# Macroscopic Expressions of Molecular Adiabatic Compressibility of Methyl and Ethyl Caprate under High Pressure and High Temperature

Authors: Fuxi Shi; Qin Zhang; Jun Chen; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The molecular compressibility, which is a macroscopic quantity to reveal the microcompressibility by additivity of molecular constitutions, is considered as a fixed value for specific organic liquids. In this study, we introduced two calculated expressions of molecular adiabatic compressibility to demonstrate its pressure and temperature dependency. The first one was developed from Wada’s constant expression based on experimental data of density and sound velocity. Secondly, by introducing the 2D fitting expressions and their partial derivative of pressure and temperature, molecular compressibility dependency was analyzed further, and a 3D fitting expression was obtained from the calculated data of the first one. The third was derived with introducing the pressure and temperature correction factors based on analogy to Lennard-Jones potential function and energy equipartition theorem. In wide range of temperatures(293<T/K<393) and pressures (0.1<P/MPa<210), which represent the typical values used in dynamic injection process for diesel engines, the calculated results consistency of three formulas demonstrated their effectiveness with the maximum 0.5384% OARD; meanwhile, the dependency on  pressure and temperature of  molecular compressibility was certified.


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## Body

## 1. Introduction

In modern diesel engine, the injection pressure of fuel spray system continuously was elevated to achieve better atomization, combustion, and emission effect and even reach 200 MPa or more, which increase vastly the dynamics sensibility of fuel compressibility. The dynamic delivery process, which was carried out under rapid variation of pressure and temperature, was strongly affected by the compressibility of fuel and its derivative properties relative to pressure and temperature. For example, density influences the conversion of volume flow rate into mass flow rate [1], whereas the compressibility (the reciprocal of bulk modulus) acts on the fuel injection timing [2, 3]. Therefore, the adaptation of injection systems to biodiesels requires an accurate knowledge of the volumetric properties of biodiesel components over a wide range of pressure and temperature [4, 5]; especially the compressibility and sound velocity values were indispensable parameters for system modeling and experimental injection rate determination [6].In order to clarify the connection between the microscopic chemical structure of organic liquids and the compressibility from investigation of sound velocity and density, many theoretical analyses and derivations have been performed to determine these properties. Rao [7] has pointed out the product of density and 7th root of isothermal compressibility kT, ρkT1/7 was highly independent of temperature and pressure. Further, molecular adiabatic compressibility, km=Mw/ρks1/7, which was proposed by Wada [8] and was calculated by summing bond increments of molecular components, was used to express the microcompressibility with a constant molar volume for many organic liquids. Schaaffs [9, 10], who firstly derived a quantitative expression to create connection between sound velocity and chemical structure of organic compounds, pointed out the properties related to sound velocity with the atomic additive property [11]. As the eigenvalues of the molecular spacing and molecular intermolecular forces, molecular compressibility was regarded as a constant.Nowadays, molecular compressibility was used to predict the speed of sound in some research [12–14]; a negative temperature correction coefficient is used to modify the influence of temperature on molecular compressibility; it is an effective solution to decrease prediction deviation when the numerical accuracy of molecular compressibility was required. Freitas et al. [13] predicted speed of sound of FAMEs by Wada’s group contribution method, but the ARD of predicted results shows regularly downward trend with elevated temperature. Later Daridon et al. [12] divided the FA esters molecule into five bigger functional groups to contribute to the molecular compressibility; meanwhile molecular compressibility was demonstrated with slight temperature dependency. These investigations provide a quantitative connection between macroproperties and molecular microstructure, but the literature on the compressibility for FA esters and biodiesels is still very scant relative to its species; most data are available only at atmospheric pressure and in a narrow range of temperature and could not match the fuel working conditions in injection system. That restricted the overall cognition on molecular compressibility. Recently, more studies [15, 16], which focused on sound velocity and density under high pressure and high temperature conditions and the modeling of these properties, gave a possibility to comprehensively understand on molecular adiabatic compressibility in P-T domain.In this paper, the calculated value of molecular compressibility was extrapolated to close working conditions of fuel injection system [17] based on [15] data. By data fitting, its dependence of pressure and temperature was formulated and analyzed at two dimensions and three dimensions. As the eigenvalues of the molecular spacing and molecular intermolecular forces, molecular compressibility was connected to molecular potential energy and thermal kinetic energy by developing a new correction formula. The consistency of calculated results derived from different principles verified effectiveness of these formulas and confirmed the dependency of molecular compressibility on pressure and temperature.


## 2. The Theory of the Molecular Compressibility

Concept of molecular compressibility was derived from the summary of the experimental data of organic liquids [8]. Wada has pointed out that if ρkS1/7 is independent of pressure, the rate of 1/ks·∂ks/∂p to 1/ρ·∂ρ/∂p was expected to be a constant n, which was 7.3 in many organic liquids. So ρks1/7 was used as a characteristic function of organic liquids; thus molecular adiabatic compressibility Mw/ρks1/7 is expressed as additivity of molecular constitution and shows an approximate fixed value for an organic liquid. But this application will bring two aspects problems. Firstly, n=7 is just an approximate value based on experimental data; this application is short of theoretical basis and must have certain error under a large range of pressure and temperature. Secondly, the difference of molecular configuration and size responding to pressure and temperature should be distinct; although Wada listed about one hundred calculated values of molecular compressibility to show additive property of bond increments, still seven or more organic liquids have appreciable discrepancies between experimental and calculated values, and the experimental conditions of data acquisition were not shown clearly.In Wada’s derivation, molecular adiabatic compressibility was expressed as(1)km=Mwρks1/7=V0ks01/7=Wada’s Const.,
where Mw, V0, ks0 are the molecular weight, the mole volume at close packed state and the compressibility of molecule itself, respectively, and shown as:
(2)Mwρ=V0+v=V0(1+k),ks=ks0(1+k)7,
where v, k are the free volume per mole and fractional free volume, respectively. For different organic liquid, V0 as the molar volume of close packed state is fixed value, ks01/7 also are shown as an almost fixed calculated value; thus Wada concluded km was approximately constant. But there exists an unreasonable assumption that V0 is unlikely to reach the close packed state at liquid state; hence V0 compressibility is variational under dynamic high pressure and high temperature. Furthermore 7th power of (1+k) in formula (2) is incorrect to express the linear superposition of compressibility between free volume and molecular size. So we can draw a conclusion that the consistent ks01/7 value is derived from tiny error with 7th root of ks0, so the formula (1) is not strictly correct and approximate algorithm under some special conditions.In addition, the formula (1) can further be developed according to the definition of adiabatic compressibility ks as follows:
(3)km=Mwρks1/7=Mwρ(-(1/V)(∂V/∂P))1/7=Mwρ((1/ρ)(∂ρ/∂P))1/7=Mwρ6/7(∂P∂ρ)1/7.So molecular adiabatic compressibility is supposed to be a function of density with pressure, and the influence of the temperature is implied in density.


## 3. The Characteristic Analysis of the Molecular Compressibility

For revealing similar law of the properties, many density and sound velocity measurements for FAMEs and FAEEs were performed and predictive methodologies were proposed to determine the sound velocity based on ultrasonic technique. Tat et al. and Tat and van Gerben [21, 22] obtained density and sound velocity of 16 FA esters in 20 to 100°C and 0.1 to 32.5 MPa and achieved only one polynomial function, which is second order in temperature and first order in pressure with different fitting coefficients, to predict density, sound velocity and isentropic bulk modulus whose dependency on temperature and pressure indicates similitude principle of change for different FA esters, (4) was used to convert the isentropic bulk modulus βs to isentropic compressibility:
(4)ks=1βs=1ρc2.Daridon et al. [12] measured sound velocity and density of seven FA esters (including ethyl and methyl, saturated and unsaturated) by using a pulse echo technique operating with 3 MHz at atmospheric pressure in 283.15~373.15 K range. The molecular compressibility, which was calculated by Wada’s group contribution method, was almost a constant for specific FA ester. the five basic microscopic radicals (CH3COO–, –CH2COO–, CH3–, –CH2–, –CH=CH–), which contributed to molecular compressibility of a FA ester according to additivity and were listed in Table 1, had a fixed value, respectively, and the predicted results for sound velocity (using (6)) showed a high prediction accuracy. This study showed that the molecular compressibility is related to molecular specific structure, and the negative temperature correction coefficient in (5) presented temperature dependence of molecular compressibility:
(5)km(T)=∑j=1nGNjkmj(1-χ(T-298.15)),(6)c=ρ3(kmMw)7/2,
where km, ks, Mw, ρ, c are molecular compressibility, isentropic compressibility, the molecular weight density, and sound velocity, respectively, kmj is the contribution of the group type-j to km which occurs Nj times in the given molecule, and χ is a constant parameter used to take into account the influence of temperature.Table 1
Group contribution values for molecular compressibility estimation at atmospheric pressure [12].
–CH3
–CH2–
–CH=CH–
CH3COO–
–CH2COO–
K
m
×
10
3
0.5097
0.35196
0.59074
1.05856
0.9061
T
0 (K)
298.15
χ
×
10
3
0.034852Freitas et al. [13] also used Wada’s model to predict the sound velocity of the fatty acid esters and biodiesels with OARDs of 0.25% and 0.45%. But these predicted results of high precision, which show evidence of effectiveness of Wada’s model, are just achieved at atmospheric pressure within a small temperature range. Latter Freitas et al. [14] developed Wada’s model to predict the sound velocity of FA esters at high pressures up to 35 MPa and pointed out that the pressure dependency of the sound velocity is approximately linear relation in the pressure range considered, in consideration of the nonlinear of ρ3 under variable pressure; thus the molecular compressibility must not be a constant. So it is necessary to extrapolate the value and tendency of molecular compressibility in high pressure and high temperature.


### 3.1. Characteristic of Molecular Isentropic Compressibility in High Pressure and High Temperature

Ndiaye et al. [15] measured sound velocity of methyl caprate and ethyl caprate at pressures up to 210 MPa in the temperature range 283.15 to 403.15 K. In addition, density was measured in 0.1 to 100 MPa and 293.15 to 393.15 K, by using a modification of Davis and Gordon’s procedure [23]; the method [24] rested on the Newton-Laplace relationships; then density was extrapolated up to 210MPa from sound velocity data (7) with an absolute average deviation of 0.07% up to 210 MPa. the isentropic and isothermal compressibilities also were presented in the P-T domain:
(7)ρ(P,T)=ρ(Pref,T)+∫PrefP1c2dP+T∫PrefPαP2CpdP,
where Cp, αp represent the isobaric heat capacity and the isobaric thermal expansion, respectively, and Pref is a reference pressure where density is known.In order to directly obtain molecular isentropic compressibilitykm from experimental data, we expressed km as a function of density and sound velocity, shown as (8).
(8)km=Mw(cρ3)2/7.Based on [15] data that sound velocity was obtained from experimental and density was obtained from calculated value by (7), km was calculated in 0.1 to 210 MPa and 303.15 to 383.15 K and the fitting results were presented with isothermal characteristics for methyl caprate and ethyl caprate in Figures 1 and 2.Figure 1
Molecular compressibility of methyl caprate (MeC10:0) as a function of pressure along various isothermal (the lines are fitting results; the dots are calculated values).Figure 2
Molecular compressibility of ethyl caprate (EeC10:0) as a function of pressure along various isothermal (the lines are fitting results; the dots are calculated values).A one-order rational fitting relation to pressure was expressed as (9), where P is pressure, other parameters A1, A2, A3 are equation coefficients, as listed in Table 2:
(9)km(P)=A1P+A2P+A3=a+bP+c.Table 2
Fitting coefficients for (9) along various isothermals.
FA ester
Methyl caprate
Ethyl caprate
Temperature
303.15 K
343.15 K
383.15 K
303.15 K
343.15 K
383.15 K
Coefficients
A1
4.597
4.612
4.632
4.985
4.992
5.004
A2
480.8
374.7
291.4
547.2
423.4
303
A3
109.7
85.55
66.58
115.5
89.39
63.96
Goodness of fit
SSE
2.25
E
-
06
2.45
E
-
06
8.37
E
-
06
1.48
E
-
06
5.76
E
-
06
8.34
E
-
06
R-square
0.9999
0.9999
0.9998
1
0.9999
0.9999
AdjustedR-square
0.9999
0.9999
0.9998
1
0.9999
0.9998
RMSE
0.000375
0.000391
0.000723
0.000304
0.0006
0.000722The rational relation (9) was achieved with satisfying goodness of fit, where the minimum R-square value is 0.9998 in all temperature domains. We may safely draw the conclusion that molecular compressibility is dependent on pressure for ethyl and methyl caprate; this dependence on pressure was revealed sufficiently with elevated pressure In Figures 1 and 2.Accordingly, the calculated results were presented with isobaric characteristics for methyl caprate and ethyl caprate in Figures3 and 4, and a two-order polynomial fitting in temperature domain was expressed as
(10)km(T)=B1T2+B2T+B3=a(T+b)2+c,
where T is temperature and other parameters B1, B2, B3 are equation coefficients, as listed in Table 3.Table 3
Fitting coefficients for (10) along various isobaric.
FA ester
Mthyl caprate
Ethyl caprate
Pressure
P
0.1
P
80
P
130
P
150
P
210
P
0.1
P
80
130
P
150
P
210
Coefficients
B1
-
2.3
E
-
06
1.6
E
-
06
1.6
E
-
06
1.6
E
-
06
1.3
E
-
06
2.2
E
-
06
3.2
E
-
06
2.6
E
-
06
2.5
E
-
06
2.2
E
-
06
B2
0.00139
−0.00056
−0.00051
−0.00047
−0.00031
−0.00094
−0.00161
−0.00117
−0.00108
−0.00094
B3
4.168
4.495
4.507
4.506
4.499
4.982
5.028
4.983
4.978
4.982
Goodness of fit
SSE
2.7
E
-
08
1.5
E
-
06
6.5
E
-
07
9.4
E
-
07
1.09
E
-
06
1.8
E
-
07
1.9
E
-
07
2.7
E
-
07
3.9
E
-
07
1.83
E
-
07
R-square
0.9998
0.9987
0.9995
0.9993
0.9992
0.9998
0.9999
0.9998
0.9997
0.9998
AdjustedR-square
0.9995
0.9974
0.9991
0.9987
0.9983
0.9997
0.9997
0.9997
0.9995
0.9997
RMSE
0.00012
0.00088
0.00057
0.00068
0.00074
0.0003
0.0003
0.00037
0.00044
0.0003Figure 3
Molecular compressibility of methyl caprate (MeC10:0) as a function of temperature along various isobaric (only calculated values were revealed).Figure 4
Molecular compressibility of ethyl caprate (EeC10:0) as a function of temperature along various isobaric (only calculated values were revealed).In Figures3 and 4, a two-order polynomial (10) was achieved with the minimum R-square value being 0.9987 in all pressure domains. At atmospheric pressure (0.1 MPa isobaric) km of methyl caprate shows a slight downtrend with two-order form described in (10) but not first-order form described in Daridon’s formula (5); however for 0.1 MPa isobaric of ethyl caprate is almost horizontal; meanwhile km show uptrend, which accelerates with increasing pressure, at isobaric with temperature elevated. km also is dependent on temperature for ethyl and methyl caprate.Furthermore to calculate molecular compressibility differencekm of ethyl caprate in isothermal line with 20 K step and in isobaric line with 10 MPa step, Figures 5 and 6 can be achieved.Figure 5
The trend of differential pressure of molecular compressibility temperature along various isothermal for ethyl caprate (only calculated values were revealed).Figure 6
The trend of differential temperature ofmolecular compressibility temperature along various isobaric for ethyl caprate (only calculated values were revealed).We can take the partial derivative with respect to pressure for (9) and to temperature for (10); thus we have
(11)(∂km∂P)T=A1P+A3-A1P+A2(P+A3)2=A1A3-A2(P+A3)2,(12)(∂km∂T)P=2B1T+B2.From Figure5, we can find that, among various isothermals, the Δkm/ΔP become bigger when the temperature was elevated gradually and almost keep fixed value above 60 MPa though nearly equal at atmospheric pressure; here temperature variation plays a more important role under high pressure. Thus the Δkm/ΔP reveal almost equality and only show dependence on temperature in high pressure, (∂km/∂P)T→Const. The result can be achieved form the calculation of (11) coefficients; the values of A1A3-A2 for various isothermal are in 23.5 to 16.2, but A3 is in about 110 to 65, when pressure is large than 60 MPa increments of (P+A3)2 acceleration.From Figure6, we can find that among various isobaric Δkm/ΔP become smaller when pressure was elevated gradually and keep almost the same value above 60 MPa with slight dependence of pressure, That is to say the possibility of being compressed of Km decreases with high pressure, (∂km/∂T)P≃Const. Meanwhile, Δkm/ΔP increase with elevated temperature, but the rate of increment declines rapidly based on 2B1 which became almost the same as above 60 MPa with 10−6 magnitude; here pressure variation reveals more contributions under low pressure.


### 3.2. The Data Fitting Formula of Molecular Compressibility in Three-Dimensional

According to the two-dimensional fitting analysis, a three-dimensional rational formula was introduced with the following expression:(13)km=a+b(T+b)2+c(T+b)2+dP+e,
where P, T are pressure and temperature and other fitting parameters a, b, c, d, e are formula coefficients, listed in Table 4.Table 4
k
m
three-dimensional fitting coefficients.
FA ester
Ethyl caprate
Methyl caprate
Coeffiecients
a
4.451
4.845
b
1.34
E
-
06
1.22
E
-
06
c
−0.0001
-
6.91
E
-
05
d
−6.793
−13.2
e
81.87
84.51
Goodness of fit
SSE
0.000307
0.000552
R-square
0.9986
0.9979
AdjustR-square
0.9986
0.9978
RMSE
0.001847
0.002476This formula has one-order rational form in pressure domain and two-order polynomial form in temperature domain; at the same time(∂km/∂P)T shows two-order rational form in pressure domain and (∂km/∂T)P shows one-order polynomial form in temperature. So it can be used to predict km without experimental data for methyl and ethyl caprate under pressures up to 210 MPa and temperature range from 283.15 to 403.15 K.


### 3.3. The Temperature and Pressure Correction of Molecular Compressibility

Combining Figures5 and 6, km shows obviously the characteristics: in lower pressure, the larger free volume will be a reasonable interpretation; in this condition the intermolecular potential mainly contributed to the pressure and the vibration of molecules is scarcely compromised; thus pressure has significant effect on it than temperature. On the contrary, in high pressure condition the molecules are much closer to the close packed state and intermolecular potential rapidly increases with tiny distance change; molecules vibration based on temperature will mainly achieve the intermolecules space and then the potential that temperature has significant effect in this time than pressure.The Lennard-Jones potential is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules [25]:
(14)VLJ=4ε[(σr)12-(σr)6]=ε[(rmr)12-2(rmr)6],
where ε is the depth of the potential well, σ is the finite distance at which the interparticle potential is zero, r is the distance between the particles, and rm is the distance at which the potential reaches its minimum. At rm, the potential function has the value -ε. However σ is determined by temperature which represents molecular kinetic energy, and σ increases with temperature. The repulsive force increase sharply with slight decrease of intermolecules distance.Nevertheless,σ was not only determined by pressure, but also implicated by the molecular configuration and size. The force situation of liquid molecules was considered in free equilibrium state under zero-pressure condition, if the gravity was neglected; here we set reference pressure Pref=0.1013 MPa, and the distance between the molecules was perceived as σ and molecules molar volume and density are set to Vref, ρref under the equilibrium state. If we assume the pressure P was a function of LJ-potential, f(P)=VLJ, and the space occupied by molecules was a cube, so Vref=σ3 and Vref/Vm=σ3/r3. A correction coefficient KP of pressure was introduced and the Lennard-Jones potential can be written as
(15)KP=f(P)=VLJε=4[(VrefVm)4-(VrefVm)2]=4[(ρρref)4-(ρρref)2].For temperature correction ofkm, energy equipartition theorem [26], which makes quantitative predictions, was considered, and it gave the total average kinetic and potential energies for a system at a given temperature in thermal equilibrium. The oscillating molecules have average energy
(16)〈H〉=〈Hkinetic〉+〈Hpotential〉=12kBT+12kBT=kBT,
where the angular brackets 〈⋯〉 denote the average of the enclosed quantity. So (ε/kBT-ε/kBTref) will be a correction factor to charge temperature effect on km(17)KT=Exp(εkBTref-εkBT),
where ε is the depth of the potential well and also represents a unit of energy. Tref, which is reference temperature, was considered as the cloud point temperature of an organic liquid under reference pressure Pref, for ethyl caprate Tref being 253.15 K and for methyl caprate Tref being 260.15 K.The Elliott, Suresh, and Donohue (ESD) equation of state accounts for the effect of the shape of a nonpolar molecule and can be extended to polymers with the addition of an extra term. The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding [27].In ESD, the shape factorc=1+3.535ω+0.533ω2, where ω is the acentric factor. And the following equation shows the energy factor ε/kB is related to the shape parameter c:
(18)εkB=Tc1+0.945(c-1)+0.134(c-1)21.023+2.225(c-1)+0.478(c-1)2.In this case, these calculation methods of relevant parameters follow EOS, and the calculation data were cited from [18, 19] and listed in Table 5 with the calculated parameters value.Table 5
Critical properties and parameters of ethyl caprate.
Fatty ester
T
c (K)
P
c (MPa)
ω
c
ε
/
k
B
ε
Ethyl caprate [18]
690.22
1.869998
0.709
3.7737413
295.34
4.08
E
-
21
Methyl caprate [19]
683
1.5701
0.596
3.2958
306.01
4.23
E
-
21By data calculation and comparison, the pressure correction factor is written asKP3/4 and the final correction formula of molecular compressibility are written as
(19)km=kmref(1+0.15(0.1KP)3/4KT1/7),
where the reference molecular compressibility, kmref, is obtained from the additivity of the subgroup of molecule under reference pressure and temperature and listed in Table 1. So if the density under specific pressure and temperature was obtained, the molecular compressibility could be calculated by (19). Because sound velocity vanishes from calculation, molecular compressibility can be obtained more conveniently.


### 3.4. The Accuracy Comparison among Molecular Compressibility Expressions

To apply (8), (13), and (19), the requirements of experimental data are different. Formula (8) needs density and sound velocity data, (19) only need density data, and (13) does not need experimental data. Further (8) is connected to dynamic precess, and (19) reflects statistical state value, and (13) derives from calculated results of (8). The change of density is more direct property associated to compressibility than sound velocity, so (19) should be more accurate. The results comparison of three formulas was showed in Figures 7 and 8 for methyl caprate and ethyl caprate, respectively. Formula (19) reveal bigger difference at whole P-T domain.Figure 7
Molecular compressibility’s values calculated from three formulas for methyl caprate along various isothermals (dots for (8), dotted lines for (13), and lines for (19)).Figure 8
Molecular compressibility’s values calculated from three formulas for ethyl caprate along various isothermals (dots for (8), dotted lines for (13), and lines for (19)).To evaluate the predictive capacity of the formulas aforementioned, the relative deviations (RDs) between two formulas ofkm were estimated according to
(20)RD(%)=kmformula-i-kmformula-jkmformula-j%.The average relative deviation (ARD) was calculated as a summation of RD overNp calculated data points at a certain temperature. The overall average relative deviation (OARD) was calculated by (21), where Ns is the number of calculated points:
(21)OARD(%)=∑nARDNs.An error comparison of calculation and prediction results among (8), (13), and (19) was presented in Table 6 for two FA esters. In the high pressure and temperature range of experimental data have an uncertainty (about 0.3% or more in AAD%) that increases uncertainty of the comparison. The OARD% information also indicates that the formulas do not introduce systematic errors. The comparison results of these methods, which were derived from different principle, show clearly the reliability of these methods when applied in extrapolation and confirm their purely predictive character. Because the molecular compressibility data is scarce under high pressure and higher temperature, it is difficult to determine which one has better performance under extended pressure and temperature ranges.Table 6
OARD % of calculated results for three methods under 303 to 383 K, 0.1 to 210 MPa.
Methyl caprate
Ethyl caprate
Formula (8) versus (13)
−0.00818
−0.00631
Formula (8) versus (19)
0.530178
0.380222
Formula (13) versus (19)
0.538393
0.386536More recently, Cunha et al. [20] proposed a new atomic contribution model to predict speed of sound, the group contribution model, which was listed in Table 1 and calculated by (5) and was replaced by seven atomic compounds: Cpar (for paraffinic carbon), COlef (for olefinic carbon), CArom (for aromatic carbon), H (for hydrogen), Osp3 (for ether oxygen), Osp2 (for ester group, being the COO decomposed into one COlef, one Osp2, and one Osp3), and OAlc (for alcohol oxygen). χa is the atom linear temperature correction parameter and a runs over all atoms; however χa have different values only for OAlc and Osp2. Adjusted Wada’s constant contributions are presented in Table 7, and Wada’s constant was calculated by
(22)km(T)=∑α=1nNαkmα(1-χα(T-298.15))
but the two methods are available only at atmospheric pressure, so we take 0.1013 MPa isobaric value to compare the accuracy of five methods; the comparison results for methyl and ethyl caprate were listed in Figures 9 and 10.Table 7
Atomic contribution values and temperature correction coefficients [20].
CPar
COlef
CArom
H
Osp3
OAlc
Osp2
103 Wada’s constant (SI)
0.0113
0.1234
0.1348
0.1694
0.1738
0.094
0.2568
103χ (K−1)
0.0191
0.0191
0.0191
0.0191
0.0191
−1.3064
0.3488Figure 9
Molecular compressibility’s values calculated from five formulas for methyl caprate at atmospheric pressure.Figure 10
Molecular compressibility’s values calculated from five formulas for ethyl caprate at atmospheric pressure.The values of (8) are obtained from the experimental speed of sound and density data, so (5) and (19) have higher consistency at atmospheric pressure. Formula (13) showed bigger difference, while (5) and (22) for ethyl caprate at 0.1 MPa isobaric.


## 3.1. Characteristic of Molecular Isentropic Compressibility in High Pressure and High Temperature

Ndiaye et al. [15] measured sound velocity of methyl caprate and ethyl caprate at pressures up to 210 MPa in the temperature range 283.15 to 403.15 K. In addition, density was measured in 0.1 to 100 MPa and 293.15 to 393.15 K, by using a modification of Davis and Gordon’s procedure [23]; the method [24] rested on the Newton-Laplace relationships; then density was extrapolated up to 210MPa from sound velocity data (7) with an absolute average deviation of 0.07% up to 210 MPa. the isentropic and isothermal compressibilities also were presented in the P-T domain:
(7)ρ(P,T)=ρ(Pref,T)+∫PrefP1c2dP+T∫PrefPαP2CpdP,
where Cp, αp represent the isobaric heat capacity and the isobaric thermal expansion, respectively, and Pref is a reference pressure where density is known.In order to directly obtain molecular isentropic compressibilitykm from experimental data, we expressed km as a function of density and sound velocity, shown as (8).
(8)km=Mw(cρ3)2/7.Based on [15] data that sound velocity was obtained from experimental and density was obtained from calculated value by (7), km was calculated in 0.1 to 210 MPa and 303.15 to 383.15 K and the fitting results were presented with isothermal characteristics for methyl caprate and ethyl caprate in Figures 1 and 2.Figure 1
Molecular compressibility of methyl caprate (MeC10:0) as a function of pressure along various isothermal (the lines are fitting results; the dots are calculated values).Figure 2
Molecular compressibility of ethyl caprate (EeC10:0) as a function of pressure along various isothermal (the lines are fitting results; the dots are calculated values).A one-order rational fitting relation to pressure was expressed as (9), where P is pressure, other parameters A1, A2, A3 are equation coefficients, as listed in Table 2:
(9)km(P)=A1P+A2P+A3=a+bP+c.Table 2
Fitting coefficients for (9) along various isothermals.
FA ester
Methyl caprate
Ethyl caprate
Temperature
303.15 K
343.15 K
383.15 K
303.15 K
343.15 K
383.15 K
Coefficients
A1
4.597
4.612
4.632
4.985
4.992
5.004
A2
480.8
374.7
291.4
547.2
423.4
303
A3
109.7
85.55
66.58
115.5
89.39
63.96
Goodness of fit
SSE
2.25
E
-
06
2.45
E
-
06
8.37
E
-
06
1.48
E
-
06
5.76
E
-
06
8.34
E
-
06
R-square
0.9999
0.9999
0.9998
1
0.9999
0.9999
AdjustedR-square
0.9999
0.9999
0.9998
1
0.9999
0.9998
RMSE
0.000375
0.000391
0.000723
0.000304
0.0006
0.000722The rational relation (9) was achieved with satisfying goodness of fit, where the minimum R-square value is 0.9998 in all temperature domains. We may safely draw the conclusion that molecular compressibility is dependent on pressure for ethyl and methyl caprate; this dependence on pressure was revealed sufficiently with elevated pressure In Figures 1 and 2.Accordingly, the calculated results were presented with isobaric characteristics for methyl caprate and ethyl caprate in Figures3 and 4, and a two-order polynomial fitting in temperature domain was expressed as
(10)km(T)=B1T2+B2T+B3=a(T+b)2+c,
where T is temperature and other parameters B1, B2, B3 are equation coefficients, as listed in Table 3.Table 3
Fitting coefficients for (10) along various isobaric.
FA ester
Mthyl caprate
Ethyl caprate
Pressure
P
0.1
P
80
P
130
P
150
P
210
P
0.1
P
80
130
P
150
P
210
Coefficients
B1
-
2.3
E
-
06
1.6
E
-
06
1.6
E
-
06
1.6
E
-
06
1.3
E
-
06
2.2
E
-
06
3.2
E
-
06
2.6
E
-
06
2.5
E
-
06
2.2
E
-
06
B2
0.00139
−0.00056
−0.00051
−0.00047
−0.00031
−0.00094
−0.00161
−0.00117
−0.00108
−0.00094
B3
4.168
4.495
4.507
4.506
4.499
4.982
5.028
4.983
4.978
4.982
Goodness of fit
SSE
2.7
E
-
08
1.5
E
-
06
6.5
E
-
07
9.4
E
-
07
1.09
E
-
06
1.8
E
-
07
1.9
E
-
07
2.7
E
-
07
3.9
E
-
07
1.83
E
-
07
R-square
0.9998
0.9987
0.9995
0.9993
0.9992
0.9998
0.9999
0.9998
0.9997
0.9998
AdjustedR-square
0.9995
0.9974
0.9991
0.9987
0.9983
0.9997
0.9997
0.9997
0.9995
0.9997
RMSE
0.00012
0.00088
0.00057
0.00068
0.00074
0.0003
0.0003
0.00037
0.00044
0.0003Figure 3
Molecular compressibility of methyl caprate (MeC10:0) as a function of temperature along various isobaric (only calculated values were revealed).Figure 4
Molecular compressibility of ethyl caprate (EeC10:0) as a function of temperature along various isobaric (only calculated values were revealed).In Figures3 and 4, a two-order polynomial (10) was achieved with the minimum R-square value being 0.9987 in all pressure domains. At atmospheric pressure (0.1 MPa isobaric) km of methyl caprate shows a slight downtrend with two-order form described in (10) but not first-order form described in Daridon’s formula (5); however for 0.1 MPa isobaric of ethyl caprate is almost horizontal; meanwhile km show uptrend, which accelerates with increasing pressure, at isobaric with temperature elevated. km also is dependent on temperature for ethyl and methyl caprate.Furthermore to calculate molecular compressibility differencekm of ethyl caprate in isothermal line with 20 K step and in isobaric line with 10 MPa step, Figures 5 and 6 can be achieved.Figure 5
The trend of differential pressure of molecular compressibility temperature along various isothermal for ethyl caprate (only calculated values were revealed).Figure 6
The trend of differential temperature ofmolecular compressibility temperature along various isobaric for ethyl caprate (only calculated values were revealed).We can take the partial derivative with respect to pressure for (9) and to temperature for (10); thus we have
(11)(∂km∂P)T=A1P+A3-A1P+A2(P+A3)2=A1A3-A2(P+A3)2,(12)(∂km∂T)P=2B1T+B2.From Figure5, we can find that, among various isothermals, the Δkm/ΔP become bigger when the temperature was elevated gradually and almost keep fixed value above 60 MPa though nearly equal at atmospheric pressure; here temperature variation plays a more important role under high pressure. Thus the Δkm/ΔP reveal almost equality and only show dependence on temperature in high pressure, (∂km/∂P)T→Const. The result can be achieved form the calculation of (11) coefficients; the values of A1A3-A2 for various isothermal are in 23.5 to 16.2, but A3 is in about 110 to 65, when pressure is large than 60 MPa increments of (P+A3)2 acceleration.From Figure6, we can find that among various isobaric Δkm/ΔP become smaller when pressure was elevated gradually and keep almost the same value above 60 MPa with slight dependence of pressure, That is to say the possibility of being compressed of Km decreases with high pressure, (∂km/∂T)P≃Const. Meanwhile, Δkm/ΔP increase with elevated temperature, but the rate of increment declines rapidly based on 2B1 which became almost the same as above 60 MPa with 10−6 magnitude; here pressure variation reveals more contributions under low pressure.


## 3.2. The Data Fitting Formula of Molecular Compressibility in Three-Dimensional

According to the two-dimensional fitting analysis, a three-dimensional rational formula was introduced with the following expression:(13)km=a+b(T+b)2+c(T+b)2+dP+e,
where P, T are pressure and temperature and other fitting parameters a, b, c, d, e are formula coefficients, listed in Table 4.Table 4
k
m
three-dimensional fitting coefficients.
FA ester
Ethyl caprate
Methyl caprate
Coeffiecients
a
4.451
4.845
b
1.34
E
-
06
1.22
E
-
06
c
−0.0001
-
6.91
E
-
05
d
−6.793
−13.2
e
81.87
84.51
Goodness of fit
SSE
0.000307
0.000552
R-square
0.9986
0.9979
AdjustR-square
0.9986
0.9978
RMSE
0.001847
0.002476This formula has one-order rational form in pressure domain and two-order polynomial form in temperature domain; at the same time(∂km/∂P)T shows two-order rational form in pressure domain and (∂km/∂T)P shows one-order polynomial form in temperature. So it can be used to predict km without experimental data for methyl and ethyl caprate under pressures up to 210 MPa and temperature range from 283.15 to 403.15 K.


## 3.3. The Temperature and Pressure Correction of Molecular Compressibility

Combining Figures5 and 6, km shows obviously the characteristics: in lower pressure, the larger free volume will be a reasonable interpretation; in this condition the intermolecular potential mainly contributed to the pressure and the vibration of molecules is scarcely compromised; thus pressure has significant effect on it than temperature. On the contrary, in high pressure condition the molecules are much closer to the close packed state and intermolecular potential rapidly increases with tiny distance change; molecules vibration based on temperature will mainly achieve the intermolecules space and then the potential that temperature has significant effect in this time than pressure.The Lennard-Jones potential is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules [25]:
(14)VLJ=4ε[(σr)12-(σr)6]=ε[(rmr)12-2(rmr)6],
where ε is the depth of the potential well, σ is the finite distance at which the interparticle potential is zero, r is the distance between the particles, and rm is the distance at which the potential reaches its minimum. At rm, the potential function has the value -ε. However σ is determined by temperature which represents molecular kinetic energy, and σ increases with temperature. The repulsive force increase sharply with slight decrease of intermolecules distance.Nevertheless,σ was not only determined by pressure, but also implicated by the molecular configuration and size. The force situation of liquid molecules was considered in free equilibrium state under zero-pressure condition, if the gravity was neglected; here we set reference pressure Pref=0.1013 MPa, and the distance between the molecules was perceived as σ and molecules molar volume and density are set to Vref, ρref under the equilibrium state. If we assume the pressure P was a function of LJ-potential, f(P)=VLJ, and the space occupied by molecules was a cube, so Vref=σ3 and Vref/Vm=σ3/r3. A correction coefficient KP of pressure was introduced and the Lennard-Jones potential can be written as
(15)KP=f(P)=VLJε=4[(VrefVm)4-(VrefVm)2]=4[(ρρref)4-(ρρref)2].For temperature correction ofkm, energy equipartition theorem [26], which makes quantitative predictions, was considered, and it gave the total average kinetic and potential energies for a system at a given temperature in thermal equilibrium. The oscillating molecules have average energy
(16)〈H〉=〈Hkinetic〉+〈Hpotential〉=12kBT+12kBT=kBT,
where the angular brackets 〈⋯〉 denote the average of the enclosed quantity. So (ε/kBT-ε/kBTref) will be a correction factor to charge temperature effect on km(17)KT=Exp(εkBTref-εkBT),
where ε is the depth of the potential well and also represents a unit of energy. Tref, which is reference temperature, was considered as the cloud point temperature of an organic liquid under reference pressure Pref, for ethyl caprate Tref being 253.15 K and for methyl caprate Tref being 260.15 K.The Elliott, Suresh, and Donohue (ESD) equation of state accounts for the effect of the shape of a nonpolar molecule and can be extended to polymers with the addition of an extra term. The EOS itself was developed through modeling computer simulations and should capture the essential physics of the size, shape, and hydrogen bonding [27].In ESD, the shape factorc=1+3.535ω+0.533ω2, where ω is the acentric factor. And the following equation shows the energy factor ε/kB is related to the shape parameter c:
(18)εkB=Tc1+0.945(c-1)+0.134(c-1)21.023+2.225(c-1)+0.478(c-1)2.In this case, these calculation methods of relevant parameters follow EOS, and the calculation data were cited from [18, 19] and listed in Table 5 with the calculated parameters value.Table 5
Critical properties and parameters of ethyl caprate.
Fatty ester
T
c (K)
P
c (MPa)
ω
c
ε
/
k
B
ε
Ethyl caprate [18]
690.22
1.869998
0.709
3.7737413
295.34
4.08
E
-
21
Methyl caprate [19]
683
1.5701
0.596
3.2958
306.01
4.23
E
-
21By data calculation and comparison, the pressure correction factor is written asKP3/4 and the final correction formula of molecular compressibility are written as
(19)km=kmref(1+0.15(0.1KP)3/4KT1/7),
where the reference molecular compressibility, kmref, is obtained from the additivity of the subgroup of molecule under reference pressure and temperature and listed in Table 1. So if the density under specific pressure and temperature was obtained, the molecular compressibility could be calculated by (19). Because sound velocity vanishes from calculation, molecular compressibility can be obtained more conveniently.


## 3.4. The Accuracy Comparison among Molecular Compressibility Expressions

To apply (8), (13), and (19), the requirements of experimental data are different. Formula (8) needs density and sound velocity data, (19) only need density data, and (13) does not need experimental data. Further (8) is connected to dynamic precess, and (19) reflects statistical state value, and (13) derives from calculated results of (8). The change of density is more direct property associated to compressibility than sound velocity, so (19) should be more accurate. The results comparison of three formulas was showed in Figures 7 and 8 for methyl caprate and ethyl caprate, respectively. Formula (19) reveal bigger difference at whole P-T domain.Figure 7
Molecular compressibility’s values calculated from three formulas for methyl caprate along various isothermals (dots for (8), dotted lines for (13), and lines for (19)).Figure 8
Molecular compressibility’s values calculated from three formulas for ethyl caprate along various isothermals (dots for (8), dotted lines for (13), and lines for (19)).To evaluate the predictive capacity of the formulas aforementioned, the relative deviations (RDs) between two formulas ofkm were estimated according to
(20)RD(%)=kmformula-i-kmformula-jkmformula-j%.The average relative deviation (ARD) was calculated as a summation of RD overNp calculated data points at a certain temperature. The overall average relative deviation (OARD) was calculated by (21), where Ns is the number of calculated points:
(21)OARD(%)=∑nARDNs.An error comparison of calculation and prediction results among (8), (13), and (19) was presented in Table 6 for two FA esters. In the high pressure and temperature range of experimental data have an uncertainty (about 0.3% or more in AAD%) that increases uncertainty of the comparison. The OARD% information also indicates that the formulas do not introduce systematic errors. The comparison results of these methods, which were derived from different principle, show clearly the reliability of these methods when applied in extrapolation and confirm their purely predictive character. Because the molecular compressibility data is scarce under high pressure and higher temperature, it is difficult to determine which one has better performance under extended pressure and temperature ranges.Table 6
OARD % of calculated results for three methods under 303 to 383 K, 0.1 to 210 MPa.
Methyl caprate
Ethyl caprate
Formula (8) versus (13)
−0.00818
−0.00631
Formula (8) versus (19)
0.530178
0.380222
Formula (13) versus (19)
0.538393
0.386536More recently, Cunha et al. [20] proposed a new atomic contribution model to predict speed of sound, the group contribution model, which was listed in Table 1 and calculated by (5) and was replaced by seven atomic compounds: Cpar (for paraffinic carbon), COlef (for olefinic carbon), CArom (for aromatic carbon), H (for hydrogen), Osp3 (for ether oxygen), Osp2 (for ester group, being the COO decomposed into one COlef, one Osp2, and one Osp3), and OAlc (for alcohol oxygen). χa is the atom linear temperature correction parameter and a runs over all atoms; however χa have different values only for OAlc and Osp2. Adjusted Wada’s constant contributions are presented in Table 7, and Wada’s constant was calculated by
(22)km(T)=∑α=1nNαkmα(1-χα(T-298.15))
but the two methods are available only at atmospheric pressure, so we take 0.1013 MPa isobaric value to compare the accuracy of five methods; the comparison results for methyl and ethyl caprate were listed in Figures 9 and 10.Table 7
Atomic contribution values and temperature correction coefficients [20].
CPar
COlef
CArom
H
Osp3
OAlc
Osp2
103 Wada’s constant (SI)
0.0113
0.1234
0.1348
0.1694
0.1738
0.094
0.2568
103χ (K−1)
0.0191
0.0191
0.0191
0.0191
0.0191
−1.3064
0.3488Figure 9
Molecular compressibility’s values calculated from five formulas for methyl caprate at atmospheric pressure.Figure 10
Molecular compressibility’s values calculated from five formulas for ethyl caprate at atmospheric pressure.The values of (8) are obtained from the experimental speed of sound and density data, so (5) and (19) have higher consistency at atmospheric pressure. Formula (13) showed bigger difference, while (5) and (22) for ethyl caprate at 0.1 MPa isobaric.


## 4. Conclusion

The molecular compressibility that was considered as a constant is challenged as a function of pressure and temperature. By data fitting a first order rational function in pressure domain and a two-order polynomial function in temperature domain were achieved. The results of 2D data fitting showed that molecular compressibility has an evident dependence of temperature and pressure. Partial derivative of pressure and temperature was introduced to analyze this dependency further. The 3D fitting results of molecular compressibility obviously showed that pressure reveals more contributions under low pressure, while temperature plays a more important role under high pressure.This dependence was correlated with molecular potential energy and kinetic energy by developing pressure and temperature correction coefficients, which are derived from Lennard-Jones potential function and energy equipartition theorem. Combining the two correction coefficients, the new molecular compressibility predicting formula is achieved. As a result, the molecular compressibility at working conditions of injection system can be calculated directly by density data in corresponding condition. With the maximum 0.5384% OARD, the consistency of calculation results demonstrated the validity of three expressions and the pressure and temperature dependence of molecular compressibility.The correction expression (Formula (19)), which contains “shape factor” of molecules and only requires density data, has conceivable prospects to be extended to other FA esters and biodiesels. But (19) needs more mathematical way to improve its accuracy; the HILO-based error correction scheme used to improve Gaussian mixture models (GMMs) [28] would be an idea worthy of learning.



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*Source: 512576-2014-01-16.xml*