Computation of Liquid Isothermal Compressibility from Density

Article Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Computation of Liquid Isothermal Compressibility from Density Measurements: An Application to Toluene Jean-Luc Daridon* and Jean-Patrick Bazile Laboratoire des Fluides Complexes et leurs Réservoirs-IPRA, UMR5150, CNRS/TOTAL/Univ Pau & Pays Adour, 64000 Pau, France ABSTRACT: A computation method was developed for estimating the compressibility and its uncertainty from density measurements carried out from atmospheric pressure to 100 MPa. For that purpose, first the ability of 19 equations for fitting density data and for calculating the isothermal compressibility was investigated. Among the equations of state tested, only 6 equations were selected for their suitability for deriving density data in the pressure range investigated. Then, a procedure for evaluating the combined standard uncertainty associated with the determination of the isothermal compressibility from the derivation of density data was tested. Finally, a numerical method was proposed for estimating the isothermal compressibility and its uncertainty from a limited number of experimental density data by considering a Monte Carlo procedure. As an example, the proposed numerical procedure was applied to liquid toluene. With this aim in mind, the density of liquid toluene was measured up to 100 MPa in the temperature range of 293.15−343.15 K by using a Ushape oscillating tube density meter.

■

by several other groups5−7 investigated the compressibility of hydrocarbons from ultrasonic speed measurements. The determination of compressibility from volumetric measurements can be made by two methods. The first consists of using the finite difference approximation in order to measure the slope of the secant line through the origin and any specified point on the volume versus the pressure curve. This measurement can be done directly by applying a pressure change and noting the variation in volume using a piezoemeter, in which the compression of the liquid is achieved either by the contraction of a bellows8 or by the movement of a solid or liquid piston.9 Given the finite difference approximation used, this technique does not lead to the determination of the tangent compressibility but only to the mean (or secant) compressibility defined by

INTRODUCTION The isothermal compressibility κT of liquid under pressure is not readily measurable as it is defined by a first-order derivative of volume. It can be determined directly from volumetric measurements or indirectly deduced from the measurement of other physical properties. In practice, the most frequently used indirect method is based on the speed of sound w, which can be measured with a high degree of accuracy, even at high pressures, and which has the advantage to be related to the isentropic compressibility κs according to the Newton−Laplace equation 1 κs = ρw 2 (1) By adding to this relation a nonadiabatic correction, which involves the isobaric expansion α and heat capacity cp, a relationship is obtained by which the isothermal compressibility can be computed from the speed of sound measurement as follows:

Tα 2 κT = κs + ρc p

κT̅ = −

(3)

where vatm is the volume of liquid at temperature T and atmospheric pressure patm, and p̃ is the differential pressure (p − patm). In practice, knowing the secant compressibility or its reciprocal, the secant bulk modulus, K̅ , is useful for calculating density at high pressure from the density at atmospheric pressure. From a thermodynamic perspective, it is more interesting to know the partial derivative and consequently the isothermal tangent compressibility

(2)

This potentiality, specific to speed of sound, has been used for obtaining isentropic and isothermal compressibility coefficients of a large number of pure liquids and liquid mixtures. In particular, Kell and Whalley1 and, recently, Trusler and Lemmon2 calculated the thermodynamic properties of water from the speed of sound, whereas Davis and Gordon3 used this method for estimating the compressibility of liquid mercury at ultrahigh pressure (GPa). Muringer et al.4 followed © XXXX American Chemical Society

1 ⎛ v − vatm ⎞ ⎜ ⎟ vatm ⎝ p ̃ ⎠

Received: February 19, 2018 Accepted: April 25, 2018

A

DOI: 10.1021/acs.jced.8b00148 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data 1 ⎛ ∂v ⎞ κT = − ⎜ ⎟ v ⎝ ∂p ⎠ T

Article

Table 1. Sample Description (4)

chemical name

CAS

source

purity

lot number

purification method

In this technique, the derivative can only be approached by reducing pressure steps and by measuring volume changes resulting from small isothermal pressure changes. Unfortunately, reducing pressure steps and volume changes drastically increases their relative uncertainties and consequently the uncertainty of compressibility. In the second volumetric method, an attempt is made to derive the isothermal tangent compressibility from density data. The measurements of density are performed at different pressures along an isotherm to allow for computing the partial derivative of volume versus pressure at constant temperature. Because of the limited number of density measurements usually carried out along each isotherm as well as the uncertainty in the density measurement, the partial derivative cannot be calculated directly from raw data by finite differences. The data set must be initially fitted by an interpolation function, 10 and subsequently, the fitted function is differentiated analytically in order to compute the partial derivative. Consequently, in this method, the uncertainty in the compressibility determination is correlated to the uncertainty in the density measurement, but it is also related to the computation procedure and, in particular, to the interpolation function used. In general, the smaller the number of experimental data and the higher the number of fitted parameters used in the interpolation function, the better is the fit of density data, but there is a higher risk of deforming the pv curve and thereby calculating inaccurate compressibility coefficients. Moreover, the higher the number of data points fitted, the better is the reliability of the derivative, but the higher is the time consumed in performing experiments. Therefore, an appropriate numerical schema must be found so as to provide an acceptable compromise between the uncertainty in compressibility determination and the time required for the measurements of density data. For this purpose, first the ability of several empirical and polynomial functions for fitting raw density data and for calculating compressibility were investigated in this article. Then, a procedure to evaluate the combined standard uncertainty associated with the compressibility estimated by fitting an interpolation function to density data was tested. Finally, a numerical method is proposed for estimating the compressibility and its uncertainty from a limited number of experimental density data by considering the most appropriate equations. As an example, the proposed numerical procedure was applied for determining the compressibility of toluene under pressure up to 100 MPa, as the derivative properties of this compound were already well investigated in the literature, and there are a large amount of compressibility data for comparison. With this aim in mind, the density of liquid toluene was measured and reported as a function of pressure along different isotherms ranging from 293.15 to 343.15 K by using a U-shape oscillating tube density meter.

toluene

108-88-3

Merck

0.999

1.08327.1000

none

vibrating U-Tube density meter (Anton-Paar) equipped with a high-pressure cell (DMA HPM). The temperature of the measuring cell was regulated by an external heat circulation thermostat, and it was measured with an Anton-Paar MKT 50 thermometer with a standard uncertainty of 0.03 K. The pressure was measured in the full pressure range with a Presens manometer with a relative standard uncertainty of 0.02%. The principle of the apparatus consists of measuring the period of oscillation of the U-tube electromagnetically excited by means of an external system and in determining, from this measurement, the density, which is related to the square of the period by a linear relation. The constants of this relation were determined by measuring the period of the tube filled with two different systems of known density. According to the method proposed by Lagourette et al.,11 the vacuum and water were considered for the calibration, and the working equation is ρ(T , p) = A(T )(τ 2(T , p) − τw2(T , p)) + ρw (T , p)

(5)

where τ is the vibration period of the U-Tube filled with the investigated mixture, τw is the vibration period with water, and ρw is the density of water taken from the equation of state of Wagner and Pruß.12 The coefficient A, considered by Lagourette et al.11 as pressure independent, is estimated for each temperature by the calibration with the vacuum and water under atmospheric pressure as follows: A (T ) =

ρ (T )

w0 2 2 τw0 (T ) − τvac (T )

(6)

where the subscripts w0 and vac refer to water under atmospheric pressure and vacuum, respectively. The combined standard uncertainty in the density measurement uM(ρ) was estimated according to the GUM of NIST13 by combining the quadratic sums of the different sources of uncertainty appearing in the working equation as follows: 2 uM (ρ) = [(τ 2 − τw2)uc(A)]2 + [2Aτuc(τ )]2

+ [2Aτwuc(τw )]2 + [u(ρw )]2

(7)

with uc2(A)

⎡ A ⎤2 ⎡ 2A2 τ ⎤2 w0 = ⎢ u(ρw0 )⎥ + ⎢ uc(τw0)⎥ ⎢⎣ ρw0 ⎥⎦ ⎥⎦ ⎣⎢ ρw0 ⎡ 2A2 τ ⎤2 vac +⎢ uc(τvac)⎥ ⎢⎣ ρw0 ⎥⎦

(8)

As density data result from the measurement of four vibration periods along independent experiments, uncertainties in temperature and pressure were taken into account in each measurement by considering the following combined standard uncertainty of the vibration period as follows:

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EXPERIMENTAL SECTION Materials. The liquid toluene used in the present work was purchased from Merck with a nominal minimum purity of 99.9%. It was used for measurement without any further treatment. Table 1 shows the details of the sample description. Experimental Method. The volumetric mass density of mixtures ρ was measured between 0.1 and 100 MPa using a

uc2(τi)

⎡⎛ ⎞ ⎤ 2 ⎡⎛ ⎞ ⎤2 ∂ τ ∂ τ i i = [u(τi)] + ⎢⎜ ⎟ u(T )⎥ + ⎢⎜ ⎟ u(p)⎥ ⎢⎣⎝ ∂T ⎠ p ⎥⎦ ⎢⎣⎝ ∂p ⎠ ⎥⎦ T 2

(9) B

DOI: 10.1021/acs.jced.8b00148 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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where subscript i refers to either the studied mixture or water or vacuum. The derivative of vibration periods (∂τi/∂T)p and (∂τi/ ∂p)T were calculated numerically from the raw data. In addition, the density uncertainty caused by sample purities xsp is taken in consideration through the following relationship: 1 − xsp usp(ρ) = ξρ (10) 3

The density data, as well as their standard uncertainties, were determined according to the method developed by Lagourette et al.,11 whereby apparatus parameter A was assumed to be independent of pressure. In order to check if the working equation based on this assumption does not generate systematic uncertainties that will increase with pressure and that will have a significant effect in compressibility computation, the raw data for both water and toluene were used to calculate the density of toluene according to the method proposed by May et al.15,16 This second method rests on a physically based model for representing both the effect of the temperature and pressure on apparatus parameters A and B according to the following relations:

Because there is no information about the nature of the impurities, a rectangular distribution was assumed, and the relative standard deviation ξ of the density of the impurities was arbitrarily set to 20%. This value was estimated by studying the variability of density of hydrocarbon compounds through the calculation of the standard deviation of a set of density data of 500 hydrocarbons by using the data compilation of Yaws.14 AMay (T , p) = A 0

BMay (T , p) = B0

1 + βτ (p − p0 ) (1 + ε1(T − T0) + ε2(T − T0)2 )(1 + αV (T − T0) + βV (p − p0 ))

1 (1 + αV(T − T0) + βV (p − p0 ))

(11)

measurement uncertainties. Therefore, it can be concluded that eq 5 does not lead to a systematic uncertainty related to an increase in pressure in the pressure range of 0.1−100 MPa. Consequently, the combined standard uncertainty of the reported density data, corresponding to the combination of the measurement uncertainty and the effects of the temperature, pressure, and purity, was estimated by

(12)

where T0 is a reference temperature chosen here to be 293.15 K, and p0 is fixed at 0.1 MPa. A0, B0, βτ, βV, ε1, ε1, and αV are apparatus parameters. These 7 parameters were determined by nonlinear regression of the vacuum and water measurements with an absolute deviation of 0.006% and a maximum deviation of 0.025%. After calibration with water, the parameters were used to determine the density of toluene from this method noted ρMay. Comparison of these densities with those determined from eq 5 does not show any systematic deviation, as can be seen in Figure 1. Moreover, the average absolute deviation and the maximum deviation between both sets of toluene density data are 0.01 and 0.028%, respectively. These values are comparable to the regression errors and are lower than the uncertainty in density caused by the propagation of

2 uc2(ρ) = uM (ρ) + usp2(ρ)

(13)

In this relation, the effects of temperature and pressure are not explicitly included but are implicitly taken into account in uM(ρ) through eq 9. From this method, the total expanded uncertainty of the density measurements at a 95% confidence level with a coverage factor of k = 2 is estimated to be 0.06% in the full p,T domain investigated. Density Measurements. Density measurements, reported in Table 2, were carried along six isotherms spaced at 10 K intervals in the temperature range of 293.15−343.15 K at pressures up to 100 MPa. The pressure measurements were performed with 10 MPa steps. However, so as to limit oscillation risk in compressibility computation at the edge of the experimental pressure domain, 3 points spaced at 2 MPa were added at both ends of the pressure range investigated. Furthermore, for one isotherm (293.15 K), density measurements were carried out every 2 MPa from atmospheric pressure up to 100 MPa to have enough data to study the effect of the pressure step on the computation method applied for calculating compressibility. These additional data are listed in Table 3. Densities in liquid toluene were measured previously under pressure by several authors. The aim of the present measurement is not to report new density data but only to get appropriate raw density measurements with well characterized uncertainties to evaluate the effectiveness of the numerical method applied for the computation of derivative properties. Therefore, comparisons between the present measurements and the densities of the literature were limited to the very precise density data determined from McLinden and Splett17 up to 30 MPa and correlated using a 20-parameter empirical equation and to the correlations proposed by Cibulka and Takagi,18 Assael. et al.,19 and Lemmon and Span.20 As these last authors report that their equation deviates from available data

Figure 1. Deviation 100(ρMay − ρ)/ρ of the experimental density value ρMay determined with the method proposed by May et al.15,16 from the density value ρ determined from eq 5 and reported in Table 2 as a function of pressure p. ■, 293.15 K; ●, 313.15 K; △, 333.15 K; and −·−, combined expanded uncertainties Uc(ρ). C

DOI: 10.1021/acs.jced.8b00148 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Table 2. Values of Density ρ at Temperatures T and Pressures p Measured in Liquid Toluene by Using U-Tube Densimetera p (MPa)

T (K)

ρ (kg·m−3)

T (K)

ρ (kg·m−3)

T (K)

ρ (kg·m−3)

0.1013 2 4 6 10 20 30 40 50 60 70 80 90 94 96 98 100 0.1013 2 4 6 10 20 30 40 50 60 70 80 90 94 96 98 100

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15

867.1 868.3 869.8 871.2 874.1 880.9 887.2 893.2 898.8 904.1 909.2 914.1 918.7 920.5 921.4 922.3 923.2 838.8 840.3 842.1 843.8 847.3 855.2 862.6 869.4 875.8 881.8 887.5 893.0 898.1 900.1 901.0 902.0 902.9

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15

857.7 858.8 860.4 861.9 865.1 872.3 879.0 885.2 891.0 896.7 901.9 907.0 911.8 913.6 914.4 915.3 916.5 829.0 830.9 832.8 834.7 838.3 846.8 854.4 861.6 868.2 874.5 880.4 885.9 891.3 893.3 894.4 895.4 896.3

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15

848.1 849.4 851.1 852.8 856.3 863.8 870.8 877.5 883.4 889.2 894.7 899.9 904.9 906.6 907.5 908.5 909.7 819.4 821.4 823.4 825.4 829.2 838.2 846.3 853.8 860.7 867.2 873.4 879.2 884.7 886.7 887.8 888.8 889.8

a

Standard uncertainties, u, are u(T) = 0.03 K, and u(p) = 0.0002p, and the combined expanded uncertainty, Uc (level of confidence = 0.95, k = 2), is Uc(ρ) = 0.0006 kg·m−3.

Table 3. Additional Density ρ Measured at Pressures p every 2 MPa Steps in Liquid Toluene at T = 293.15 Ka p (MPa)

ρ (kg·m−3)

p (MPa)

ρ (kg·m−3)

p (MPa)

ρ (kg·m−3)

8 12 14 16 18 22 24 26 28 32 34 36

872.6 875.5 876.9 878.2 879.6 882.2 883.5 884.7 886.0 888.5 889.6 890.8

38 42 44 46 48 52 54 56 58 62 64 66

892.0 894.3 895.4 896.5 897.7 899.8 900.9 902.0 903.1 905.2 906.2 907.2

68 72 74 76 78 82 84 86 88 92

908.2 910.2 911.2 912.2 913.1 915.0 915.9 916.9 917.8 919.6

ranges: 0.1−50 MPa (medium pressure range) and 50−100 MPa (high pressure range). The measurements reported here are in very good agreement with the equation of McLinden and Splett.17 The absolute average deviation (AAD) between this equation and the data reported in Table 2 was 0.02%, with a maximum that reaches 0.05%. The comparison with the correlation developed by Cibulka and Takagi18 gave an AAD of 0.01% and a maximum deviation of 0.04% in the MP range, whereas in the HP range, AAD% was 0.05%, and MD was 0.09%. Deviations of 0.07 and 0.12% were observed on average with the correlation developed by Assael et al.19 in the MP and HP ranges, respectively. Finally, the present measurements deviate on average from the correlation of Lemmon and Span20 by 0.03 and 0.06% in the MP and HP ranges, respectively. The maximum deviation was 0.05% in the MP range and 0.1% in the HP range. The deviations obtained as a function of pressure at one temperature are plotted in Figure 2. It can be seen in this figure that the maximum deviation corresponds to the higher pressure. Taking into account the uncertainties in the correlation calculations, these statistical results are in agreement with the estimations of our experimental uncertainties represented in the figure by the dot-chain line.

a Standard uncertainties, u, are u(T) = 0.03 K and u(p) = 0.0002p, and the combined expanded uncertainty, Uc (level of confidence = 0.95, k = 2), is Uc(ρ) = 0.0006 kg·m−3.

by no more than 0.075% up to 50 MPa and 0.15% up to 175 MPa, the comparisons were performed in two different pressure D

DOI: 10.1021/acs.jced.8b00148 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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curve that may not depict the actual behavior between the tabulated points, even though it passes exactly through the fitted points. Consequently, interpolation is inappropriate for calculating derivative properties from finite experimental data points. In this situation it is preferable to use an approximation method in which the fitted function aims at smoothing the experimental points by a continuous function that passes as close as possible to the original data, taking care to keep the smooth curve within the error bar of the original data. In this method, the form of the fitted function plays an essential role for the goodness of the fit but also, and perhaps most importantly, for the suitability of the derivative properties calculation. Uncertainty Evaluation. The uncertainty associated with the result of the compressibility computation by derivation of an empirical equation of state fitted to a discrete set of experimental density data has two distinct sources. The first one comes from the propagation of experimental errors during both the fitting procedure and derivation. It mainly depends on the standard uncertainty of the input data. The second is related to the absence of a known fundamental equation of state for representing the liquid state and to the uncertainty surrounding the choice of the equation used. It is therefore model dependent. A statistical method consistent with the principles of the guide to the expression of the uncertainty in measurement (GUM) was considered for evaluating the first contribution corresponding to the propagation of experimental uncertainties in compressibility derivation from a given equation of state. This procedure, based on a Monte Carlo method, consists first in generating a large number N of data sets with the same number of points and with the same pressures as the original experimental points but with density values randomly perturbed around the experimental measurements. For that purpose, a discrete pseudorandom distribution is generated for each density data by using a normal probability distribution function centered on the experimental density value. Its standard deviation is fixed equal to the standard uncertainty in density measurement uM(ρ). The uncertainty caused by sample impurities was not taken into consideration in this part as it corresponds to a systematic uncertainty that does not act on the derivation process. Its contribution must be added independently as for density. Then, each set of generated data point is used for fitting an isothermal equation of state and for calculating its derivatives. This fit is carried out by a simple least-squares fitting method in which all data are weighted equally. This procedure generates a distribution of N compressibility values for each experimental condition. The moments of the output distributions are calculated so as to estimate both the compressibilities and their standard uncertainties. The uncertainty associated with the functional form of the equation of state used for fitting the data was quantified by deploying several isothermal equations with different forms. After determining the compressibility values from the different equations of state, the standard deviation of the distribution is calculated and taken as the standard uncertainty arising from uncertainty of the functional form of the true equation of state. The equations considered for this stage must all fit the density data points within their uncertainty. Isothermal Equations of State. Many studies have been undertaken for the purpose of obtaining an equation able to describe the volumetric properties of liquid from theoretical

Figure 2. Deviation 100(ρcor − ρexp)/ρexp of the density value ρcor obtained with correlations from density value ρexp reported in Table 2 as a function of pressure p at T = 293.15 K. ○, McLinden;17 ◆, Cibulka and Takagi;18 ■, Assael et al.;19 △, Lemmon and Span;20 and −·−, expanded uncertainty.

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COMPRESSIBILITY COMPUTATION Procedure. The determination of the isothermal compressibility from a set of experimental density data points consists of calculating the values of the partial derivative of density with respect to pressure. As no theoretical functional form exists for representing the compressibility of liquid with reliability, this derivation must be performed numerically. The easiest way for calculating numerical derivatives is the finite difference method, as it comes straight from the definition of the derivative. It consists of approximating the derivative to the slope of the secant line (SL) through two adjacent experimental points at pressure pi and pi+1: SlopeSL =

ρ(pi + 1 , T ) − ρ(pi , T ) pi + 1 − pi

(14)

This slope converges to the slope of the tangent line when the experimental points get near to one another. Consequently, the truncation error made when approximating the true derivative by the slope of the secant line can be reduced by decreasing the pressure step. Unfortunately, this operation leads to a numerical subtraction of two almost equivalent numbers resulting in a huge increase of the relative uncertainty, which makes such a method impractical with a set of unconnected experimental data points. To overcome this difficulty related to the incompatibility between the minimization of the truncation error and the reduction of the experimental uncertainty, the derivation of the experimental density data must be performed in two steps. The first aims at fitting a function to the experimental data. The second consists in differentiating the fitted function to evaluate the derivative of density with respect to pressure at each pressure target. There are many different techniques for fitting a function to data points. They differ from each other by the aim of the fit: interpolation or approximation. Interpolation function fits, such as polynomial or piecewise polynomials, are suitable for plotting curves through discrete data points and then calculating the value of the fitted function at an arbitrary point. However, in this method, the distribution of data, as well as their uncertainties, can affect the shape of the interpolation E

DOI: 10.1021/acs.jced.8b00148 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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Table 4. List of the Nineteen Equations of State Tested number of parameter

typea

2

3

3 + ρatm fixed

3

2

3

3 + ρatm fixed

3

2 + ρatm fixed

2

3 + ρatm fixed

3

equation name eq 1-MS3-V eq 2-MS3-ρ eq 3-MS2-P

form

vr̃ = Bp ̃ + Cp ̃ + Dp ̃

ρ̃r = Bp ̃ + Cp ̃ + Dp ̃ p ̃ = Bρr ̃ +

Cρr2̃ Cρr2̃

eq 4-MS3-P

p ̃ = Bρr ̃ +

eq 5-MS3-P

p ̃ = Bln ρr +

+ Dρr3̃ C(ln ρr )2 + D(ln ρr )3 2 3

3 + ρatm fixed

3

v = A + Bp ̃ + Cp ̃ + Dp ̃

4

3

ρ = A + Bp ̃ + Cp 2̃ + Dp3̃

4

3

eq 8-PO2-P

p ̃ = A + Bρ + Cρ2

3

3

eq 9-PO3-P

2

4

2

bp ̃ 1 + cp ̃

2 + ρatm fixed

3

A−l

3

1

3

3

3

1

3

1

4

2

4

1

3

3

4

1

3

1

eq 6-PO3-V eq 7-PO3-ρ

p ̃ = A + Bρ + Cρ + Dρ

eq 10-DOW-ρ

ρ̃r =

eq 11-HUD-P

p̃ =

eq 12-L-SBM

ρ=

eq 13-L-TBM

ρ = A(1 + Bp ̃ )C

eq 14-L-DER eq 15-Q-SBM eq 16-Q-TBM eq 17-Q-TBM eq 18-Q-DER eq 19-R-DER

exp(B + C(A − l)) with l = v1/3

l2 A + Bp ̃ 1 + Dp ̃

(

p̃ + B B

(

ρ = A − Cln ρ=

3

−1

))

A + Bp ̃ + ACp 2̃ 1 + Dp ̃ + Cp 2̃

ρ=A

(

1 + Bp ̃ 1 + Dp ̃

ρ = Aexp

(

C

)

Bp ̃ 1 + Cp ̃

⎛ ρ = ⎜A − Cln ⎝

(

(

)

p̃ + B Dp ̃ + B

)⎞⎠

−1

⎟

−1

( p̃ +B B ) − Dp̃)

ρ = A − C ln

Type 1: Retained in the final computation procedure. Type 2: Rejected from the final procedure as being too sensitive to input density uncertainty. Type 3: Rejected from the final procedure as being not suitable for derivation at extremities of the experimental domain. a

smooth shape of the compression curve in the pressure range investigated. Therefore, a third-degree polynomial was only considered for representing the relative compression as a function of p̃. This first equation, noted eq 1-MS3-V, is given in Table 4. Two other Maclaurin series were also tested. One (eq 2MS3-ρ) expresses the relative density change ρ̃r = (ρ − ρatm)/ ρatm as a function pressure p̃, whereas the other expands the relative pressure p ̃ in terms of relative density changes ρ̃r , as suggested by Bridgman25 and by Davis and Gordon.3 For this last case, it has been observed by Bridgman25 that a quadratic form can correctly fit the data. Therefore, two series of this type were investigated: one (eq 3-MS2-P) of degree two that involves only two fitting parameters and the other of degree three (eq 4-MS3-P). A similar expression involving the logarithm of the relative density ρr= ρ/ρatm instead of ρ̃r was also reported by Davis and Gordon.3 A cubic form (eq 5-MS3-P) of this type was tested here. The use of the compressionv ̃ or the relative density change ρ̃ or other related properties like ln(ρ̃r ) that become zero at atmospheric pressure has the advantage to remove one fitting coefficient. Moreover, it allows working with smaller values than original quantities v and ρ during the fitting procedure, and consequently, it reduces the relative deviation between experimental and calculated densities after adding the atmospheric values and therefore improves the representation of density data. However, forming such a quantity by a shift in

considerations, but none of them has led to the development of a fundamental equation of state that can suitably represent liquid density and its derivatives. Consequently, empirical equations must be used to describe the influence of pressure on liquid density along isotherms from atmospheric to several hundred MPa. For that purpose, several equations have been proposed in the literature. Hayward,21 MacDonald,22 and Sun et al.23 have carried out comparative studies in order to test the ability of these equations to represent the compressive properties of liquids under high pressure by doing some tests based on experimental density data of water, mercury, aromatic hydrocarbons, and alcohols. Their conclusions were significantly different from each other, and no equation appears as superior by comparison with others after reading all these reviews. Consequently, we have investigated here several equations of state in order to determine the influence of their mathematical form on the computation of compressibility of toluene. The simplest method for building an isothermal equation of state consists in expanding the compression24 ṽ = (vatm − v) or the relative compression vr̃ = (vatm − v)/vatm by a finite Maclaurin series in relative pressure p ̃ = p − patm. By carrying out tests with second-, third-, and fourth-order series, it has been seen that a second-order series with only two parameters is not able to fit the experimental data within the experimental uncertainty, whereas a good fit is obtained with a degree of three. Furthermore, it has been noted that few improvements are obtained with higher degree polynomials, according to the F

DOI: 10.1021/acs.jced.8b00148 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

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pressure in a “narrow” pressure range, three simple equations were proposed for modeling liquid compressibility. The first one concerns the secant bulk modulus K̅ defined by v p̃ K̅ = atm vatm − v (17)

volumetric properties may significantly increase the relative uncertainty attached to this quantity, particularly near atmospheric pressure, as can be seen in Figure 3. This

It was expressed as a linear function of pressure by Tait28 according to the following equation, called original Tait’s equation:

Π + p̃ (18) A As shown in Figure 4, the secant bulk modulus calculated from the experimental density data of toluene at 293.15 K K̅ =

Figure 3. Comparison at T = 293.15 K between linear behavior predicted by Hudleston’s equation27 and experimental data of ln(l2P̃ / (latm − l)) as a function of (latm − l) with l = v1/3. ●, Experimental data; , Hudleston’s equation;27 and error bars indicate the expanded uncertainty corresponding to a 95% confidence level.

drawback, which affects the fitting procedure, is usually not noticeable by only comparing density calculations. To the contrary, it could seriously affect the derivative calculations around p ̃ = 0. Therefore, for derivation of compressibility, it could be preferable to fit experimental data to a polynomial function involving original volumetric properties v and ρ instead of ṽ and ρ̃. For that purpose, four cubic polynomials (eq 6-PO3-V, eq 7-PO3-ρ, eq 8-PO2-P, and eq 9-PO3-P) were considered here and are listed in Table 4. Other forms than polynomials can be considered to describe the volumetric behavior of liquids as a function of pressure. Dowson and Higginson26 observed that the relative density of mineral oils can be closely fitted by a simple rational function pr̃ =

bp ̃ 1 + cp ̃

Figure 4. Comparison between the experimental secant bulk modulus K̅ and linear or quadratic equations at T = 293.15 K as a function of pressure p. ●, Experimental data; - - -, linear equation; , quadratic equation; and error bars indicate the expanded uncertainty corresponding to a 95% confidence level.

seems to follow this linear trend within the pressure range concerned. However, the high uncertainty attached to first points biases the representation. By rearranging this equation explicitly in terms of density, one obtains a rational function of pressure p,̃ referenced as eq 12-L-SBM in Table 4. This equation in terms of density takes a form very similar to eq 15. The second simple equation expresses the tangent bulk modulus by a linear function of the relative pressure as follows:

(15)

This expression is referenced in Table 4 as eq 10-DOW-ρ. Hudleston27 has shown by postulating a law relating molecular interaction force to the distance between molecules that the volumetric behavior of normal liquids can be described at high pressures by using the following equation: ⎛ l 2p ̃ ln⎜ ⎝ latm −

⎞ ⎟ = b + c(latm − l) with l = v1/3 l⎠

⎛ ∂p ⎞ K = −v⎜ ⎟ = a + bp ̃ ⎝ ∂v ⎠T

(16)

It can be observed in Figure 3, where ln(l2 p̃/(latm − l)) is plotted as a function of latm − l, that the linear behavior predicted by Hudleston’s equation27 can depict the experimental measurements within the error bars. Therefore, this equation has also been taken into consideration in the present study. It is called eq 11-HUD-P in Table 4, where it was written explicitly in terms of relative pressure. An alternative approach for developing an isothermal equation of state consists in expressing the compressibility of its reciprocal as a function of pressure instead of density or volume. According to the nearly linear variation of the reciprocal of the compressibility, the bulk modulus, K, versus

(19)

Integration of such an equation with respect to pressure yields another nonlinear expression called Murnaghan’s equation29 ⎛ v ⎞ 1 ⎛ a ⎞ ln⎜ ⎟ = ln⎜ ⎟ b ⎝ a + bp ̃ ⎠ ⎝ vatm ⎠

(20)

This equation, named eq 13-L-TBM in Table 4, was reformulated so as to express density explicitly in terms of pressure. The third simple equation consists of considering the variation of the derivative (∂p/∂v) with respect to pressure G

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pressure with respect to volume can also be extended to high pressure by either adding a quadratic term as follows:

by the same expression as the original Tait’s equation for secant bulk modulus ⎛ ∂p ⎞ Π + p̃ ⎜ ⎟ =− ⎝ ∂v ⎠T A

⎛ ∂p ⎞ ⎜ ⎟ = a + bp ̃ + cp 2̃ ⎝ ∂v ⎠T

(21)

By calculating numerically, this derivative using a symmetric difference quotient of raw data measured at regular interval of 2 MPa and by plotting the values as a function of pressure in Figure 5, it can be seen that this linear assumption seems

(24)

or by expressing it by a simple rational form as follows:

⎛ ∂p ⎞ a + bp ̃ ⎜ ⎟ = ⎝ ∂v ⎠T 1 + cp ̃

24

(25)

These expressions can be integrated to give rise to two additional isothermal equations named eq 18-Q-DER and eq 19-R-DER. Table 4 thus lists a total of 19 isothermal empirical equations. In this table, all of the equations are written explicitly either in terms of volumetric properties (ρ, v, ρ̃ ,v )̃ or in terms of pressure p. As can be seen in Table 4, only 2−4 parameters are involved in the equations selected. This limited number of fitting parameter was deemed necessary to avoid oscillation of the fitted function between the experimental points and therefore to ensure smooth derivatives.

■

RESULTS AND DISCUSSION To assess the capacity of the equations listed in Table 4 to represent the volumetric property of liquid toluene as a function of pressure, the quality of fit was first investigated by comparing the average absolute deviation (AAD%) and the maximum deviation (MD%) obtained between the calculations performed with the different fitted equations and the experimental data. The fits were performed for each isotherm by considering a simple unweighted least-squares method, in which no uncertainty is associated with the measurements. For most of the equations, the least-squares fit was done by considering the pressure as the x-coordinate, whereas a volumetric property is associated with the y-coordinate. Therefore, the goodness of fit is evaluated by calculating the relative deviation between the calculated and experimental density values at a given pressure. For Hudleston’s equation27 as well as for eqs eq 3-MS2-P, eq 4-MS3-P, eq 8-PO2-P, and eq 9-PO3-P that cannot be expressed explicitly in terms of density, the fit was carried out by minimizing the merit function corresponding to the sum of squares of the pressure deviation. However, as for other equations, the goodness of the fit is checked by calculating the density deviation so as to apply the comparison to the same property and therefore ensure a consistent basis for comparing the deviations. In that case, the equation must be previously solved in order to calculate the density given by the equation at a fixed pressure. The results of this comparative study are summarized in Table 5, which reports the average absolute and maximum deviations calculated from the full set of data and from the measurements carried out along the 293.15 K isotherm with 2 MPa steps. They clearly show that all equations have the ability to describe the volumetric behavior within the pressure range investigated. Whatever the equation, the maximum deviation is always lower than the estimated relative uncertainty. Moreover, the average absolute deviation is more than ten times lower than the experimental uncertainty. Despite the significant number of data (51) used for describing the 293.15 K isotherm, the average deviation is not higher than that for other isotherms. Taking into account the parameters-to-data ratio (2/51 to 4/51), this good agreement between the fit and experiment shows the capacities of all the empirical equations selected to represent the volumetric behavior of toluene, but

Figure 5. Comparison between the linear expectation of the so-called Tait equation and the experimental values of the symmetric numerical derivative (Δp/Δv)T with Δp = 2 × 2 MPa at T = 293.15 K for pressure p from 0.1 to 100 MPa. ●, Experimental data; , linear expectation; and error bars indicate the expanded uncertainty corresponding to a 95% confidence level.

reasonable and is in agreement with the numerical values given the high uncertainty inherent to the numerical differentiation of density data. The integration by Tammann30 of this form gave rise to the so-called Tait equation that is named eq 14-L-DER in Table 4. The extension of the pressure domain imposes to modify such simple linear expressions for either the secant or tangent bulk modulus. This can be done by adding a quadratic term in the expression of the bulk modulus. In this case, the secant bulk modulus can be expressed by the following form: K̅ = a + bp ̃ + cp 2̃

(22)

Figure 4 shows that the addition of this second order term leads to an improvement of the representation of the pressure effect on the bulk modulus in comparison to the linear secant bulk modulus. Despite the fact that both fitted equations pass through the error bars, the quadratic equation passes closer to the experimental points than the linear equation at low pressure. Rearrangement of this relation enables one to write the density as a rational function of the pressure. This equation is noted eq 15-Q-SBM in Table 4. In the same way, the tangent bulk modulus can be expressed by the following expression: K = a + bp ̃ + cp 2̃

(23)

Integration of this quadratic function gives rise to 3 different equations of state according to the sign of the discriminant. Only the positive and zero solutions were considered here. The corresponding equations are listed in Table 4 as eq 16-Q-TBM and eq 17-Q-TBM. The linear form of the derivative of H

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therefore, it does not notably differ from it over the pressure range investigated. Since all of the equations selected are able to represent the experimental density data within their experimental uncertainties, they were all differentiated to calculate the compressibility at each experimental condition. The results are plotted as a function of pressure in Figure 6. This figure shows that all of the calculated curves overlap in the intermediate pressure range that cover the pressures between 20 and 80 MPa. Beyond this domain, the curves are further apart, and the group of curves widens at the extremes of low and high pressure. This result points out the influence of the functional form of the equation used to fit the density data. It has nearly no influence on the intermediate pressure domain, but it has a considerable effect at the edges. So as to quantify such an effect, the mean kT and standard deviations σκ of the obtained compressibility data sets were calculated. The relative standard deviation (RSD% = 100σκT/kT) was plotted as a function of pressure in Figure 7, and the deviation of the different equations from the mean is shown in Figure 8. Again, the overall situation varies from one region to another. In the intermediate region, the relative standard deviation is less than 0.5%. Beyond this region, the relative standard deviation increases considerably and reaches 1.5% at atmospheric pressure and 1.8% at 100 MPa. Examination of the deviations plotted in Figure 8 shows that there is no agreement between the compressibility calculated by the different equations at the extremity of the investigated pressure domain. Some equations deviate by more than 4% from the mean value at higher pressure. Although all equations have the capacity to fit very well the density data, the relative standard deviation plotted in Figure 7 and the deviations of Figure 8 show that they are not all suitable for deriving compressibility. The intercomparison test does not help to distinguish which equations are suitable and which ones are not. Another

Table 5. Density Deviation between the Experimental Data Reported in Tables 2 and 3 and the Calculations of the Fitted Equations Listed in Table 4 293.15 K with 2 MPa steps equation name eq eq eq eq eq eq eq eq eq eq eq eq eq eq eq eq eq eq eq a

1-MS3-V 2-MS3-ρ 3-MS2-P 4-MS3-P 5-MS3-P 6-PO3-V 7-PO3-ρ 8-PO2-P 9-PO3-P 10-DOW-ρ 11-HUD-P 12-L-SBM 13-L-TBM 14-L-DER 15-Q-SBM 16-Q-TBM 17-Q-TBM 18-Q-DER 19-R-DER

AAD% 4.7 4.0 4.2 2.6 3.8 4.1 3.6 3.8 2.8 7.8 2.7 5.9 3.4 3.0 2.8 2.7 5.8 2.7 2.7

× × × × × × × × × × × × × × × × × × ×

a

−3

10 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3

b

MD% 1.1 1.0 9.8 7.6 1.5 1.0 9.7 1.6 8.9 1.8 7.9 1.7 9.6 8.6 8.1 8.3 1.7 8.4 8.5

× × × × × × × × × × × × × × × × × × ×

−2

10 10−2 10−3 10−3 10−2 10−2 10−3 10−2 10−3 10−2 10−3 10−2 10−3 10−3 10−3 10−3 10−2 10−3 10−3

all data AAD% 7.6 6.7 8.9 5.0 5.5 7.2 6.5 7.7 5.4 1.0 5.2 9.4 5.9 5.5 5.4 5.2 9.4 5.3 5.7

× × × × × × × × × × × × × × × × × × ×

AAD%: Relative average absolute deviation. maximum deviation.

a

−3

10 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−2 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 b

MD%b 3.8 3.8 4.9 3.8 3.8 2.7 2.8 4.5 3.3 3.9 3.6 3.4 3.1 3.2 3.0 3.4 3.4 3.3 3.1

× × × × × × × × × × × × × × × × × × ×

10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2

MD%: Relative

even more it highlights the smoothness of the experimental data. The best fit is obtained with eq 4-MS3-P proposed by Davis and Gordon3 and with Hudleston’s equation27 (eq 11HUD-P), whereas the worst situation is found with equation eq 10-DOW-ρ and with eq 12-L-SBM and eq 17-Q-TBM that yield the same deviations. In fact it appears, by expanding eq 17-Q-TBM in a Taylor series, that eq 12-L-SBM corresponds to the linear approximation to eq 17-Q-TBM around p ̃ = 0, and

Figure 6. Isothermal compressibility κT in liquid toluene derived from density measurements at T = 293.15 K by using various equations for fitting density data as a function of pressure p. I

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equations (eq 10-DOW-ρ and eq 12-L-SBM) and eq 17-QTBM underestimated the compressibility at the higher pressures. Figure 10 shows that the third-degree Maclaurin series (eq 1-MS3-V, eq 2-MS3-ρ, eq 4-MS3-P, and eq 5-MS3P) underestimated the compressibility at the lower pressure. Finally, it can be seen in Figure 10 that polynomial eq 8-PO2-P overestimates compressibility at low pressure. Consequently, these ten equations, called type 3 in Table 4, must be removed from the computation procedure in order to select a coherent group of equations for deriving compressibility. The 9 remaining equations can represent the compressibility in the full experimental range. Furthermore, it can be noted from Figures 9 and 10 that some of these equations (eq 3-MS2-P, eq 11-HUD-P, eq 16-Q-TBM, and eq 18-Q-DER) can provide a good estimation of compressibility by extrapolating density data in the high pressure domain. Some others (eq 11-HUD-P, eq 13-L-TBM, eq 14-L-DER, eq 16-Q-TBM, eq 18-Q-DER, and eq 19-R-DER) can calculate compressibility by extrapolation in the low pressure domain. The compressibility set calculated from this reduced set of 9 equations exhibits relative standard deviations that reach, at most, 0.8%, as can be seen in Figure 7. This standard deviation can be associated with the standard uncertainty uForm(κT) arising from the unknown form of the true equation of state. In order to estimate the propagation of experimental uncertainties in the compressibility derivation uDer(κT) from a given equation of state, N data sets were generated by randomly perturbing the experimental values with a normal probability distribution function. For each experimental condition, the mean of the distribution was fixed to the experimental value, whereas the standard deviation was set to the combined standard uncertainty in the density measurement uM(ρ). From every data set randomly generated, compressibility was calculated to obtain a compressibility distribution for each experimental condition. As can be seen in Figure 11 for one equation (eq 13-L-TBM) and one experimental condition, the

Figure 7. RSD% of the isothermal compressibility data determined as a function of pressure p from various fitting equations at T = 293.15 K. ●, Set of the 19 equations listed in Table 4 and ○, set of the nine equations of type 1 and 2 in Table 4.

criterion was used to pinpoint which of the equations are not applicable for calculating the compressibility at the extreme conditions. It consists of fitting the same equations to two different pressure ranges in order to change the edge conditions. The first range corresponds to the domain from 0.1 to 60 MPa, whereas the second covers the pressures from 40 to 100 MPa. The compressibilities calculated within these shortened ranges were compared from the compressibility data estimated from the full set of data with a standard deviation of 0.4% between 20 and 80 MPa. The results are shown in Figures 9 and 10. Observation of Figure 9 clearly shows that the polynomial forms, which express volume (eq 1-MS3-V and eq 6-PO3-V) or density (eq 2-MS3-ρ and eq 7-PO3-ρ) in terms of pressure, are not suitable for calculating the compressibility, as they greatly overestimate the compressibility at the maximum pressure. At the opposite, the linear secant bulk modulus type

Figure 8. Deviation between the isothermal compressibility kT differentiated from various fitting equations and the mean isothermal compressibility ⟨kT⟩ at T = 293.15 K as a function of pressure p. J

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Figure 9. Isothermal compressibility kT calculated at T = 293.15 K within a shortened pressure p range of 0.1−60 MPa. Reference data were determined from the full data set with a RSD 0.4% between 40 and 80 MPa. Error bars indicate 2 × RSD for the reference data.

Figure 10. Isothermal compressibility kT calculated at T = 293.15 K within the shortened pressure p range of 40−100 MPa. Reference data were determined from the full data set with a RSD 0.4% between 40 and 80 MPa. Error bars indicate 2 × RSD for the reference data.

probability distribution for the output compressibility data. Therefore, the compressibility is taken from the mean. The standard uncertainty uDer(κT) associated with the propagation

resulting compressibility distribution has a Gaussian shape. That means that the propagation of the normal distribution through the computation procedure provides the same type of K

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Figure 13. Ratios of the RSDs, RSD(κT)/RSD(ρ), between the output and the input data in the Monte Carlo procedure as a function of pressure p. Green ---, eq 3-MS2-P; red ---, eq 9-PO3-P; blue , eq 11HUD-P; and black ···, eq 14-L-DER.

Figure 11. Relative frequency histogram of the isothermal compressibility kT distribution obtained from eq 13-L-TBM at T = 293.15 K and p = 20 MPa when the input density data are randomly perturbed by a Gaussian distribution of N = 5000 points for each experimental data. , Normal distribution.

in the pressure range of 20−80 MPa, and the maximum standard deviations observed at both ends of the experimental domain. Therefore, the comparison between equations was carried out by comparing the standard deviations at 0.1, 50, and 100 MPa. The values obtained for each of the nine selected equations are listed in Table 6. The magnitude of the standard deviations varies considerably from one equation to another. The equations can be separated in two groups. The first one, called type 2 in Table 4, represents the equations affected by the fluctuation of density during the fitting procedure and, as a result, are poorly suited to derive compressibility. The polynomial functions, eq 3-MS2-P and eq 9-PO3-P, and the rational equation, eq 15-Q-SBM, fall into this category. They exhibit huge standard deviations for compressibility at 0.1 and 100 MPa. Their RSD ratio between output and input data can be higher than 10 at atmospheric pressure. This very high value shows the great sensitivity of compressibility computation to the density uncertainty when working with such functions. In order to determine the influence of the experimental pressure step on the procedure, the calculations were repeated with different pressure steps of 5 and 10 MPa. The relative standard deviation obtained with these steps are added in Table 6. As an example, Figure 14 shows the RSD% obtained with eq 14-LDER using different pressure steps for measuring density. From this figure, it can be noted that the RSD% is only slightly affected by the experimental mesh size between 20 and 80 MPa. Moreover, using a 10 MPa step instead of 2 MPa divides the number of points by 5 but leads to a change in the compressibility calculation of only 0.3% at the maximum (corresponding to atmospheric pressure). A similar trend is observed with all equations. As can be seen in Figure 14, adding 3 points after the lower bound at 2, 4, and 6 MPa and 3 points before the upper bound at 94, 96, and 98 MPa in the 10 MPa steps frame reduces the RSD values that become similar to those calculated with 5 MPa steps. This configuration, which gives a good compromise between the number of experimental points (17 for an isotherm) and the RSD values (lower than 1% in the worst case), was adopted in our computation procedure. To investigate the influence of the input standard deviation on compressibility data computed with one of these retained equations, calculations were repeated by considering different relative standard deviation on input density data (RSD(ρ)) from 0.017 up to 0.23% (corresponding to 2 kg·m−3). The results obtained with eq 14-L-DER shown in Figure 15 reveal that the standard deviation of the compressibility calculated

of measurement uncertainties in the derivation procedure is taken from the standard deviation of the distribution. The number N of data sets was fixed to 5000 after carrying out various tests with different Monte Carlo trials. Figure 12 shows,

Figure 12. Influence of the number N of trials for the Monte Carlo procedure used to determine the isothermal compressibility kT, and its standard deviation σκ obtained with eq 14-L-DER at T = 293.15 K and atmospheric pressure. Blue ●, isothermal compressibility and red ●, standard deviation.

as an example, the evolution of both the mean and the standard deviation of the distribution as a function of N. It can be observed that both these quantities stabilize beyond N = 3000. The standard deviations obtained by this procedure were determined for each equation, and the ratio of the RSD% between the output and the input data RSD(κT)/RSD(ρ) were calculated in order to quantify the propagation of the probability distribution function through the numerical derivation procedure. The results are plotted in Figure 13 as a function of pressure for four different equations, eq 3-MS2-P, eq 9-PO3-P, eq 11HUD-P, and eq 14-L-DER, as examples. Examination of this figure reveals the same trend for all equations with a minimum L

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Table 6. Comparison of the RSD% at 0.1, 50, and 100 MPa Obtained for the Isothermal Compressibility from Different Experimental Pressure Steps 2 MPa steps equation name eq eq eq eq eq eq eq eq eq

3-MS2-P 9-PO3-P 11-HUD-P 13-L-TBM 14-L-DER 15-Q-SBM 16-Q-TBM 18-Q-DER 19-R-DER

5 MPa steps

10 + 3 × 2 MPa steps

10 MPa steps

0.1 MPa

50 MPa

100 MPa

0.1 MPa

50 MPa

100 MPa

0.1 MPa

50 MPa

100 MPa

0.1 MPa

50 MPa

100 MPa

1.3 2.2 0.6 0.6 0.6 1.2 0.5 0.6 0.5

0.2 0.3 0.1 0.1 0.1 0.3 0.1 0.2 0.1

0.5 1.0 0.4 0.4 0.4 1.3 0.5 0.5 0.5

1.3 2.4 0.9 0.8 0.8 1.6 0.7 0.8 0.7

0.2 0.4 0.2 0.2 0.2 0.5 0.2 0.2 0.2

0.6 1.4 0.6 0.6 0.6 1.8 0.7 0.7 0.8

1.4 2.7 1.2 0.9 1.1 2.0 1.0 1.0 1.0

0.3 0.6 0.3 0.3 0.3 0.6 0.3 0.3 0.3

0.7 1.8 0.8 0.8 0.8 2.3 0.9 0.9 1.0

1.4 2.7 0.9 0.7 0.8 1.8 0.7 0.8 0.8

0.2 0.6 0.2 0.2 0.2 0.6 0.2 0.2 0.2

0.6 1.5 0.6 0.6 0.6 1.9 0.7 0.7 0.8

Figure 14. RSD% for the isothermal compressibility obtained with equation eq 14-L-DER at T = 293.15 K as a function of pressure p using different pressure steps. Blue ●, 2 MPa; red ⧫, 5 MPa; green ▲, 10 MPa; and black ○, 10 MPa + 3× 2 MPa.

Figure 16. RSD(κT) for the isothermal compressibility obtained with different equations as a function of the relative standard deviation RSD(ρ) of the input density. Red −+−, eq 9-PO3-P; red −■−, eq 15Q-SBM; red −×−, eq 3-MS2-P; black −△−, eq 11-HUD-P; green −, eq 14-L-DER; red −−−, eq 18-Q-DER; blue −·−, eq 16-Q-SBM; black −◊−, eq 19-R-DER; and blue −●−, eq 13-L-TBM.

output directly proportional to the RSD of the input. Whatever the pressure, the same trend is observed. Only the value of slope changes from one pressure to another. Eq 13-L-TBM, which deviates below the linear trend, is the least sensitive to density perturbation during the fitting procedure. Therefore, this equation appears as the most suitable for deriving compressibility from volumetric measurement when uncertainty in density data becomes significant. Therefore, this equation could be selected for determining the compressibility and the standard uncertainty in the compressibility derivation uDer(κT,eq13) and ucomp(κT) could be obtained by adding both sources of uncertainty in quadrature as follows:

Figure 15. RSD% for the isothermal compressibility obtained with equation eq 14-L-DER at T = 293.15 K as a function of pressure p by considering different standard deviations σρ on input density data. Black , 0.15 kg·m−3; blue - - - -, 0.3 kg·m−3; red −·−·, 0.5 kg·m−3; black −●−, 1.0 kg·m−3; blue −◊−, 1.5 kg·m−3; and green −▲−, 2.0 kg·m−3.

2 2 2 ucomp (κT ) = uDer (κT,eq13) + uForm (κT)

(26)

However, it was preferred to evaluate ucomp(κT) in a single Monte Carlo procedure. To combine both sources of uncertainty, the six selected equations corresponding to type 1 in Table 4 were randomly used at each trial of the Monte Carlo method developed. As can be observed in Figure 17, the resulting compressibility distribution has a Gaussian shape that allows for estimating simultaneously the compressibility and the combined standard uncertainty in the compressibility computation ucomp(κT) from the mean and the standard deviation of the distribution, respectively. This computation procedure based on the random use of six different equations, i.e., eq 11HUD-P, eq 13-L-TBM, eq 14-L-DER, eq 16-Q-TBM, eq 18-Q-

increases greatly as the standard deviation of input data increases. Here again, this trend is magnified at both ends of the experimental pressure range. By plotting in Figure 16 the RSD of the output compressibility data as a function of the RSD of the input density data at atmospheric pressure, it can be seen that equations eq 3-MS2-P, eq 9-PO3-P, and eq 15-Q-SBM exhibit a huge increase of RSD(κT) with RSD(ρ). Consequently, these three equations do not appear to be suited to deriving compressibility, and they must be removed from the computation procedure. Apart from eq 13-L-TBM, all of the remaining equations display a linear response with a RSD of the M

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measured the isothermal compressibility of toluene at atmospheric pressure using an ultracentrifugation technique. The deviations of the literature data from the values reported in this work are listed in Table 8 and shown in Figure 18. Chain-dotted lines corresponding to ± Uc (κT) were added in the figure so as to help in the comparison between literature and our data within their expended uncertainty. Literature data are scattered and spread from −2.5 to 4.5%. The oldest values of Tyrer32 are systematically higher than other data. Apart from these data and those of Shraiber,37 and Jacobson,34 most of the literature data agree with our values within ± Uc (κT). Our data agree best with the measurement of Aicart et al.,39 Garbajosa et al.,40 Diaz Pena et al.,41 and Gamboa et al.43 carried out by a direct piezometric technique with a claimed uncertainty of 0.5%. A percentage deviation of 0.44 is observed on average with all of these data. Four isothermal compressibility data sets on compressed liquid toluene have been reported in the literature. Easteal and Wooff45 determined isothermal compressibility up to 275 MPa using volumetric measurements carried out with a bellows volutometer. Sun et al.46 obtained compressibility data from the speed of sound measurement carried out up to 260 MPa in an extended temperature range. Finally, Zéberg-Mikkelsen et al.47 and Moravkova et al.48 reported data obtained from the differentiation of density data measured using a vibrating Utube density meter up to 40 MPa. Unfortunately, none of these high pressure works gave the uncertainty associated with their determination. Comparison of these data with the values given here are summarized in Table 9 and plotted as a function of pressure in Figure 19 at a given temperature as an example. Also reported in Table 9 and shown in Figure 19 are the deviations with the correlations of McLinden and Splett,17 Cibulka and Takagi,18 Assael et al.,19 and Lemmon and Span.20 As in Figure 18, the magnitude of the expended uncertainty is represented by chain-dotted curves. Among the literature data sets, the data of Sun et al.46 and Zeberg-Michelsen and Andersen47 agree with ours within the expended uncertainty. In Figure 19, it can be noted that the data of Moravkova et al.48 are systematically lower than ours. The Easteal and Wooff45 data at ambient pressure agree much better with our data than the data at the higher pressure, where the deviation reaches 6%. The comparison of the correlations shows that the equation of state developed by Lemmon and Span20 is in agreement with the values reported here within the entire region of investigation. Larger deviations are observed with other correlations at higher pressures. The derivation of the very precise 20-parameter equation leads to isothermal compressibility values in very good agreement with the data reported here above 1 MPa, but a significant deviation is observed at atmospheric pressure with a maximum deviation of 7%. This result confirms the difficulty to represent the pressure influence on density at the lower pressure of the experimental domain. Whatever the pressure, it can be observed that a broad spread of literature data makes the comparison very difficult. However, the positive or negative nature of the deviation plotted in Figure 19 shows that our data, used for the baseline, are located between the different literature sets. Finally, by considering for each set of literature data an uncertainty similar to those estimated here, it can be seen that our confidence interval overlaps all literature intervals with a confidence level of 95%, with the exception of Easteal and Wooff45 data at a higher pressure. All of these comparisons show the suitability of the

Figure 17. Frequency histogram at T = 323.15 K and P = 50 MPa of the isothermal compressibility κT distributions obtained from the computation method retained and from the six equations considered in the procedure. Blue ●, computed distribution obtained by using randomly six different equations; −·−, normal distribution; red −−⧫−−, eq 11-HUD-P; black +, eq 13-L-TBM; red −○−, eq 14-L-DER; green −△−, eq 16-Q-SBM; black −−▲−−, eq 18-QDER; and blue −□−, eq 19-R-DER.

DER, and eq 19-R-DER, was used for differentiating the density data measured every 2 MPa at the extremities and 10 MPa in the rest of the domain at 293.15 K, as well as all of the other experimental temperatures. The results are summarized in Table 7. The expanded uncertainty Uc (κT) given in the table was obtained by considering a coverage factor of k = 2 and by adding (in quadrature) to ucomp(κT) the uncertainty usp(κT) caused by sample impurities. In order to remain consistent with the calculation of the uncertainty in density, this last contribution was estimated from the same type of relation as eq 10. Toluene was chosen for this study because its volumetric properties have been investigated by many groups, and there are many available data of compressibility at ambient pressure in the open literature.31−44 Moreover, some high pressure data already exist in the temperature range investigated.45−48 Finally, several correlations17−20 have been developed for representing density as a function of pressure, and therefore they allow for calculating compressibility over an extended range of pressure. Thirteen sources of data have been used for comparison at atmospheric pressure. These data were mainly obtained by three different techniques. Shraiber,37 Aicart et al.,39 Garbajosa et al.,40 Diaz Pena et al.,41 and Gamboa et al.43 reported data obtained by direct measurement of the change in volume with respect to pressure at fixed temperature using piezometers. Freyer et al.,33 Jacobson,34 Rajagopal et al.,36 Reddy et al.,38 Tamura et al.,42 and Malakondaiah et al.44 determined the isothermal compressibility using an acoustic technique. This method consists of the measuring speed of sound and density in order to determine first the isentropic compressibility using eq 1. The isothermal compressibility is then derived from eq 2 by using additional measurements of heat capacity and isobaric expansion. Tyrer32 employed a hybrid method that consisted of using a piezometer to measure the isentropic compressibility. From this measurement, the isothermal compressibility was then deduced through eq 2. Finally, Richard and Rogers35 N

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Table 7. Values of Compressibility κT at Temperatures T and Pressures p Determined in Liquid Toluene by the Proposed Computation Method P (MPa)

T (K)

κT (GPa−1)

Uc (κT) (GPa−1)

T (K)

κT (GPa−1)

Uc (κT) (GPa−1)

T (K)

κT (GPa−1)

Uc (κT) (GPa−1)

0.1013 2 4 6 10 20 30 40 50 60 70 80 90 94 96 98 100 0.1013 2 4 6 10 20 30 40 50 60 70 80 90 94 96 98 100

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15

0.878 0.863 0.847 0.833 0.805 0.744 0.692 0.648 0.609 0.575 0.546 0.519 0.495 0.486 0.482 0.478 0.474 1.084 1.062 1.040 1.019 0.979 0.892 0.821 0.760 0.708 0.663 0.624 0.589 0.558 0.547 0.541 0.536 0.531

0.017 0.015 0.014 0.012 0.010 0.006 0.004 0.003 0.003 0.004 0.004 0.005 0.006 0.007 0.007 0.007 0.008 0.020 0.018 0.016 0.014 0.011 0.006 0.004 0.003 0.003 0.004 0.004 0.005 0.006 0.007 0.007 0.007 0.007

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15

0.942 0.925 0.908 0.891 0.860 0.792 0.734 0.684 0.641 0.604 0.570 0.541 0.514 0.504 0.499 0.495 0.490 1.177 1.150 1.124 1.099 1.052 0.951 0.869 0.801 0.743 0.694 0.651 0.613 0.580 0.568 0.562 0.556 0.551

0.018 0.016 0.014 0.013 0.010 0.006 0.004 0.003 0.003 0.004 0.004 0.005 0.006 0.007 0.007 0.007 0.007 0.022 0.020 0.017 0.015 0.011 0.006 0.005 0.004 0.004 0.004 0.004 0.006 0.007 0.008 0.008 0.008 0.009

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15

1.013 0.993 0.973 0.954 0.919 0.841 0.776 0.720 0.673 0.632 0.596 0.564 0.535 0.525 0.519 0.514 0.509 1.276 1.245 1.214 1.185 1.130 1.015 0.922 0.845 0.781 0.727 0.679 0.638 0.602 0.589 0.583 0.576 0.570

0.018 0.016 0.015 0.013 0.011 0.006 0.004 0.003 0.003 0.004 0.004 0.005 0.007 0.007 0.007 0.008 0.008 0.026 0.022 0.019 0.016 0.012 0.006 0.005 0.004 0.004 0.004 0.005 0.006 0.007 0.008 0.008 0.008 0.009

critical temperatures. The computation procedure can be either extended to higher pressures or to critical conditions. However, the extension to higher pressures involves the equation of state with more parameters that can generate perturbations at the lower pressure. It thus appears to be preferable in this case to split the fitting pressure range in two parts that overlap: the low pressure domain from 0.1 to 100 MPa and the higher domain (beyond 80 MPa). In the same way, the extension to the critical conditions requires changing the equations of state so as to ensure the representation of the high compressibility of fluids in these conditions. Finally, the different tests performed in this work have suggested to use 17 experimental density measurements for describing an isotherm between 0.1 and 100 MPa; however, it may be required for some applications to carry out fewer measurements. In this case, it is recommended to limit the equations of state to equations involving only 3 parameters.

Table 8. Deviations of the Compressibility Data Reported in Table 7 from Literature Data at Atmospheric Pressure ref

technique

32 33 34 35 36 37 38 39 40 41 42 43 44

piezometric (ks) acoustic acoustic ultracentrifugation acoustic piezometric acoustic piezometric piezometric piezometric acoustic piezometric acousticc

comparison range 293−343 293−323 293 K 293 K 298−323 293−343 298 K 298−333 298−333 298−333 293−303 298−333 298−313

AAD%a

MD%b

3.4 0.8 2.4 1.9 0.5 1.6 1.7 0.5 0.5 0.4 1.4 0.5 0.5

4 1.9 2.4 1.9 0.7 3.9 1.7 1.0 0.8 0.9 1.7 0.9 0.5

K K

K K K K K K K K

a AAD%: Relative average absolute deviation. maximum deviation.

b

■

MD%: Relative

CONCLUSION A computation procedure that allows for determining the liquid compressibility from density measurements has been set out in this article. The ability of 19 empirical liquid equations of state were first tested to both fit density data and represent the isothermal compressibility. Although all of the equations investigated had the capacity to fit the density data with a

proposed method for determining isothermal compressibility of liquids under pressure. The method was developed for hydrocarbon liquids between 0.1 and 100 MPa and for temperature conditions far from the O

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Figure 18. Deviation 100(kT,lit − kT,exp)/kT,exp of the isothermal compressibility of the literature kT,lit from the experimental compressibility kT,exp reported in Table 7 as a function of temperature T at atmospheric pressure. Green ▲, Tyrer;32 red ■, Freyer et al.;33 black −, Jacobson;34 black *, Richard and Rogers;35 red ◊, Rajagopal et al.;36 yellow ⧫, Shraiber;37 black □, Reddy et al.;38 black ●, Aicart et al.;39 blue +, Garbajosa et al.;40 black ○, Diaz Pena et al.;41 blue △, Tamura et al.;42 red ×, Gamboa et al.;43 blue −, Malakondaiah et al.;44 and black −·−·−, expanded uncertainty.

Figure 19. Deviation 100(kT,lit − kT,exp)/kT,exp of the isothermal compressibility of the literature k T,lit from the experimental compressibility kT,exp reported in Table 7 at T = 303 K as a function of pressure p. Black ●, Easteal and Wooff;45 black ○, Sun et al.;46 blue +, Zéberg-Mikkelsen;47 green ▲, Moravkova et al.;48 red ×, McLinden and Splett;17 yellow ⧫, Cibulka and Takagi;18 red ■, Assael et al.;19 blue △, Lemmon and Span;20 and −·−·−, expanded uncertainty.

experimental measurements by considering a normal probability distribution function; • to consider randomly 6 different equations with 3 and 4 parameters (eq 11-HUD-P, eq 13-L-TBM, eq 14-L-DER, eq 16-Q-TBM, eq 18-Q-DER, and eq 19-R-DER) for fitting the perturbed data point in the Monte Carlo procedure and thus to calculate the derivative of density with respect to pressure for each trial; • to calculate the mean and the standard deviation of the resulting compressibility distribution to evaluate both the compressibility and its uncertainty. The expanded uncertainty of compressibility determined by the proposed method is on the order of 1% in the middle part of the experimental domain 20−80 MPa and 2% at both extremities of the domain. The method proposed here for deriving density data with respect to pressure could be used in the same way to evaluate the density derivative with respect to temperature and therefore to determine the isobaric expansion of liquids under pressure.

Table 9. Deviations of the Compressibility Data Reported in Table 7 from Literature Data under Pressure

a

ref

technique

comparison range

45

density derivation

46

acoustic

47

density derivation

48

density derivation

17

20-parameter eq

17

20-parameter eq

18

correlation

19

correlation

20

correlation

293−323 K 0.1−100 MPa 293−313 K 0.1−100 MPa 293−353 K 5−40 MPa 298−323 K 0.1−40 MPa 293−343 K 0.1 MPa 293−343 K 2−30 MPa 293−343 K 0.1−100 MPa 293−343 K 0.1−100 MPa 293−343 K 0.1−100 MPa

AAD%: Relative average absolute deviation. maximum deviation.

AAD%a

b

MD%b

3

5.9

0.8

1.5

0.4

1.1

1.6

3.2

5.8

7.3

0.8

1.7

0.9

1.9

1.3

2.4

0.8

2

■

AUTHOR INFORMATION

Corresponding Author

*[email protected].

MD%: Relative

ORCID

Jean-Luc Daridon: 0000-0002-0522-0075 Notes

The authors declare no competing financial interest.

■

maximum deviation lower than the experimental uncertainty, only six equations were able to represent the compressibility suitably at both extremes of the experimental domain. For determining compressibility between 0.1 and 100 MPa with suitability it was proposed: • to carry out 17 density measurements evenly distributed throughout the pressure range, every 2 MPa steps close to extremities and every 10 MPa steps beyond; • to use a Monte Carlo method with 5000 trials in which the density data are randomly perturbed around the

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Q

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