Research Article Macroscopic Expressions of Molecular ...

Research ArticleMacroscopic Expressions of Molecular AdiabaticCompressibility of Methyl and Ethyl Caprate underHigh Pressure and High Temperature

Fuxi Shi1 Qin Zhang2 Jun Chen1 and Hamid Reza Karimi4

1 College of Mechanical and Electronic Engineering Northwest AampF University No 22 Xinong Road Yangling XirsquoanShaanxi 712100 China

2Department of Biological Systems Engineering Washington State University Room 105 24106 N Bunn Road Prosser WA USA4Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Jun Chen chenjun jdxynwsuafeducn

Received 18 November 2013 Accepted 18 December 2013 Published 14 January 2014

Academic Editor Ming Liu

Copyright copy 2014 Fuxi Shi et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The molecular compressibility which is a macroscopic quantity to reveal the microcompressibility by additivity of molecularconstitutions is considered as a fixed value for specific organic liquids In this study we introduced two calculated expressionsof molecular adiabatic compressibility to demonstrate its pressure and temperature dependency The first one was developed fromWadarsquos constant expression based on experimental data of density and sound velocity Secondly by introducing the 2D fittingexpressions and their partial derivative of pressure and temperature molecular compressibility dependency was analyzed furtherand a 3Dfitting expressionwas obtained from the calculated data of the first oneThe thirdwas derivedwith introducing the pressureand temperature correction factors based on analogy to Lennard-Jones potential function and energy equipartition theorem Inwide range of temperatures (293 lt TK lt 393) and pressures (01 lt PMPa lt 210) which represent the typical values used indynamic injection process for diesel engines the calculated results consistency of three formulas demonstrated their effectivenesswith the maximum 05384 OARD meanwhile the dependency on pressure and temperature of molecular compressibility wascertified

1 Introduction

In modern diesel engine the injection pressure of fuel spraysystem continuously was elevated to achieve better atom-ization combustion and emission effect and even reach200MPa or more which increase vastly the dynamics sen-sibility of fuel compressibility The dynamic delivery processwhich was carried out under rapid variation of pressure andtemperature was strongly affected by the compressibility offuel and its derivative properties relative to pressure andtemperature For example density influences the conversionof volume flow rate into mass flow rate [1] whereas thecompressibility (the reciprocal of bulk modulus) acts on thefuel injection timing [2 3]Therefore the adaptation of injec-tion systems to biodiesels requires an accurate knowledgeof the volumetric properties of biodiesel components over a

wide range of pressure and temperature [4 5] especially thecompressibility and sound velocity values were indispensableparameters for system modeling and experimental injectionrate determination [6]

In order to clarify the connection between the micro-scopic chemical structure of organic liquids and the com-pressibility from investigation of sound velocity and densitymany theoretical analyses and derivations have been per-formed to determine these properties Rao [7] has pointedout the product of density and 7th root of isothermal com-pressibility 119896

119879 12058811989617119879

was highly independent of temperatureand pressure Further molecular adiabatic compressibility119896119898= 11987211990812058811989617119904

which was proposed by Wada [8] and wascalculated by summing bond increments of molecular com-ponents was used to express the microcompressibility with

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 512576 10 pageshttpdxdoiorg1011552014512576

2 Abstract and Applied Analysis

a constant molar volume for many organic liquids Schaaffs[9 10] who firstly derived a quantitative expression to createconnection between sound velocity and chemical structureof organic compounds pointed out the properties related tosound velocity with the atomic additive property [11] As theeigenvalues of the molecular spacing and molecular inter-molecular forces molecular compressibility was regarded asa constant

Nowadays molecular compressibility was used to predictthe speed of sound in some research [12ndash14] a negative tem-perature correction coefficient is used tomodify the influenceof temperature onmolecular compressibility it is an effectivesolution to decrease prediction deviationwhen the numericalaccuracy of molecular compressibility was required Freitaset al [13] predicted speed of sound of FAMEs by Wadarsquosgroup contributionmethod but the ARD of predicted resultsshows regularly downward trend with elevated temperatureLater Daridon et al [12] divided the FA esters molecule intofive bigger functional groups to contribute to the molecularcompressibility meanwhile molecular compressibility wasdemonstrated with slight temperature dependency Theseinvestigations provide a quantitative connection betweenmacroproperties and molecular microstructure but the lit-erature on the compressibility for FA esters and biodiesels isstill very scant relative to its species most data are availableonly at atmospheric pressure and in a narrow range oftemperature and could notmatch the fuel working conditionsin injection system That restricted the overall cognition onmolecular compressibility Recently more studies [15 16]which focused on sound velocity and density under highpressure and high temperature conditions and the modelingof these properties gave a possibility to comprehensivelyunderstand on molecular adiabatic compressibility in 119875-119879domain

In this paper the calculated value of molecular com-pressibility was extrapolated to close working conditions offuel injection system [17] based on [15] data By data fittingits dependence of pressure and temperature was formulatedand analyzed at two dimensions and three dimensions Asthe eigenvalues of the molecular spacing and molecularintermolecular forces molecular compressibility was con-nected to molecular potential energy and thermal kineticenergy by developing a new correction formula The consis-tency of calculated results derived from different principlesverified effectiveness of these formulas and confirmed thedependency of molecular compressibility on pressure andtemperature

2 The Theory ofthe Molecular Compressibility

Concept of molecular compressibility was derived from thesummary of the experimental data of organic liquids [8]Wada has pointed out that if 12058811989617

119878is independent of pressure

the rate of 1119896119904sdot 120597119896119904120597119901 to 1120588 sdot 120597120588120597119901 was expected to

be a constant 119899 which was 73 in many organic liquidsSo 12058811989617

119904was used as a characteristic function of organic

liquids thus molecular adiabatic compressibility11987211990812058811989617119904

is

expressed as additivity of molecular constitution and showsan approximate fixed value for an organic liquid But thisapplication will bring two aspects problems Firstly 119899 = 7

is just an approximate value based on experimental datathis application is short of theoretical basis and must havecertain error under a large range of pressure and temperatureSecondly the difference of molecular configuration and sizeresponding to pressure and temperature should be distinctalthoughWada listed about one hundred calculated values ofmolecular compressibility to show additive property of bondincrements still seven or more organic liquids have appre-ciable discrepancies between experimental and calculatedvalues and the experimental conditions of data acquisitionwere not shown clearly

InWadarsquos derivation molecular adiabatic compressibilitywas expressed as

119896119898=

119872119908

12058811989617

119904

=1198810

11989617

1199040

=Wadarsquos Const (1)

where1198721199081198810 1198961199040are the molecular weight the mole volume

at close packed state and the compressibility of moleculeitself respectively and shown as

119872119908

120588= 1198810+ V = 119881

0 (1 + 119896)

119896119904= 1198961199040(1 + 119896)

7

(2)

where V 119896 are the free volume per mole and fractionalfree volume respectively For different organic liquid 119881

0

as the molar volume of close packed state is fixed value11989617

1199040also are shown as an almost fixed calculated value

thus Wada concluded 119896119898was approximately constant But

there exists an unreasonable assumption that 1198810is unlikely

to reach the close packed state at liquid state hence 1198810

compressibility is variational under dynamic high pressureand high temperature Furthermore 7th power of (1 + 119896) informula (2) is incorrect to express the linear superpositionof compressibility between free volume and molecular sizeSo we can draw a conclusion that the consistent 11989617

1199040value is

derived from tiny error with 7th root of 1198961199040 so the formula (1)

is not strictly correct and approximate algorithm under somespecial conditions

In addition the formula (1) can further be developedaccording to the definition of adiabatic compressibility 119896

119904as

follows

119896119898=

119872119908

12058811989617

119904

=119872119908

120588(minus (1119881) (120597119881120597119875))17

=119872119908

120588((1120588) (120597120588120597119875))17

=119872119908

12058867(120597119875

120597120588)

17

(3)

So molecular adiabatic compressibility is supposed to bea function of density with pressure and the influence of thetemperature is implied in density

Abstract and Applied Analysis 3

Table 1 Group contribution values for molecular compressibility estimation at atmospheric pressure [12]

ndashCH3 ndashCH2ndash ndashCH=CHndash CH3COOndash ndashCH2COOndash119870119898times 103 05097 035196 059074 105856 09061

1198790(K) 29815

120594 times 103 0034852

3 The Characteristic Analysis ofthe Molecular Compressibility

For revealing similar law of the properties many density andsound velocity measurements for FAMEs and FAEEs wereperformed and predictive methodologies were proposed todetermine the sound velocity based on ultrasonic techniqueTat et al and Tat and van Gerben [21 22] obtained densityand sound velocity of 16 FA esters in 20 to 100∘C and 01 to325MPa and achieved only one polynomial function whichis second order in temperature and first order in pressurewith different fitting coefficients to predict density soundvelocity and isentropic bulk modulus whose dependency ontemperature and pressure indicates similitude principle ofchange for different FA esters (4) was used to convert theisentropic bulk modulus 120573

119904to isentropic compressibility

119896119904=1

120573119904

=1

1205881198882 (4)

Daridon et al [12] measured sound velocity and densityof seven FA esters (including ethyl and methyl saturated andunsaturated) by using a pulse echo technique operating with3MHz at atmospheric pressure in 28315sim37315 K rangeThemolecular compressibility which was calculated by Wadarsquosgroup contribution method was almost a constant for spe-cific FA ester the five basic microscopic radicals (CH

3COOndash

ndashCH2COOndash CH

3ndash ndashCH

2ndash ndashCH=CHndash) which contributed

to molecular compressibility of a FA ester according to addi-tivity andwere listed in Table 1 had a fixed value respectivelyand the predicted results for sound velocity (using (6))showed a high prediction accuracy This study showed thatthe molecular compressibility is related to molecular specificstructure and the negative temperature correction coefficientin (5) presented temperature dependence of molecular com-pressibility

119896119898 (119879) =

119899119866

sum119895=1

119873119895119896119898119895(1 minus 120594 (119879 minus 29815)) (5)

119888 = 1205883(119896119898

119872119908

)

72

(6)

where 119896119898 119896119904 119872119908 120588 119888 are molecular compressibility isen-

tropic compressibility the molecular weight density andsound velocity respectively 119896

119898119895is the contribution of the

group type-119895 to 119896119898

which occurs 119873119895times in the given

molecule and 120594 is a constant parameter used to take intoaccount the influence of temperature

Freitas et al [13] also used Wadarsquos model to predict thesound velocity of the fatty acid esters and biodiesels with

OARDs of 025 and 045 But these predicted resultsof high precision which show evidence of effectiveness ofWadarsquos model are just achieved at atmospheric pressurewithin a small temperature range Latter Freitas et al [14]developed Wadarsquos model to predict the sound velocity of FAesters at high pressures up to 35MPa and pointed out that thepressure dependency of the sound velocity is approximatelylinear relation in the pressure range considered in consid-eration of the nonlinear of 1205883 under variable pressure thusthe molecular compressibility must not be a constant So it isnecessary to extrapolate the value and tendency of molecularcompressibility in high pressure and high temperature

31 Characteristic of Molecular Isentropic Compressibility inHigh Pressure and High Temperature Ndiaye et al [15]measured sound velocity of methyl caprate and ethyl caprateat pressures up to 210MPa in the temperature range 28315 to40315 K In addition density was measured in 01 to 100MPaand 29315 to 39315 K by using a modification of Davisand Gordonrsquos procedure [23] the method [24] rested on theNewton-Laplace relationships then density was extrapolatedup to 210MPa from sound velocity data (7) with an absoluteaverage deviation of 007 up to 210MPa the isentropic andisothermal compressibilities also were presented in the 119875-119879domain

120588 (119875 119879) = 120588 (119875ref 119879) + int119875

119875ref

1

1198882119889119875 + 119879int

119875

119875ref

1205722119875

119862119901

119889119875 (7)

where119862119901 120572119901represent the isobaric heat capacity and the iso-

baric thermal expansion respectively and 119875ref is a referencepressure where density is known

In order to directly obtain molecular isentropic com-pressibility 119896

119898from experimental data we expressed 119896

119898as

a function of density and sound velocity shown as (8)

119896119898= 119872119908(119888

1205883)

27

(8)

Based on [15] data that sound velocity was obtained fromexperimental and density was obtained from calculated valueby (7) 119896

119898was calculated in 01 to 210MPa and 30315 to

38315 K and the fitting results were presented with isother-mal characteristics for methyl caprate and ethyl caprate inFigures 1 and 2

A one-order rational fitting relation to pressure wasexpressed as (9) where 119875 is pressure other parameters 119860

1

1198602 1198603are equation coefficients as listed in Table 2

119896119898 (119875) =

1198601119875 + 119860

2

119875 + 1198603

= 119886 +119887

119875 + 119888 (9)


Table 2 Fitting coefficients for (9) along various isothermals

FA ester Methyl caprate Ethyl caprateTemperature 30315 K 34315 K 38315 K 30315 K 34315 K 38315 KCoefficients1198601

4597 4612 4632 4985 4992 50041198602

4808 3747 2914 5472 4234 3031198603

1097 8555 6658 1155 8939 6396Goodness of fit

SSE 225119864 minus 06 245119864 minus 06 837119864 minus 06 148119864 minus 06 576119864 minus 06 834119864 minus 06

R-square 09999 09999 09998 1 09999 09999Adjusted R-square 09999 09999 09998 1 09999 09998RMSE 0000375 0000391 0000723 0000304 00006 0000722

435

44

445

45

455

46

0 30 60 90 120 150 180 210

P (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)

Figure 1 Molecular compressibility of methyl caprate (MeC100) asa function of pressure along various isothermal (the lines are fittingresults the dots are calculated values)

The rational relation (9) was achieved with satisfyinggoodness of fit where the minimum 119877-square value is09998 in all temperature domains We may safely draw theconclusion that molecular compressibility is dependent onpressure for ethyl and methyl caprate this dependence onpressure was revealed sufficiently with elevated pressure InFigures 1 and 2

Accordingly the calculated results were presented withisobaric characteristics for methyl caprate and ethyl capratein Figures 3 and 4 and a two-order polynomial fitting intemperature domain was expressed as

119896119898 (119879) = 1198611119879

2+ 1198612119879 + 119861

3= 119886(119879 + 119887)

2+ 119888 (10)

where 119879 is temperature and other parameters 1198611 1198612 1198613are

equation coefficients as listed in Table 3In Figures 3 and 4 a two-order polynomial (10) was

achieved with the minimum 119877-square value being 09987in all pressure domains At atmospheric pressure (01MPaisobaric) 119896

119898of methyl caprate shows a slight downtrend

4725

4775

4825

4875

4925

0 30 60 90 120 150 180 210T (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)

Figure 2 Molecular compressibility of ethyl caprate (EeC100) as afunction of pressure along various isothermal (the lines are fittingresults the dots are calculated values)

with two-order form described in (10) but not first-orderform described inDaridonrsquos formula (5) however for 01MPaisobaric of ethyl caprate is almost horizontal meanwhile 119896

119898

show uptrend which accelerates with increasing pressure atisobaric with temperature elevated 119896

119898also is dependent on

temperature for ethyl and methyl caprateFurthermore to calculate molecular compressibility dif-

ference 119896119898of ethyl caprate in isothermal line with 20K step

and in isobaric line with 10MPa step Figures 5 and 6 can beachieved

We can take the partial derivative with respect to pressurefor (9) and to temperature for (10) thus we have

(120597119896119898

120597119875)119879

=1198601

119875 + 1198603

minus1198601119875 + 119860

2

(119875 + 1198603)2=11986011198603minus 1198602

(119875 + 1198603)2 (11)

(120597119896119898

120597119879)119875

= 21198611119879 + 119861

2 (12)


Table 3 Fitting coefficients for (10) along various isobaric

FA ester Mthyl caprate Ethyl capratePressure 11987501 11987580 119875130 119875150 119875210 11987501 11987580 130 119875150 119875210

Coefficients1198611

minus23119864 minus 06 16119864 minus 06 16119864 minus 06 16119864 minus 06 13119864 minus 06 22119864 minus 06 32119864 minus 06 26119864 minus 06 25119864 minus 06 22119864 minus 06

1198612

000139 minus000056 minus000051 minus000047 minus000031 minus000094 minus000161 minus000117 minus000108 minus0000941198613

4168 4495 4507 4506 4499 4982 5028 4983 4978 4982Goodness of fit

SSE 27119864 minus 08 15119864 minus 06 65119864 minus 07 94119864 minus 07 109119864 minus 06 18119864 minus 07 19119864 minus 07 27119864 minus 07 39119864 minus 07 183119864 minus 07

R-square 09998 09987 09995 09993 09992 09998 09999 09998 09997 09998Adjusted R-square 09995 09974 09991 09987 09983 09997 09997 09997 09995 09997RMSE 000012 000088 000057 000068 000074 00003 00003 000037 000044 00003

435

44

445

45

455

46

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (K)

km(103)

Figure 3 Molecular compressibility of methyl caprate (MeC100)as a function of temperature along various isobaric (only calculatedvalues were revealed)

From Figure 5 we can find that among various isother-mals the Δ119896

119898Δ119875 become bigger when the temperature

was elevated gradually and almost keep fixed value above60MPa though nearly equal at atmospheric pressure heretemperature variation plays a more important role underhigh pressure Thus the Δ119896

119898Δ119875 reveal almost equality and

only show dependence on temperature in high pressure(120597119896119898120597119875)119879rarr Const The result can be achieved form the

calculation of (11) coefficients the values of 11986011198603minus 1198602for

various isothermal are in 235 to 162 but 1198603is in about

110 to 65 when pressure is large than 60MPa increments of(119875 + 119860

3)2 acceleration

From Figure 6 we can find that among various isobaricΔ119896119898Δ119875 become smaller when pressure was elevated grad-

ually and keep almost the same value above 60MPa with

4725

4775

4825

4875

4925

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (MPa)

km(103)

Figure 4 Molecular compressibility of ethyl caprate (EeC100) asa function of temperature along various isobaric (only calculatedvalues were revealed)

slight dependence of pressure That is to say the possibilityof being compressed of 119870

119898decreases with high pressure

(120597119896119898120597119879)119875

≃ Const Meanwhile Δ119896119898Δ119875 increase with

elevated temperature but the rate of increment declinesrapidly based on 2119861

1which became almost the same as above

60MPa with 10minus6 magnitude here pressure variation revealsmore contributions under low pressure

32 The Data Fitting Formula of Molecular Compressibilityin Three-Dimensional According to the two-dimensionalfitting analysis a three-dimensional rational formula wasintroduced with the following expression

119896119898= 119886 + 119887(119879 + 119887)

2+119888(119879 + 119887)

2+ 119889

119875 + 119890 (13)


0

0002

0004

0006

0008

001

0012

0014

0016

0018

0 30 60 90 120 150 180 210

3231534315

3631538315

998779km998779

T

P (MPa)

Figure 5 The trend of differential pressure of molecular compress-ibility temperature along various isothermal for ethyl caprate (onlycalculated values were revealed)

0

00005

0001

00015

0002

00025

0003

00035

300 320 340 360 380T (K)

102030405060708090

100110120130140150170190210

998779km998779

P

Figure 6 The trend of differential temperature ofmolecular com-pressibility temperature along various isobaric for ethyl caprate(only calculated values were revealed)

where 119875 119879 are pressure and temperature and other fittingparameters 119886 119887 119888 119889 119890 are formula coefficients listed inTable 4

This formula has one-order rational form in pressuredomain and two-order polynomial form in temperaturedomain at the same time (120597119896

119898120597119875)119879shows two-order ratio-

nal form in pressure domain and (120597119896119898120597119879)119875shows one-order

polynomial form in temperature So it can be used to predict

Table 4 119896119898three-dimensional fitting coefficients

FA ester Ethyl caprate Methyl caprateCoeffiecients119886 4451 4845119887 134119864 minus 06 122119864 minus 06

119888 minus00001 minus691119864 minus 05

119889 minus6793 minus132119890 8187 8451

Goodness of fitSSE 0000307 0000552R-square 09986 09979Adjust R-square 09986 09978RMSE 0001847 0002476

119896119898without experimental data for methyl and ethyl caprate

under pressures up to 210MPa and temperature range from28315 to 40315 K

33 The Temperature and Pressure Correction of Molecu-lar Compressibility Combining Figures 5 and 6 119896

119898shows

obviously the characteristics in lower pressure the largerfree volume will be a reasonable interpretation in thiscondition the intermolecular potential mainly contributedto the pressure and the vibration of molecules is scarcelycompromised thus pressure has significant effect on it thantemperature On the contrary in high pressure conditionthe molecules are much closer to the close packed stateand intermolecular potential rapidly increases with tinydistance change molecules vibration based on temperaturewill mainly achieve the intermolecules space and then thepotential that temperature has significant effect in this timethan pressure

The Lennard-Jones potential is a mathematically simplemodel that approximates the interaction between a pair ofneutral atoms or molecules [25]

119881LJ = 4120576 [(120590

119903)12

minus (120590

119903)6

] = 120576 [(119903119898

119903)12

minus 2(119903119898

119903)6

] (14)

where 120576 is the depth of the potential well 120590 is the finitedistance at which the interparticle potential is zero 119903 is thedistance between the particles and 119903

119898is the distance at

which the potential reaches its minimum At 119903119898 the potential

function has the value minus120576 However 120590 is determined bytemperature which represents molecular kinetic energy and120590 increases with temperature The repulsive force increasesharply with slight decrease of intermolecules distance

Nevertheless 120590 was not only determined by pressurebut also implicated by the molecular configuration and sizeThe force situation of liquid molecules was considered infree equilibrium state under zero-pressure condition if thegravity was neglected here we set reference pressure 119875ref =01013MPa and the distance between the molecules wasperceived as 120590 and molecules molar volume and density areset to 119881ref 120588ref under the equilibrium state If we assume thepressure 119875 was a function of LJ-potential 119891(119875) = 119881LJ andthe space occupied by molecules was a cube so 119881ref = 120590

3 and


Table 5 Critical properties and parameters of ethyl caprate

Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576

Ethyl caprate [18] 69022 1869998 0709 37737413 29534 408119864 minus 21

Methyl caprate [19] 683 15701 0596 32958 30601 423119864 minus 21

119881ref119881119898 = 12059031199033 A correction coefficient 119870

119875of pressure was

introduced and the Lennard-Jones potential can be written as

119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)

For temperature correction of 119896119898 energy equipartition

theorem [26] which makes quantitative predictions wasconsidered and it gave the total average kinetic and potentialenergies for a system at a given temperature in thermalequilibriumThe oscillating molecules have average energy

⟨119867⟩ = ⟨119867kinetic⟩ + ⟨119867potential⟩ =1

2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)

where the angular brackets ⟨sdot sdot sdot ⟩ denote the average of theenclosed quantity So (120576119896

119861119879 minus 120576119896

119861119879ref) will be a correction

factor to charge temperature effect on 119896119898

119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)

where 120576 is the depth of the potential well and also representsa unit of energy 119879ref which is reference temperature wasconsidered as the cloud point temperature of an organicliquid under reference pressure 119875ref for ethyl caprate 119879refbeing 25315 K and for methyl caprate 119879ref being 26015 K

The Elliott Suresh and Donohue (ESD) equation of stateaccounts for the effect of the shape of a nonpolar moleculeand can be extended to polymers with the addition of an extraterm The EOS itself was developed through modeling com-puter simulations and should capture the essential physics ofthe size shape and hydrogen bonding [27]

In ESD the shape factor 119888 = 1 + 3535120596 + 05331205962 where120596 is the acentric factor And the following equation shows theenergy factor 120576119896

119861is related to the shape parameter 119888

120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)

In this case these calculationmethods of relevant param-eters follow EOS and the calculation data were cited from[18 19] and listed in Table 5 with the calculated parametersvalue

By data calculation and comparison the pressure correc-tion factor is written as11987034

119875and the final correction formula

of molecular compressibility are written as

119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)

Corr303Corr343Corr383Fit303Fit343

Fit383303153431538315

km(103)

Figure 7 Molecular compressibilityrsquos values calculated from threeformulas for methyl caprate along various isothermals (dots for (8)dotted lines for (13) and lines for (19))

where the reference molecular compressibility 119896119898ref is

obtained from the additivity of the subgroup of moleculeunder reference pressure and temperature and listed inTable 1 So if the density under specific pressure and tem-perature was obtained the molecular compressibility couldbe calculated by (19) Because sound velocity vanishes fromcalculation molecular compressibility can be obtained moreconveniently

34TheAccuracyComparison amongMolecular Compressibil-ity Expressions To apply (8) (13) and (19) the requirementsof experimental data are different Formula (8) needs densityand sound velocity data (19) only need density data and (13)does not need experimental data Further (8) is connectedto dynamic precess and (19) reflects statistical state valueand (13) derives from calculated results of (8) The change ofdensity is more direct property associated to compressibilitythan sound velocity so (19) should be more accurate Theresults comparison of three formulas was showed in Figures7 and 8 for methyl caprate and ethyl caprate respectivelyFormula (19) reveal bigger difference at whole 119875-119879 domain

To evaluate the predictive capacity of the formulasaforementioned the relative deviations (RDs) between twoformulas of 119896

119898were estimated according to

RD () =119896119898formula-119894 minus 119896119898formula-119895

119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)

Figure 8 Molecular compressibilityrsquos values calculated from threeformulas for ethyl caprate along various isothermals (dots for (8)dotted lines for (13) and lines for (19))

Table 6 OARD of calculated results for three methods under 303to 383K 01 to 210MPa

Methyl caprate Ethyl caprateFormula (8) versus (13) minus000818 minus000631Formula (8) versus (19) 0530178 0380222Formula (13) versus (19) 0538393 0386536

The average relative deviation (ARD) was calculated as asummation of RD over119873

119901calculated data points at a certain

temperature The overall average relative deviation (OARD)was calculated by (21) where119873

119904is the number of calculated

points

OARD () =sum119899ARD119873119904

(21)

An error comparison of calculation and prediction resultsamong (8) (13) and (19) was presented in Table 6 for twoFA esters In the high pressure and temperature range ofexperimental data have an uncertainty (about 03 or morein AAD) that increases uncertainty of the comparison TheOARD information also indicates that the formulas donot introduce systematic errors The comparison results ofthese methods which were derived from different principleshow clearly the reliability of these methods when applied inextrapolation and confirm their purely predictive characterBecause the molecular compressibility data is scarce underhigh pressure and higher temperature it is difficult todetermine which one has better performance under extendedpressure and temperature ranges

More recently Cunha et al [20] proposed a new atomiccontribution model to predict speed of sound the groupcontributionmodel whichwas listed inTable 1 and calculatedby (5) and was replaced by seven atomic compounds Cpar(for paraffinic carbon) COlef (for olefinic carbon) CArom (foraromatic carbon) H (for hydrogen) Osp3 (for ether oxygen)

435

436

437

438

439

44

295 325 355 385

Formula 13 Formula 7Formula 10 Formula 4Formula 16

T (K)

km(103)

Figure 9 Molecular compressibilityrsquos values calculated from fiveformulas for methyl caprate at atmospheric pressure

472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385

Formula 13Formula 7Formula 10

Formula 4Formula 16

T (K)

km(103)

Figure 10 Molecular compressibilityrsquos values calculated from fiveformulas for ethyl caprate at atmospheric pressure

Osp2 (for ester group being the COO decomposed into oneCOlef one Osp2 and one Osp3) and OAlc (for alcohol oxygen)120594119886is the atom linear temperature correction parameter and

119886 runs over all atoms however 120594119886have different values only

for OAlc andOsp2 AdjustedWadarsquos constant contributions arepresented in Table 7 and Wadarsquos constant was calculated by

119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)

but the two methods are available only at atmosphericpressure so we take 01013MPa isobaric value to compare theaccuracy of five methods the comparison results for methyland ethyl caprate were listed in Figures 9 and 10

The values of (8) are obtained from the experimentalspeed of sound and density data so (5) and (19) have higherconsistency at atmospheric pressure Formula (13) showedbigger difference while (5) and (22) for ethyl caprate at01MPa isobaric


Table 7 Atomic contribution values and temperature correction coefficients [20]

CPar COlef CArom H Osp3 OAlc Osp2

103 Wadarsquos constant (SI) 00113 01234 01348 01694 01738 0094 02568103120594 (Kminus1) 00191 00191 00191 00191 00191 minus13064 03488

4 Conclusion

The molecular compressibility that was considered as a con-stant is challenged as a function of pressure and temperatureBy data fitting a first order rational function in pressuredomain and a two-order polynomial function in temperaturedomain were achieved The results of 2D data fitting showedthat molecular compressibility has an evident dependenceof temperature and pressure Partial derivative of pressureand temperature was introduced to analyze this dependencyfurther The 3D fitting results of molecular compressibilityobviously showed that pressure reveals more contributionsunder low pressure while temperature plays a more impor-tant role under high pressure

This dependence was correlated with molecular poten-tial energy and kinetic energy by developing pressure andtemperature correction coefficients which are derived fromLennard-Jones potential function and energy equipartitiontheorem Combining the two correction coefficients the newmolecular compressibility predicting formula is achieved Asa result the molecular compressibility at working conditionsof injection system can be calculated directly by density datain corresponding condition With the maximum 05384OARD the consistency of calculation results demonstratedthe validity of three expressions and the pressure and tem-perature dependence of molecular compressibility

The correction expression (Formula (19)) which con-tains ldquoshape factorrdquo of molecules and only requires densitydata has conceivable prospects to be extended to other FAesters and biodiesels But (19) needs more mathematical wayto improve its accuracy the HILO-based error correctionscheme used to improve Gaussian mixture models (GMMs)[28] would be an idea worthy of learning

Conflict of Interests

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

References

[1] F Boudy andP Seers ldquoImpact of physical properties of biodieselon the injection process in a common-rail direct injectionsystemrdquo Energy Conversion andManagement vol 50 no 12 pp2905ndash2912 2009

[2] A L Boehman D Morris J Szybist and E Esen ldquoThe impactof the bulk modulus of diesel fuels on fuel injection timingrdquoEnergy amp Fuels vol 18 no 6 pp 1877ndash1882 2004

[3] M E Tat and J H van Gerpen ldquoMeasurement of biodieselsound velocity and its impact on injection timingrdquo Final Reportfor National Renewable Energy Laboratory ACG-8-18066-112000

[4] G Knothe ldquolsquoDesignerrsquo biodiesel optimizing fatty ester compo-sition to improve fuel propertiesrdquo Energy amp Fuels vol 22 no 2pp 1358ndash1364 2008

[5] G Knothe C A Sharp and T W Ryan III ldquoExhaust emissionsof biodiesel petrodiesel neat methyl esters and alkanes in anew technology enginerdquo Energy amp Fuels vol 20 no 1 pp 403ndash408 2006

[6] R Payri F J Salvador J Gimeno and G Bracho ldquoThe effectof temperature and pressure on thermodynamic properties ofdiesel and biodiesel fuelsrdquo Fuel vol 90 no 3 pp 1172ndash1180 2011

[7] M R Rao ldquoThe adiabatic compressibility of liquidsrdquo TheJournal of Chemical Physics vol 14 p 699 1946

[8] Y Wada ldquoOn the relation between compressibility and molalvolume of organic liquidsrdquo Journal of the Physical Society ofJapan vol 4 no 4-6 pp 280ndash283 1949

[9] W Schaaffs ldquoSchllgeschwindigkeit und Molekulstuktur inFlussigkeitenrdquo Zeitschrift fur Physikalische Chemie vol 196 p413 1951

[10] W Schaaffs ldquoDer ultraschall und die struktur der flussigkeitenrdquoIl Nuovo Cimento vol 7 no 2 supplement pp 286ndash295 1950

[11] W Schaaffs ldquoDie Additivitiatsgesgtze der schllgeschwindigkeitin flussigkeitenrdquo Ergebnisse der Exakten Naturwiss vol 25 p109 1951

[12] J L Daridon J A P Coutinho E H I Ndiaye and M L LParedes ldquoNovel data and a group contribution method for theprediction of the sound velocity and isentropic Compressibilityof pure fatty acids methyl and ethyl estersrdquo Fuel vol 105 pp466ndash470 2013

[13] S V D Freitas D L Cunha R A Reis et al ldquoApplicationof wadarsquos group contribution method to the prediction of thesound velocity of biodieselrdquo Energy amp Fuels vol 27 no 3 pp1365ndash1370 2013

[14] S V D Freitas A Santos M L C J Moita et al ldquoMeasurementand prediction of speeds of sound of fatty acid ethyl esters andethylic biodieselsrdquo Fuel vol 108 pp 840ndash845 2013

[15] E H I Ndiaye D Nasri and J L Daridon ldquoSound velocitydensity and derivative properties of fatty acid methyl and ethylesters under high pressure methyl caprate and ethyl capraterdquoJournal of ChemicalampEngineeringData vol 57 no 10 pp 2667ndash2676 2012

[16] S V D Freitas M L L Paredes J L Daridon A S Lima andJ A P Coutinho ldquoMeasurement and prediction of the speed ofsound of biodiesel fuelsrdquo Fuel vol 103 pp 1018ndash1022 2013

[17] F Shi and J Chen ldquoInfluence of injection temperature onatomization characteristics of biodieselrdquo Transactions of theChinese Society of Agricultural Machinery vol 44 no 7 pp 33ndash38 2013

[18] M C Costa L A D Boros M L S Batista J A P CoutinhoM A Kraenbuhl and A J A Meirelles ldquoPhase diagrams ofmixtures of ethyl palmitate with fatty acid ethyl estersrdquo Fuel vol91 no 1 pp 177ndash181 2012

[19] O C DidzMeasurement and Modelling Methodology for HeavyOil and Bitumen Vapour Pressure University of Calgary 2012


[20] D L Cunha J A P Coutinho R A Reis and M L L ParedesldquoAn atomic contribution model for the prediction of speed ofsoundrdquo Fluid Phase Equilibria vol 358 pp 108ndash113 2013

[21] M E Tat J H Van Gerpen S Soylu M Canakci A Monyemand S Wormley ldquoThe sound velocity and isentropic bulkmodulus of biodiesel at 21∘C from atmospheric pressure to 35MPardquo Journal of the American Oil Chemistsrsquo Society vol 77 no3 pp 285ndash289 2000

[22] M E Tat and J H van Gerpen ldquoSpeed of sound and isentropicbulk modulus of alkyl monoesters at elevated temperatures andpressuresrdquo Journal of the AmericanOil Chemistsrsquo Society vol 80no 12 pp 1249ndash1256 2003

[23] L A Davis and R B Gordon ldquoCompression of mercury at highpressurerdquo The Journal of Chemical Physics vol 46 no 7 pp2650ndash2660 1967

[24] J L Daridon B Lagourette and J P Grolier ldquoExperimentalmeasurements of the speed of sound in n-Hexane from 293to 373 K and up to 150 MPardquo International Journal of Thermo-physics vol 19 no 1 pp 145ndash160 1998

[25] J E Lennard-Jones ldquoOn the determination of molecular fieldsIIFrom the equation of state of a gasrdquo Proceedings of the RoyalSociety of London A vol 106 no 738 pp 463ndash477 1924

[26] K Huang Statistical Mechanics John Wiley amp Sons 2ndedition 1987

[27] J R Elliott Jr S J Suresh and M D Donohue ldquoA simpleequation of state for nonspherical and associating moleculesrdquoIndustrial amp Engineering Chemistry Research vol 29 no 7 pp1476ndash1485 1990

[28] J H Park and H K Kim ldquoDual-microphone voice activitydetection incorporating gaussian mixture models with anerror correction scheme in non-stationary noise environmentsrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 6 pp 2533ndash2542 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a constant molar volume for many organic liquids Schaaffs[9 10] who firstly derived a quantitative expression to createconnection between sound velocity and chemical structureof organic compounds pointed out the properties related tosound velocity with the atomic additive property [11] As theeigenvalues of the molecular spacing and molecular inter-molecular forces molecular compressibility was regarded asa constant

Nowadays molecular compressibility was used to predictthe speed of sound in some research [12ndash14] a negative tem-perature correction coefficient is used tomodify the influenceof temperature onmolecular compressibility it is an effectivesolution to decrease prediction deviationwhen the numericalaccuracy of molecular compressibility was required Freitaset al [13] predicted speed of sound of FAMEs by Wadarsquosgroup contributionmethod but the ARD of predicted resultsshows regularly downward trend with elevated temperatureLater Daridon et al [12] divided the FA esters molecule intofive bigger functional groups to contribute to the molecularcompressibility meanwhile molecular compressibility wasdemonstrated with slight temperature dependency Theseinvestigations provide a quantitative connection betweenmacroproperties and molecular microstructure but the lit-erature on the compressibility for FA esters and biodiesels isstill very scant relative to its species most data are availableonly at atmospheric pressure and in a narrow range oftemperature and could notmatch the fuel working conditionsin injection system That restricted the overall cognition onmolecular compressibility Recently more studies [15 16]which focused on sound velocity and density under highpressure and high temperature conditions and the modelingof these properties gave a possibility to comprehensivelyunderstand on molecular adiabatic compressibility in 119875-119879domain

In this paper the calculated value of molecular com-pressibility was extrapolated to close working conditions offuel injection system [17] based on [15] data By data fittingits dependence of pressure and temperature was formulatedand analyzed at two dimensions and three dimensions Asthe eigenvalues of the molecular spacing and molecularintermolecular forces molecular compressibility was con-nected to molecular potential energy and thermal kineticenergy by developing a new correction formula The consis-tency of calculated results derived from different principlesverified effectiveness of these formulas and confirmed thedependency of molecular compressibility on pressure andtemperature

2 The Theory ofthe Molecular Compressibility

Concept of molecular compressibility was derived from thesummary of the experimental data of organic liquids [8]Wada has pointed out that if 12058811989617

119878is independent of pressure

the rate of 1119896119904sdot 120597119896119904120597119901 to 1120588 sdot 120597120588120597119901 was expected to

be a constant 119899 which was 73 in many organic liquidsSo 12058811989617

119904was used as a characteristic function of organic

liquids thus molecular adiabatic compressibility11987211990812058811989617119904

is

expressed as additivity of molecular constitution and showsan approximate fixed value for an organic liquid But thisapplication will bring two aspects problems Firstly 119899 = 7

is just an approximate value based on experimental datathis application is short of theoretical basis and must havecertain error under a large range of pressure and temperatureSecondly the difference of molecular configuration and sizeresponding to pressure and temperature should be distinctalthoughWada listed about one hundred calculated values ofmolecular compressibility to show additive property of bondincrements still seven or more organic liquids have appre-ciable discrepancies between experimental and calculatedvalues and the experimental conditions of data acquisitionwere not shown clearly

InWadarsquos derivation molecular adiabatic compressibilitywas expressed as

119896119898=

119872119908

12058811989617

119904

=1198810

11989617

1199040

=Wadarsquos Const (1)

where1198721199081198810 1198961199040are the molecular weight the mole volume

at close packed state and the compressibility of moleculeitself respectively and shown as

119872119908

120588= 1198810+ V = 119881

0 (1 + 119896)

119896119904= 1198961199040(1 + 119896)

7

(2)

where V 119896 are the free volume per mole and fractionalfree volume respectively For different organic liquid 119881

0

as the molar volume of close packed state is fixed value11989617

1199040also are shown as an almost fixed calculated value

thus Wada concluded 119896119898was approximately constant But

there exists an unreasonable assumption that 1198810is unlikely

to reach the close packed state at liquid state hence 1198810

compressibility is variational under dynamic high pressureand high temperature Furthermore 7th power of (1 + 119896) informula (2) is incorrect to express the linear superpositionof compressibility between free volume and molecular sizeSo we can draw a conclusion that the consistent 11989617

1199040value is

derived from tiny error with 7th root of 1198961199040 so the formula (1)

is not strictly correct and approximate algorithm under somespecial conditions

In addition the formula (1) can further be developedaccording to the definition of adiabatic compressibility 119896

119904as

follows

119896119898=

119872119908

12058811989617

119904

=119872119908

120588(minus (1119881) (120597119881120597119875))17

=119872119908

120588((1120588) (120597120588120597119875))17

=119872119908

12058867(120597119875

120597120588)

17

(3)

So molecular adiabatic compressibility is supposed to bea function of density with pressure and the influence of thetemperature is implied in density




1198790(K) 29815

120594 times 103 0034852




119896119904=1

120573119904

=1

1205881198882 (4)


3COOndash

ndashCH2COOndash CH

3ndash ndashCH



119896119898 (119879) =

119899119866

sum119895=1

119873119895119896119898119895(1 minus 120594 (119879 minus 29815)) (5)

119888 = 1205883(119896119898

119872119908

)

72

(6)




group type-119895 to 119896119898






120588 (119875 119879) = 120588 (119875ref 119879) + int119875

119875ref

1

1198882119889119875 + 119879int

119875

119875ref

1205722119875

119862119901

119889119875 (7)





119898as


119896119898= 119872119908(119888

1205883)

27

(8)





1


119896119898 (119875) =

1198601119875 + 119860

2

119875 + 1198603

= 119886 +119887

119875 + 119888 (9)




4597 4612 4632 4985 4992 50041198602

4808 3747 2914 5472 4234 3031198603

1097 8555 6658 1155 8939 6396Goodness of fit



435

44

445

45

455

46

0 30 60 90 120 150 180 210

P (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)




119896119898 (119879) = 1198611119879

2+ 1198612119879 + 119861

3= 119886(119879 + 119887)

2+ 119888 (10)





4725

4775

4825

4875

4925

0 30 60 90 120 150 180 210T (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)



119898







(120597119896119898

120597119875)119879

=1198601

119875 + 1198603

minus1198601119875 + 119860

2

(119875 + 1198603)2=11986011198603minus 1198602

(119875 + 1198603)2 (11)

(120597119896119898

120597119879)119875

= 21198611119879 + 119861

2 (12)




Coefficients1198611


1198612


4168 4495 4507 4506 4499 4982 5028 4983 4978 4982Goodness of fit



435

44

445

45

455

46

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (K)

km(103)










3)2 acceleration



4725

4775

4825

4875

4925

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (MPa)

km(103)




(120597119896119898120597119879)119875






119896119898= 119886 + 119887(119879 + 119887)

2+119888(119879 + 119887)

2+ 119889

119875 + 119890 (13)


0

0002

0004

0006

0008

001

0012

0014

0016

0018

0 30 60 90 120 150 180 210

3231534315

3631538315

998779km998779

T

P (MPa)


0

00005

0001

00015

0002

00025

0003

00035

300 320 340 360 380T (K)

102030405060708090

100110120130140150170190210

998779km998779

P










119889 minus6793 minus132119890 8187 8451





119898shows



119881LJ = 4120576 [(120590

119903)12

minus (120590

119903)6

] = 120576 [(119903119898

119903)12

minus 2(119903119898

119903)6

] (14)






3 and



Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576






119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)




2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)


119861119879 minus 120576119896



119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)





120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)





119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)


Fit383303153431538315

km(103)








119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198790(K) 29815

120594 times 103 0034852




119896119904=1

120573119904

=1

1205881198882 (4)


3COOndash

ndashCH2COOndash CH

3ndash ndashCH



119896119898 (119879) =

119899119866

sum119895=1

119873119895119896119898119895(1 minus 120594 (119879 minus 29815)) (5)

119888 = 1205883(119896119898

119872119908

)

72

(6)




group type-119895 to 119896119898






120588 (119875 119879) = 120588 (119875ref 119879) + int119875

119875ref

1

1198882119889119875 + 119879int

119875

119875ref

1205722119875

119862119901

119889119875 (7)





119898as


119896119898= 119872119908(119888

1205883)

27

(8)





1


119896119898 (119875) =

1198601119875 + 119860

2

119875 + 1198603

= 119886 +119887

119875 + 119888 (9)




4597 4612 4632 4985 4992 50041198602

4808 3747 2914 5472 4234 3031198603

1097 8555 6658 1155 8939 6396Goodness of fit



435

44

445

45

455

46

0 30 60 90 120 150 180 210

P (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)




119896119898 (119879) = 1198611119879

2+ 1198612119879 + 119861

3= 119886(119879 + 119887)

2+ 119888 (10)





4725

4775

4825

4875

4925

0 30 60 90 120 150 180 210T (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)



119898







(120597119896119898

120597119875)119879

=1198601

119875 + 1198603

minus1198601119875 + 119860

2

(119875 + 1198603)2=11986011198603minus 1198602

(119875 + 1198603)2 (11)

(120597119896119898

120597119879)119875

= 21198611119879 + 119861

2 (12)




Coefficients1198611


1198612


4168 4495 4507 4506 4499 4982 5028 4983 4978 4982Goodness of fit



435

44

445

45

455

46

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (K)

km(103)










3)2 acceleration



4725

4775

4825

4875

4925

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (MPa)

km(103)




(120597119896119898120597119879)119875






119896119898= 119886 + 119887(119879 + 119887)

2+119888(119879 + 119887)

2+ 119889

119875 + 119890 (13)


0

0002

0004

0006

0008

001

0012

0014

0016

0018

0 30 60 90 120 150 180 210

3231534315

3631538315

998779km998779

T

P (MPa)


0

00005

0001

00015

0002

00025

0003

00035

300 320 340 360 380T (K)

102030405060708090

100110120130140150170190210

998779km998779

P










119889 minus6793 minus132119890 8187 8451





119898shows



119881LJ = 4120576 [(120590

119903)12

minus (120590

119903)6

] = 120576 [(119903119898

119903)12

minus 2(119903119898

119903)6

] (14)






3 and



Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576






119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)




2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)


119861119879 minus 120576119896



119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)





120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)





119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)


Fit383303153431538315

km(103)








119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4597 4612 4632 4985 4992 50041198602

4808 3747 2914 5472 4234 3031198603

1097 8555 6658 1155 8939 6396Goodness of fit



435

44

445

45

455

46

0 30 60 90 120 150 180 210

P (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)




119896119898 (119879) = 1198611119879

2+ 1198612119879 + 119861

3= 119886(119879 + 119887)

2+ 119888 (10)





4725

4775

4825

4875

4925

0 30 60 90 120 150 180 210T (MPa)

30315 3231534315 3631538315 F30315F34315 F38315

km(103)



119898







(120597119896119898

120597119875)119879

=1198601

119875 + 1198603

minus1198601119875 + 119860

2

(119875 + 1198603)2=11986011198603minus 1198602

(119875 + 1198603)2 (11)

(120597119896119898

120597119879)119875

= 21198611119879 + 119861

2 (12)




Coefficients1198611


1198612


4168 4495 4507 4506 4499 4982 5028 4983 4978 4982Goodness of fit



435

44

445

45

455

46

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (K)

km(103)










3)2 acceleration



4725

4775

4825

4875

4925

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (MPa)

km(103)




(120597119896119898120597119879)119875






119896119898= 119886 + 119887(119879 + 119887)

2+119888(119879 + 119887)

2+ 119889

119875 + 119890 (13)


0

0002

0004

0006

0008

001

0012

0014

0016

0018

0 30 60 90 120 150 180 210

3231534315

3631538315

998779km998779

T

P (MPa)


0

00005

0001

00015

0002

00025

0003

00035

300 320 340 360 380T (K)

102030405060708090

100110120130140150170190210

998779km998779

P










119889 minus6793 minus132119890 8187 8451





119898shows



119881LJ = 4120576 [(120590

119903)12

minus (120590

119903)6

] = 120576 [(119903119898

119903)12

minus 2(119903119898

119903)6

] (14)






3 and



Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576






119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)




2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)


119861119879 minus 120576119896



119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)





120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)





119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)


Fit383303153431538315

km(103)








119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Coefficients1198611


1198612


4168 4495 4507 4506 4499 4982 5028 4983 4978 4982Goodness of fit



435

44

445

45

455

46

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (K)

km(103)










3)2 acceleration



4725

4775

4825

4875

4925

300 320 340 360 380

P01P10P20P30P40P50P60P70P80P90

P100P110P120P130P140P150P170P190P210

T (MPa)

km(103)




(120597119896119898120597119879)119875






119896119898= 119886 + 119887(119879 + 119887)

2+119888(119879 + 119887)

2+ 119889

119875 + 119890 (13)


0

0002

0004

0006

0008

001

0012

0014

0016

0018

0 30 60 90 120 150 180 210

3231534315

3631538315

998779km998779

T

P (MPa)


0

00005

0001

00015

0002

00025

0003

00035

300 320 340 360 380T (K)

102030405060708090

100110120130140150170190210

998779km998779

P










119889 minus6793 minus132119890 8187 8451





119898shows



119881LJ = 4120576 [(120590

119903)12

minus (120590

119903)6

] = 120576 [(119903119898

119903)12

minus 2(119903119898

119903)6

] (14)






3 and



Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576






119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)




2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)


119861119879 minus 120576119896



119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)





120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)





119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)


Fit383303153431538315

km(103)








119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0002

0004

0006

0008

001

0012

0014

0016

0018

0 30 60 90 120 150 180 210

3231534315

3631538315

998779km998779

T

P (MPa)


0

00005

0001

00015

0002

00025

0003

00035

300 320 340 360 380T (K)

102030405060708090

100110120130140150170190210

998779km998779

P










119889 minus6793 minus132119890 8187 8451





119898shows



119881LJ = 4120576 [(120590

119903)12

minus (120590

119903)6

] = 120576 [(119903119898

119903)12

minus 2(119903119898

119903)6

] (14)






3 and



Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576






119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)




2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)


119861119879 minus 120576119896



119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)





120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)





119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)


Fit383303153431538315

km(103)








119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Fatty ester 119879119888(K) 119875

119888(MPa) 120596 c 120576119896

119861120576






119870119875= 119891 (119875) =

119881LJ

120576= 4 [(

119881ref119881119898

)

4

minus (119881ref119881119898

)

2

]

= 4 [(120588

120588ref)

4

minus (120588

120588ref)

2

]

(15)




2119896119861119879 +

1

2119896119861119879 = 119896

119861119879

(16)


119861119879 minus 120576119896



119870119879= Exp( 120576

119896119861119879ref

minus120576

119896119861119879) (17)





120576

119896119861

= 119879119888

1 + 0945 (119888 minus 1) + 0134(119888 minus 1)2

1023 + 2225 (119888 minus 1) + 0478(119888 minus 1)2 (18)





119896119898= 119896119898ref(1 +

015(01119870119875)34

11987017

119879

) (19)

435

44

445

45

455

46

0 30 60 90 120 150 180 210P (MPa)


Fit383303153431538315

km(103)








119896119898formula-119895

(20)


P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










P (MPa)

472

477

482

487

492

0 30 60 90 120 150 180 210


Fit383303153431538315

km(103)








points

OARD () =sum119899ARD119873119904

(21)



435

436

437

438

439

44

295 325 355 385


T (K)

km(103)


472

473

474

475

476

477

478

479

48

295 305 315 325 335 345 355 365 375 385


Formula 4Formula 16

T (K)

km(103)





119896119898 (119879) =

119899

sum120572=1

119873120572119896119898120572(1 minus 120594

120572 (119879 minus 29815)) (22)







4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













4 Conclusion






References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Macroscopic Expressions of Molecular ...

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