0 17885835 90 140 0 316822N2 0 8729420 70 115 0 193704CH4 0 6137508 100 180 0 328518CO2 0 17196269 220 290 0 531772

(13) now looks like

(120597119888V

120597120588)119879

= minus119877

119872120588[2119879119886 + 119879

2(120597119886

120597119879)120588

] (17)

According to the calculus one may also write

(120597119886

120597119879)120601

= (120597119886

120597119879)120588

+ (120597119886

120597120588)119879

(120597120588

120597119879)120601

(18)

If (17) and (18) are combined one obtains

(120597119886

120597119879)120601

= minus1

119879[119872120588

119877119879(120597119888V

120597120588)119879

+ 2119886] + (120597119886

120597120588)119879

(120597120588

120597119879)120601

(19)

Now (12) (15) and (19) may be solved simultaneously for 119885and 119888V in the range of 119901 and 119879 in which accurate values of thespeed of sound are available

When the integration is complete 119888119901 is calculated fromthe following relation

119888119901 = 119888V1198721199062

119885119877119879[1 minus119901

119885(120597119885

120597119901)119879

] (20)

which is obtained from (3) when the density is replaced withthe compressibility factor

3 Results and Discussion

The speed of sound is usually available in the form 119906 =119906(119879 119901) and not in the form 119906 = 119906(119879 120588) However thenumerical integration is performed with 119879 and 120588120588119904 as theindependent variables rather than 119879 and 119901119901119904 becausemore accurate results are obtained (Bijedic and Neimarlija[10]) For that reason the lines of constant 120601 are generatedin advance by the Peng-Robinson equation of state (Pengand Robinson [14]) except the line 120601 = 10 which isgenerated by the corresponding reference equation of stateSince the compressibility factor is calculated before the speedof sound is required in each integration step the latter isinterpolated along the isotherms with respect to the pressure(119901 = 119885120588119877119879119872)Then119872times119873 partial derivatives (120597120588120597119879)

120601are

estimated from the interpolating polynomial in the LagrangeformThe total number of the isotherms119872 is 7 to 10 and thetotal number of the lines with constant 120601119873 is 10 The valuesof (120597120588120597119879)

120601off the isotherms are interpolated along the lines

of constant 120601 with respect to the temperatureThe calculation procedure is initialized by specifying

values of 119885 and 119886 along the isotherm 1198790 with the highest

Table 2 120588-119879-119901 ranges covered for the transcritical domain


Ar 0 17885835 140 280 0 975263N2 0 8729420 115 255 0 639904CH4 0 6137508 180 320 0 914395CO2 0 17196269 290 430 0 1184715

Table 3 The calculated and the reference (120597119885120597119879)120588 at 1198790

Substance 119901 [MPa] (120597119885120597119879)120588 times 103 [Kminus1]

Calculated Reference

Ar1198790 = 140K

081075631499369020729260253872002904232031682270

098731968129453392344911758890

098731968129455392404913458933

N21198790 = 115 K

047881110893211812462020154083001780123019370410

110372207933152443495582965622

110372208133158443645586565692

CH41198790 = 180K

085981061579633021691930263854802997768032851810

084671684125078332684157551380

084671684125079332734158951412

CO21198790 = 290K

150051402729930037114140446854005024763053177280

069731378320445270043355338461

069731378320445270053355538472

Table 4 The maximum RD in the subcritical domain

Substance RD []119885 119888V 119888119901

Ar +00014minus00001

+00012minus00532

+00016minus00801

N2+00025minus00000

+00135minus00875

+00142minus01085

CH4+00021minus00004

+00352minus00482

+00361minus00787

CO2+00018minus00003

+00073minus00211

+00074minus00562

temperature (see Table 1) at 10 equidistant values of 120601 [0110] The values of 119885 are generated by the correspondingreference equation of state while values of 119886 are derived fromthe speed of sound by solving the set of (12) and (13) in 119879-120588domain in the transcritical temperature range (see Tables 2and 3) Then the partial derivatives (120597119885120597120588)

119879and (120597119886120597120588)

119879

are estimated along 1198790 from the interpolating polynomial in


Table 5 AAD in the subcritical domain

Substance AAD []119885 119888V 119888119901

Ar 00001 00025 00037N2 00003 00048 00061CH4 00003 00062 00078CO2 00005 00048 00069Mean 00003 00046 00061

the Lagrange form Next 119888V is calculated from (12) and thepartial derivatives (120597119888V120597120588)119879 are estimated along 1198790 also fromthe interpolating polynomial in the Lagrange form Nowvalues of 119885 and 119886 are calculated from (15) and (19) along theisotherm1198791 = 1198790minusΔ119879 by the Runge-Kutta-Verner fifth-orderand six-order method with the adaptive step-size Δ119879 (Hullet al [15]) The whole procedure is repeated until the lowesttemperature of the range is reached (see Table 1)

The results are assessed by means of the relative deviation(RD) and the absolute average deviation (AAD) which arecomputed from the following relations

RD =119883cal minus 119883ref119883ref

times 100 []

AAD = 100

(119872 minus 1)119873

119872

sum119894=2

119873

sum119895=1

1003816100381610038161003816119883cal minus 119883ref1003816100381610038161003816

119883ref []

(21)

where 119883cal is the calculated value and 119883ref is the referencevalue of 119885 119888V and 119888119901 (Tegeler et al [16] Span et al [17]Setzmann and Wagner [18] and Span and Wagner [19])

As one could anticipate the worst results are obtainedalong the saturation line (120601 = 10) because it exhibits thehighest nonlinearity of 119885 with respect to 119879 and because allthe partial derivatives with respect to the density are theworstexactly along this boundary line (Rungersquos phenomenon)For that reason the maximum relative deviations given inTable 4 actually refer to the saturation line While theseresults are worse than the rest they are still in the limits ofthe experimental uncertainties of the direct measurements(see Table 8) However the mean AADs of the results are 2orders ofmagnitude (1-2 along the saturation line) better thanthe experimental uncertainties of the corresponding directmeasurements (see Tables 5 and 6)

4 Conclusions

Thecompressibility factor and the heat capacity (the isochoricand the isobaric) of a vapor in the subcritical temperaturerange (including the saturation line) may be derived from thespeed of soundwith theAADs twoorders ofmagnitude betterthan the experimental uncertainties of the correspondingdirect measurements This can be accomplished on accountof just a few data points of the compressibility factor obtainedfrom the direct measurement of the density Moreover thetemperature derivatives of the compressibility factor neededfor initialization of the numerical integration may also be

Table 6 AAD along the saturation line

Substance AAD []119885 119888V 119888119901

Ar 00004 00093 00142N2 00009 00180 00235CH4 00009 00155 00220CO2 00004 00093 00160Mean 00007 00130 00189

Table 7 AAD in the transcritical domain

Substance AAD []119885 119888V 119888119901

Ar 00031 00573 00680N2 00039 00671 00672CH4 00016 00435 00957CO2 00009 00222 00943Mean 00024 00475 00813

Table 8 The uncertainty of the reference data

Substance Uncertainty []119885 119888V 119888119901

Ar plusmn002 plusmn030 plusmn030N2 plusmn002 plusmn030 plusmn030CH4 plusmn003 plusmn100 plusmn100CO2 plusmn003 plusmn100 plusmn100

derived from the speed of sound Practically the wholegaseous phase from the triple point to far above the criticalpoint and from the ideal gas limit to the saturation linemay becoveredwith just a few thermal data points fromother source

Appendix

See Tables 1 2 3 4 5 6 7 and 8

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M B Ewing and A R H Goodwin ldquoAn apparatus based on aspherical resonator formeasuring the speed of sound in gases athigh pressures Results for argon at temperatures between 255 Kand 300K and at pressures up to 7MPardquoThe Journal of ChemicalThermodynamics vol 24 no 5 pp 531ndash547 1992

[2] A F Estrada-Alexanders and J P M Trusler ldquoThe speed ofsound in gaseous argon at temperatures between 110 K and450K and at pressures up to 19MPardquo The Journal of ChemicalThermodynamics vol 27 no 10 pp 1075ndash1089 1995

[3] M F Costa Gomes and J P M Trusler ldquoThe speed of soundin nitrogen at temperatures between 119879 = 250K and 119879 =


350K and at pressures up to 30 MPardquo Journal of ChemicalThermodynamics vol 30 no 5 pp 527ndash534 1998

[4] J P M Trusler and M Zarari ldquoThe speed of sound andderived thermodynamic properties of methane at temperaturesbetween 275 K and 375 K and pressures up to 10 MPardquo TheJournal of ChemicalThermodynamics vol 24 no 9 pp 973ndash9911992

[5] A F Estrada-Alexanders and J P M Trusler ldquoSpeed of soundin carbon dioxide at temperatures between (220 and 450) K andpressures up to 14 MPardquo Journal of Chemical Thermodynamicsvol 30 no 12 pp 1589ndash1601 1998

[6] J P M Trusler W A Wakeham and M P Zarari ldquoModelintermolecular potentials and virial coefficients determinedfrom the speed of soundrdquo Molecular Physics vol 90 no 5 pp695ndash703 1997

[7] J F Estela-Uribe and J P M Trusler ldquoAcoustic and volumetricvirial coefficients of nitrogenrdquo International Journal of Thermo-physics vol 21 no 5 pp 1033ndash1044 2000

[8] A F Estrada-Alexanders J P M Trusler and M P ZararildquoDetermination of thermodynamic properties from the speedof soundrdquo International Journal of Thermophysics vol 16 no 3pp 663ndash673 1995

[9] M Bijedic and N Neimarlija ldquoThermodynamic propertiesof gases from speed-of-sound measurementsrdquo InternationalJournal of Thermophysics vol 28 no 1 pp 268ndash278 2007

[10] M Bijedic and N Neimarlija ldquoThermodynamic properties ofcarbon dioxide derived from the speed of soundrdquo Journal of theIranian Chemical Society vol 5 no 2 pp 286ndash295 2008

[11] M Bijedic and N Neimarlija ldquoSpeed of sound as a source ofaccurate thermodynamic properties of gasesrdquo Latin AmericanApplied Research vol 43 no 4 pp 393ndash398 2013

[12] J PM TruslerPhysical Acoustics andMetrology of Fluids AdamHilger Bristol UK 1991

[13] M J Moran and H N Shapiro Fundamentals of EngineeringThermodynamics John Wiley amp Sons Chichester UK 5thedition 2006

[14] D Y Peng andD B Robinson ldquoA new two-constant equation ofstaterdquo Industrial and Engineering Chemistry Fundamentals vol15 no 1 pp 59ndash64 1976

[15] T E Hull W H Enright and K R Jackson ldquoUsers guide forDVERKmdasha subroutine for solving non-stiff ODEsrdquo Tech RepUniversity of Toronto Toronto Canada 1976

[16] C Tegeler R Span and W Wagner ldquoA new equation of statefor argon covering the fluid region for temperatures from themelting line to 700 K at pressures up to 1000 MPardquo Journal ofPhysical and Chemical Reference Data vol 28 no 3 pp 779ndash850 1999

[17] R Span E W Lemmon R T Jacobsen W Wagner and AYokozeki ldquoA reference equation of state for the thermodynamicproperties of nitrogen for temperatures from 63151 to 1000 Kand pressures to 2200 MPardquo Journal of Physical and ChemicalReference Data vol 29 no 6 pp 1361ndash1433 2000

[18] U Setzmann andWWagner ldquoA new equation of state and tablesof thermodynamic properties for methane covering the rangefrom the melting line to 625K at pressures up to 1000MPardquoJournal of Physical and Chemical Reference Data vol 20 no 6pp 1061ndash1155 1991

[19] R Span and W Wagner ldquoA new equation of state for carbondioxide covering the fluid region from the triple-point temper-ature to 1100K at pressures up to 800MPardquo Journal of Physicaland Chemical Reference Data vol 25 no 6 pp 1509ndash1596 1996

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AerodynamicsJournal of

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ThermodynamicsJournal of


the thermodynamic properties of the saturated vapor arederived as well While the derived properties of a gas in thetranscritical temperature range have a very good agreementwith the corresponding reference data they are not discussedin this paper (see Table 7)

2 Theory

The thermodynamic speed of sound (the mechanical distur-bance of small amplitude and low frequency) is connectedto othermacroscopic thermodynamic properties through thefollowing fundamental relation (Trusler [12])

1199062= (120597119901

120597120588)119904

(1)

where 119906 is the speed of sound 119901 is the pressure 120588 is thedensity and 119904 is the specific entropy However since 119904 is not ameasurable property it is convenient to avoid it This can beaccomplished with the help of the following thermodynamicrelation (Moran and Shapiro [13])

(120597119901

120597120588)119904

=119888119901

119888V(120597119901

120597120588)119879

(2)

where 119879 is the thermodynamic temperature 119888119901 is the specificheat capacity at constant pressure and 119888V is the specific heatcapacity at constant volume Now (1) takes the followingform

1199062=119888119901

119888V(120597119901

120597120588)119879

(3)

The isobaric heat capacity may be eliminated from (3) onaccount of the following thermodynamic relation (Moranand Shapiro [13])

119888119901 minus 119888V = minus119879(120597V120597119879)

2

119901

(120597119901

120597V)119879

(4)

where V is the specific volume If the specific volume isreplaced with the density (4) becomes

119888119901 minus 119888V =119879

1205882(120597120588

120597119879)

2

119901

(120597119901

120597120588)119879

(5)

According to the chain rule from the calculus one may write

(120597119901

120597120588)119879

(120597120588

120597119879)119901

(120597119879

120597119901)120588

= minus1 (6)

From (6) it follows that

(120597120588

120597119879)119901

= minus(120597120588

120597119901)119879

(120597119901

120597119879)120588

(7)

If the right hand side of (7) is introduced into (5) one obtains

119888119901 = 119888V +119879

1205882(120597120588

120597119901)119879

(120597119901

120597119879)

2

120588

(8)

Introducing the right hand side of (8) into (3) one finallyobtains

1199062= (120597119901

120597120588)119879

+119879

1205882119888V(120597119901

120597119879)

2

120588

(9)

In (9) not only the density but also the isochoric heat capacityis unknown Therefore another equation which connectsthe heat capacity with the thermal properties is necessaryFor this purpose the following thermodynamic relation issuitable

(120597119888V

120597120588)119879

= minus119879

1205882(1205972119901

1205971198792)120588

(10)

If the density is replaced with a less varying compressibilityfactor

119885 =119872119901

120588119877119879 (11)

where 119885 is the compressibility factor 119872 is the molar massand 119877 is the universal gas constant (9) and (10) afterrearrangement take the following forms respectively

(120597119885

120597120588)119879

=1

1205881198721199062

119877119879minus 119885 minus119877

119872119888V[119885 + 119879(

120597119885

120597119879)120588

]

2

(12)

(120597119888V

120597120588)119879

= minus119877

119872120588[2119879(120597119885

120597119879)120588

+ 1198792(1205972119885

1205971198792)120588

] (13)

Equations (12) and (13) may be solved simultaneously for 119885and 119888V in the range of 119901 and 119879 in which accurate values ofthe speed of sound are available (Bijedic and Neimarlija [11])This set of equations is designed to be solved in rectangular119879-120588 domain

If the subcritical vapor phase is considered it is con-venient to introduce new independent variable instead ofthe density For this purpose the ratio of the density andits maximum value at the observed temperature (at thesaturation) is suitable that is

120601 =120588

120588119904 (14)

where 120601 is the new independent variable and 120588119904 is the densityof the saturated vapor at the observed temperature In such away new rectangular 119879-120601 domain is obtained which allowsone to cover the temperature range from the critical pointdown to the triple point and the density range from the idealgas limit up to the saturation line

According to the calculus one may write

(120597119885

120597119879)120601

= (120597119885

120597119879)120588

+ (120597119885

120597120588)119879

(120597120588

120597119879)120601

(15)

If new variable 119886 is defined as

119886 = (120597119885

120597119879)120588

(16)




Ar 0 17885835 90 140 0 316822N2 0 8729420 70 115 0 193704CH4 0 6137508 100 180 0 328518CO2 0 17196269 220 290 0 531772

(13) now looks like

(120597119888V

120597120588)119879

= minus119877

119872120588[2119879119886 + 119879

2(120597119886

120597119879)120588

] (17)


(120597119886

120597119879)120601

= (120597119886

120597119879)120588

+ (120597119886

120597120588)119879

(120597120588

120597119879)120601

(18)


(120597119886

120597119879)120601

= minus1

119879[119872120588

119877119879(120597119888V

120597120588)119879

+ 2119886] + (120597119886

120597120588)119879

(120597120588

120597119879)120601

(19)



119888119901 = 119888V1198721199062

119885119877119879[1 minus119901

119885(120597119885

120597119901)119879

] (20)




120601are







Ar 0 17885835 140 280 0 975263N2 0 8729420 115 255 0 639904CH4 0 6137508 180 320 0 914395CO2 0 17196269 290 430 0 1184715




Ar1198790 = 140K

081075631499369020729260253872002904232031682270

098731968129453392344911758890

098731968129455392404913458933

N21198790 = 115 K

047881110893211812462020154083001780123019370410

110372207933152443495582965622

110372208133158443645586565692

CH41198790 = 180K

085981061579633021691930263854802997768032851810

084671684125078332684157551380

084671684125079332734158951412

CO21198790 = 290K

150051402729930037114140446854005024763053177280

069731378320445270043355338461

069731378320445270053355538472


Substance RD []119885 119888V 119888119901

Ar +00014minus00001

+00012minus00532

+00016minus00801

N2+00025minus00000

+00135minus00875

+00142minus01085

CH4+00021minus00004

+00352minus00482

+00361minus00787

CO2+00018minus00003

+00073minus00211

+00074minus00562


119879and (120597119886120597120588)

119879




Substance AAD []119885 119888V 119888119901

Ar 00001 00025 00037N2 00003 00048 00061CH4 00003 00062 00078CO2 00005 00048 00069Mean 00003 00046 00061




times 100 []

AAD = 100

(119872 minus 1)119873

119872

sum119894=2

119873

sum119895=1

1003816100381610038161003816119883cal minus 119883ref1003816100381610038161003816

119883ref []

(21)



4 Conclusions



Substance AAD []119885 119888V 119888119901

Ar 00004 00093 00142N2 00009 00180 00235CH4 00009 00155 00220CO2 00004 00093 00160Mean 00007 00130 00189


Substance AAD []119885 119888V 119888119901

Ar 00031 00573 00680N2 00039 00671 00672CH4 00016 00435 00957CO2 00009 00222 00943Mean 00024 00475 00813





Appendix




References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Ar 0 17885835 90 140 0 316822N2 0 8729420 70 115 0 193704CH4 0 6137508 100 180 0 328518CO2 0 17196269 220 290 0 531772

(13) now looks like

(120597119888V

120597120588)119879

= minus119877

119872120588[2119879119886 + 119879

2(120597119886

120597119879)120588

] (17)


(120597119886

120597119879)120601

= (120597119886

120597119879)120588

+ (120597119886

120597120588)119879

(120597120588

120597119879)120601

(18)


(120597119886

120597119879)120601

= minus1

119879[119872120588

119877119879(120597119888V

120597120588)119879

+ 2119886] + (120597119886

120597120588)119879

(120597120588

120597119879)120601

(19)



119888119901 = 119888V1198721199062

119885119877119879[1 minus119901

119885(120597119885

120597119901)119879

] (20)




120601are







Ar 0 17885835 140 280 0 975263N2 0 8729420 115 255 0 639904CH4 0 6137508 180 320 0 914395CO2 0 17196269 290 430 0 1184715




Ar1198790 = 140K

081075631499369020729260253872002904232031682270

098731968129453392344911758890

098731968129455392404913458933

N21198790 = 115 K

047881110893211812462020154083001780123019370410

110372207933152443495582965622

110372208133158443645586565692

CH41198790 = 180K

085981061579633021691930263854802997768032851810

084671684125078332684157551380

084671684125079332734158951412

CO21198790 = 290K

150051402729930037114140446854005024763053177280

069731378320445270043355338461

069731378320445270053355538472


Substance RD []119885 119888V 119888119901

Ar +00014minus00001

+00012minus00532

+00016minus00801

N2+00025minus00000

+00135minus00875

+00142minus01085

CH4+00021minus00004

+00352minus00482

+00361minus00787

CO2+00018minus00003

+00073minus00211

+00074minus00562


119879and (120597119886120597120588)

119879




Substance AAD []119885 119888V 119888119901

Ar 00001 00025 00037N2 00003 00048 00061CH4 00003 00062 00078CO2 00005 00048 00069Mean 00003 00046 00061




times 100 []

AAD = 100

(119872 minus 1)119873

119872

sum119894=2

119873

sum119895=1

1003816100381610038161003816119883cal minus 119883ref1003816100381610038161003816

119883ref []

(21)



4 Conclusions



Substance AAD []119885 119888V 119888119901

Ar 00004 00093 00142N2 00009 00180 00235CH4 00009 00155 00220CO2 00004 00093 00160Mean 00007 00130 00189


Substance AAD []119885 119888V 119888119901

Ar 00031 00573 00680N2 00039 00671 00672CH4 00016 00435 00957CO2 00009 00222 00943Mean 00024 00475 00813





Appendix




References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Substance AAD []119885 119888V 119888119901

Ar 00001 00025 00037N2 00003 00048 00061CH4 00003 00062 00078CO2 00005 00048 00069Mean 00003 00046 00061




times 100 []

AAD = 100

(119872 minus 1)119873

119872

sum119894=2

119873

sum119895=1

1003816100381610038161003816119883cal minus 119883ref1003816100381610038161003816

119883ref []

(21)



4 Conclusions



Substance AAD []119885 119888V 119888119901

Ar 00004 00093 00142N2 00009 00180 00235CH4 00009 00155 00220CO2 00004 00093 00160Mean 00007 00130 00189


Substance AAD []119885 119888V 119888119901

Ar 00031 00573 00680N2 00039 00671 00672CH4 00016 00435 00957CO2 00009 00222 00943Mean 00024 00475 00813





Appendix




References



























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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