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)
4 Abstract and Applied Analysis
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)
Abstract and Applied Analysis 5
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)
6 Abstract and Applied Analysis
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
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Journal of
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Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Abstract and Applied Analysis
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)
Abstract and Applied Analysis 5
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)
6 Abstract and Applied Analysis
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
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Abstract and Applied Analysis
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)
Abstract and Applied Analysis 5
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)
6 Abstract and Applied Analysis
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
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
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)
Abstract and Applied Analysis 5
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)
6 Abstract and Applied Analysis
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
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
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)
6 Abstract and Applied Analysis
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
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
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
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
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)
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
P (MPa)
472
477
482
487
492
0 30 60 90 120 150 180 210
Corr303Corr343Corr383Fit303Fit343
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
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
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
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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