The pressure and temperature dependence of volume and viscosity of four Diesel fuels

Fuel 135 (2014) 112–119

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The pressure and temperature dependence of volume and viscosity of four Diesel fuels Scott Bair ⇑ Georgia Institute of Technology, Center for High-Pressure Rheology, George W. Woodruff School of Mechanical Engineering, Atlanta, GA 30332-0405, United States

h i g h l i g h t s  The volume and viscosity of 4 Diesel fuels were measured to 350 MPa and to 160 °C.  The Tait and Murnaghan equations represented the pVT response.  The improved Yasutomi model did not fit the data well.  The Doolittle free-volume equation was accurate only for one fuel.  The viscosities lie across a transition in the response of viscosity to T and p.

a r t i c l e

i n f o

Article history: Received 7 May 2014 Received in revised form 9 June 2014 Accepted 12 June 2014 Available online 2 July 2014 Keywords: Diesel fuels Density Viscosity Compressibility Bulk modulus

a b s t r a c t The relative volumes and the viscosities of four Diesel fuels have been measured experimentally to pressures up to 350 MPa and temperatures to 160 °C. The experimental liquids were an extra low viscosity reference fuel, an ethanol based blend, neat biodiesel and 20% biodiesel. The Tait and Murnaghan equations of state represented the pressure–volume–temperature response equally well. The improved Yasutomi model for supercooling liquids, which accurately represents the temperature and pressure dependence of the viscosity of lubricating oils, did not fit the data well except for neat biodiesel. Surprisingly, the Doolittle free-volume equation was accurate only for the low viscosity reference fuel. The reason for the correlation difficulties may be illuminated by the behavior of the thermodynamic scaling of the viscosities. The Stickel analysis of the normalized Ashurst–Hoover parameter indicates that, for all liquids except the reference fuel, the measured viscosities lie across the transition in the response of viscosity to temperature and pressure. Consequently, only the comprehensive normalized Ashurst–Hoover scaling model successfully fits all data. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The compressibility of Diesel fuels affects the injection timing in pump-line-nozzle injection systems. The more compressible fuel will build pressure more slowly and be injected later [1,2]. Common-rail injection does not share this sensitivity to bulk modulus. However, compressibility and viscosity at high pressure influence atomization [3,4] in common-rail injection. There have been many recent experimental characterizations of the high pressure viscosity of biodiesels [4,5] and suggested correlations. Duncan and coworkers [5] employed a Tait–Litovitz model with six parameters to correlate the viscosity of biodiesel with temperature and pressure to 140 MPa. Freitas and coworkers [6]

⇑ Tel.: +1 404 894 3273. E-mail address: [email protected] http://dx.doi.org/10.1016/j.fuel.2014.06.035 0016-2361/Ó 2014 Elsevier Ltd. All rights reserved.

described the viscosities of mixtures of diesel and bioediesel with the Grundberg–Nissan model. Pumps and injectors are protected from wear by the hydrodynamic lubrication which is provided by the fuel viscosity [7]. Therefore, the viscosity at the pressures of a tribological contact is important. An experimental measurement program was undertaken in support of modeling of the cavitation in Diesel fuel injection. The conditions for measurements were initially pressures of 0.1, 50, 100, 150, 200, 250 and 350 MPa and temperatures of 40, 80 and 150 °C. It became necessary to modify this program due to the physical nature of some of the samples. The experimental materials are a series of four fuels provided by Cummins. They are (1) 16A ELV, Viscor Calibration Fluid, 0.910 mm2/s at 20 °C, density 0.7523 g/cm3 at 20 °C, low viscosity version of 16 A. (2) ED95, ethanol 95% blend, remaining additives from SEKALB BioFuels & Chemicals AB, 811 kg/m3 at 15 °C.

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S. Bair / Fuel 135 (2014) 112–119

(3) B20, Chevron Phillips Chemical, biodiesel, 20% fatty acid methyl ester, remaining regular Diesel fuel. (4) B100, Integrity Biofuels, biodiesel fatty acid methyl ester.

3.0%

The relative volume was measured with a metal bellows piezometer [8]. The pressure–volume–temperature data were utilized in two equations of state, Tait and Murnaghan [8]. The low shear viscosity was measured using a falling cylinder viscometer [9]. This viscometer has been described many times and is ordinarily employed for measurement of the various pressure–viscosity coefficients of lubricating oils. It has been recently used to characterize the temperature and pressure dependence of the refrigerant R134a [10]. The equation of state along with the viscosity were employed to calculate the interaction parameter in a thermodynamic scaling parameter [11], which was incorporated into an accurate temperature and pressure correlation of viscosity. In addition, two more viscosity correlations were applied to the data, the Doolittle freevolume model [8] and the improved Yasutomi [12] model, with limited success.

1.0%

2. Volume measurements 2.1. LVDT-based piezometer

Deviaon from NIST, 2012 for nonane

Deviaon

2.0%

Pressure / MPa

0.0%

0

50

100

150

200

250

300

- 1.0%

350 313K 353K

- 2.0%

423K

- 3.0% Fig. 2. Test of the piezometer calibration with n-nonane.

p ¼ 0Þ, of the fuels referenced to 40 °C and atmospheric pressure are listed in the tables (see Tables 1–4). The relative volumes of all samples are compared in Fig. 3 at 80 °C. The biodiesels are clearly less compressible than the reference fuel or the ethanol blend. 2.4. The equations of state

The change in length of a metal bellows can be measured in an LVDT (linear variable differential transformer) based falling body viscometer by detecting the total travel of a sinker within a closed container provided that one surface which arrests the sinker is attached to the movable end of the bellows. The relative volume, V=V R ; V R ¼ VðT ¼ T R ; p ¼ 0Þ is reported with uncertainty of 0.1%. The sample volume may not be less than 0.84 of the volume captured when the sample was installed at ambient temperature and pressure. Also, the volume must not exceed 1.105 times the volume captured when the sample was installed or the bellows will be permanently deformed. There is also a viscosity limit which has not yet been reached for the LVDT-based instrument.

Here, K 0 is the isothermal bulk modulus, K, at ambient pressure or p ¼ 0 and K 00 is the pressure rate of change of K at ambient pressure or p ¼ 0. The isothermal bulk modulus, KðT; pÞ ¼ qj@p=@ qjT¼const ¼ Vj@p=@VjT¼const for this model is

2.2. Calibrations

K¼

The accuracy for the determination of relative volume was tested using the NIST web book values for water and n-nonane [13]. The deviations from NIST are shown in the plots below (see Figs. 1 and 2).

On the other hand, in some cases for which the Tait form becomes overly soft at high pressure, the Murnaghan EoS [8] is preferred

2.3. Volume measurements Relatively low boiling temperatures of 16A ELV and ED95 required that a different schedule of temperatures and pressures be adopted. The relative volumes, V=V R ¼ VðT; pÞ=VðT R ¼ 40  C;

3.0%

 1

  1 p ð1 þ K 00 Þ ½K 0 þ pð1 þ K 00 Þ 0 ln 1 þ K0 1 þ K0

ð2Þ

   K10 V q0 K 00 0 ¼ ¼ 1þ p V0 q K0

ð3Þ

The isothermal bulk modulus for this model is K ¼ K 0 þ K 00 p. For any isothermal form, K 0 ðTÞ can be described by

K 0 ¼ K 00 expðbK TÞ

16A ELV

0.0% 0 -1.0%

ð1Þ

ð4Þ

Table 1 Relative volume of 16A ELV, V=V R ¼ VðT; pÞ=VðT R ¼ 40  C;p ¼ 0Þ.

1.0%

Deviaon

  V 1 p q 0 ¼ 0 ¼1 ln 1 þ ð1 þ K Þ 0 V0 K0 q 1 þ K 00

Temperature is absolute in Eq. (4). The volume at ambient pressure, V 0 , must also vary with temperature and it is generally held that a linear dependence of ambient pressure density on

Deviaon from NIST for water

2.0%

Of the simple isothermal equations of state (EoS) that can be written as V ¼ VðpÞ, the Tait equation [8] is often held to be the most accurate and even accurate for extrapolation to very high pressures.

50

100

150

200

250

300

Pressure / MPa 296K 323K

-2.0%

363K -3.0% Fig. 1. Test of the piezometer calibration with water.

350

V/VR

p/MPa

T = 40 °C

T = 80 °C

T = 105 °C

0 50 100 150 200 250 300 350

1.0000 0.9489 0.9160 0.8920 0.8717 0.8566

1.0500 0.9811 0.9420 0.9155 0.8937 0.8757 0.8603

1.0829

T = 150 °C 1.0397 0.9872 0.9521 0.9270 0.9062 0.8735

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S. Bair / Fuel 135 (2014) 112–119

Table 2 Relative volume of ED95, V=V R ¼ VðT; pÞ=VðT R ¼ 40  C; p ¼ 0Þ. ED95

V/VR

p/MPa

T = 26 °C

T = 40 °C

0 8 50 100 150 200 250 350

0.9868

1.0000

T = 80 °C 1.0323 0.9867 0.9516 0.9258 0.9052 0.8868

0.9511 0.9226 0.9001 0.8813 0.8655

T = 150 °C

1.0628 1.0102 0.9746 0.9476 0.9253 0.8919

Table 3 Relative volume of B20, V=V R ¼ VðT; pÞ=VðT R ¼ 40  C; p ¼ 0Þ. B20

V/VR

p/MPa

T = 40 °C

T = 80 °C

T = 150 °C

0 50 100 150 200 250 350

1.0000 0.9650 0.9406 0.9211 0.9048 0.8922 0.8694

1.0351 0.9919 0.9633 0.9413 0.9236 0.9084 0.8850

1.1026 1.0362 0.9991 0.9722 0.9509 0.9339 0.9074

Table 4 Relative volume of B100, V=V R ¼ VðT; pÞ=VðT R ¼ 40  C; p ¼ 0Þ. B100

V/VR

p/MPa

T = 40 °C

T = 80 °C

T = 150 °C

0 50 100 150 200 250 350

1.0000 0.9665 0.9418 0.9225 0.9061 0.8911

1.0364 0.9953 0.9662 0.9438 0.9261 0.9104 0.8839

1.0990 1.0407 1.0055 0.9780 0.9564 0.9385 0.9109

1.1

Tait at 80°C

1.05 ED95

Relave Volume

V0 ¼ 1 þ aV ðT  T R Þ VR

ð5Þ

Here V R is the reference volume at the reference temperature, T R ¼ 40 °C and ambient pressure. The parameters were fitted to the measured relative volumes using both EoS. The relative standard deviation was 0.06–0.34% with similar results for each equation, although the average was slightly smaller for the Tait EoS. The parameters are given in Tables 5 and 6 for Tait and Murnaghan respectively. Also reported is the P 1=2 standard deviation of relative volume, ½ ðV corr =V meas  1Þ2 =N . The Murnaghan and Tait forms are compared in Fig. 4 where it is clear that for these pressures the two forms give the same result. The bulk modulus, KðT; pÞ, may be readily calculated. This has been done using the Murnaghan EoS for 100 MPa and plotted versus temperature in Fig. 5. The correlation for Number 2 Diesel has been taken from previous work [14]. 2.5. Problems associated with ethanol The poorer fit of the equations of state to the volumes for ED95 may be caused by the hydrogen bonding of the ethanol which results in property relations atypical of organic liquids. An example of such atypical behavior is water. However, the measurement conditions approach the vapor dome for ethanol. The pressure/volume isotherm for the critical temperature, 214 °C, must curve to become parallel to the volume axis at the critical density [15]. In other words, the compressibility is infinite at the critical point. This affects any isotherms as they approach the vapor dome. To investigate further, the Tait equation for ED95 is compared with equations of state from references [16,17] for pure ethanol in Fig. 6. The agreement is quite good for all conditions except for 150 °C at low pressure. The lowest pressure for the present measurements at 150 °C is 50 MPa. It seems unlikely that a simple EoS for compressed liquids will be accurate for ED95 under these high temperature, low pressure conditions and that more data will be required for conditions approaching the coexistence curve to parameterize the more appropriate correlations. Or, the equation of state for ethanol could be substituted with little error at pressures to 100 MPa on the basis of Fig. 6.

Table 5 Parameters of the Tait equation of state.

16A ELV

1

temperature is accurate; however such a relationship will often lead to crossing of isotherms at very high pressure. To avoid this difficulty the ambient pressure volume is made linear with temperature.

B20

Sample

16A ELV

ED95

B20

B100

B100

K 00 K 00 =GPa

9.677 6.267 0.006894

8.353 3.838 0.004680

10.700 7.438 0.005971

9.437 5.643 0.004829

1.260E03

1.190E03

9.217E04

8.966E04

0.11%

0.33%

0.06%

0.06%

bK =K 1

0.95

aV =K 1 Std. Dev.

0.9 Table 6 Parameters of the Murnaghan equation of state.

0.85

0.8

0

100

200

300

400

Pressure / MPa Fig. 3. Comparing the samples at 80 °C for the Tait equation.

Sample

16A ELV

ED95

B20

B100

K 00 K 00 =GPa

8.458 6.703 0.006932

7.323 3.852 0.004552

9.718 7.743 0.005997

8.583 5.821 0.004843

1.260E03

1.176E03

9.217E04

8.964E04

0.13%

0.34%

0.06%

0.06%

bK =K 1 aV =K 1 Std. Dev.

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S. Bair / Fuel 135 (2014) 112–119

1.15

1.2

B20

ED95 at 313K ED95 at 353K

1.15

1.1

ED95 at 423K Tait 313K

Ethanol by Sauermann et al.

Tait 423K Murn 313K

Relave Volume

Relave Volume

1.1

Tait 353K

1.05

Murn 353K

1

Murn 423K

0.95

Ethanol by Dillon & Penoncello

1.05

1

0.95

0.9

0.9

0.85 0.85 0.8 0.8

0

100

200

300

20%

p=100 MPa

200

DEHS 97% Deviaon from Paredas et al., 2012

15%

16A ELV ED95

313K

Deviaon

10%

B20 B100

2

150

Pressure / MPa

2.6

2.2

100

Fig. 6. The pressure dependence of the relative volume of ED95 compared with that of ethanol as reported in [16,17].

Fig. 4. Comparing the Tait and Murnaghan equations.

Bulk Modulus at 100 MPa / GPa

50

400

Pressure / MPa

2.4

0

No.2 Diesel

343K

398K

5% 0% 0

1.8

50

100

150

Pressure / MPa

-5% 1.6 -10% 1.4

Fig. 7. Calibration check using the viscosity correlation of [18].

1.2 1

0

50

100

150

200

250

Temperature /°C Fig. 5. Comparing the bulk moduli of the samples at p = 100 MPa for the Murnaghan equation.

All four samples are compared at 80 °C in Fig. 8. The viscosity of Number 2 Diesel has been added to Fig. 8 for comparison. The viscosity relation for Number 2 Diesel was obtained from the Doolittle equation parameterized in [14]. 4. Viscosity correlations 4.1. The improved Yasutomi correlation

3. Viscosity measurements A falling cylinder viscometer, which has been used in many studies [9] of the effect of pressure and temperature on viscosity, was used with a sinker which applies a shear stress of 1.2 Pa. The calibration of this sinker has been validated many times before. A recent comparison with the correlation of Paredas et al. [18] for di (2ethylhexyl) sebacate (DEHS) is shown in Fig. 7. The viscosities are listed in Tables 7–10. Thixotropy, timedependent viscosity, was observed for B100 at 40 °C and 350 MPa. The solid structure ‘‘melted’’ as the pressure was reduced to 290 MPa.

It is desirable to have a correlation for the viscosity to estimate the viscosity at temperatures and pressures other than those selected for measurements. The improved Yasutomi correlation [12] is capable of reproducing the inflection in the log(viscosity) versus pressure response of supercooled liquids such as lubricating oils. The Yasutomi model is a pressure modification of the Williams–Landel–Ferry model for temperature dependence

l ¼ lg exp



2:303C 1 ðT  T g ÞF C 2 þ ðT  T g ÞF

 ð6Þ

where T g ðpÞ is the glass transition temperature which varies with pressure as T g ¼ T g0 þ A1 lnð1 þ A2 pÞ. The dimensionless relative

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S. Bair / Fuel 135 (2014) 112–119

Table 7 Viscosity of 16A ELV.

25

80°C

16A ELV 40 Viscosity/mPa s

80

107

0 50 100 150 200 250 350

0.545 0.897 1.317 1.89 2.58 3.44 6.06

0.370 0.590 0.844 1.155 1.53 1.95 3.15

0.282 0.456 0.662

150

16A ELV

20

ED95 B20

0.352 0.495 0.657 0.834 1.043 1.520

Table 8 Viscosity of ED95.

Viscosity / mPas

T/°C p/MPa

B100

15

No.2 Diesel

10

5

ED95 T/°C p/MPa

22 Viscosity/mPa s

40

0 10 50 100 150 200 250 350

1.748

1.217

2.33 2.93 3.64

80

150

0 0.629 0.801 0.994 1.192 1.373 1.608 2.09

1.579 1.97 2.36 2.87 3.39 4.57

0.252 0.336 0.425 0.506 0.595 0.694 0.864

0

100

200

300

400

Pressure / MPa Fig. 8. Comparing all samples at 80 °C.

Table 11 Parameters of the improved Yasutomi model.

Table 9 Viscosity of B20. B20 T/°C p/MPa

40 Viscosity/mPa s

80

150

0 50 100 150 200 250 350

2.41 4.33 7.44 12.08 19.1 30.3 72.7

1.170 1.98 3.10 4.62 6.66 9.73 18.91

0.552 0.920 1.316 1.791 2.40 3.19 5.17

Table 10 Viscosity of B100.

lubricant viscosity. Values shown in red indicate non-physical behavior, glass temperature less than absolute zero. In fact, the fits to some data, especially at 150 °C, were very poor with some deviations of about 10%, except for B100 which was acceptable. This may not be surprising since these are not all supercooling liquids. 4.2. The Doolittle free volume correlation

B100 T/°C p/MPa

40 Viscosity/mPa s

80

150

0 50 100 150 200 250 350

3.53 5.92 9.31 13.91 20.3 29.5 Thixotropic

1.737 2.81 4.12 5.70 7.76 10.45 18.12

0.794 1.254 1.768 2.37 3.03 3.83 5.77

thermal expansivity of the free volume, FðpÞ, is given by a new b empirical expression, F ¼ ð1 þ b1 pÞ 2 . Due to the relative importance of the low pressures, the data at 0.1, 50 and 100 MPa were weighted at times 16, times 4 and times 2, respectively. The glass viscosity, lg , was assumed to have a universal value of 1012 Pa s. The other parameters are listed in Table 11 along with P the standard deviation of the relative viscosity, ½ ðlcorr =lmeas  1Þ2 =N1=2 . Experience has shown that the standard deviation of the relative viscosity may be expected to be no more than 1% per decade of viscosity data for the most accurate correlations of

The most widely used model for the variation of the low-shear viscosity with temperature and high pressure has been the free volume model, using the Doolittle equation for the higher molecular mass liquids.

"

l ¼ lR exp B

V 1R VR

V1 V 1R V  VV1R VR



1 1  VV1R R

!# ð7Þ

It is assumed that the occupied volume, V 1 , depends linearly on temperature and is independent of pressure. The occupied volume has a value of V 1R at the reference state.

V1 ¼ 1 þ eðT  T R Þ V 1R

ð8Þ

For convenience, we write R0 ¼ V 1R =V R . Here, V=V R is obtained from the Tait equation (1) above. The parameters are given in Table 12. The fits to data are satisfactory for only 16A ELV. For ED95 the lowest pressure at 150 °C was excluded due the problem of representing volume at that condition. In general, an accurate correlation can reduce the standard deviation to about 1 percent per decade of viscosity.

117

S. Bair / Fuel 135 (2014) 112–119 Table 12 Parameters of the Doolittle model.

Table 13 Parameters of the Bair and Laesecke model.

Sample

16A ELV

ED95

B20

B100

Sample

16A ELV

ED95

B20

B100

lR =mPa s

0.5581 2.1197 0.6006 2.838E04 2.2%

1.1675 9.0785 0.3193 1.538E03 5.6%

2.3088 1.5166 0.72095 4.619E04 8.8%

3.3860 2.4009 0.65201 5.215E04 9.8%

A=mPa s B C q Q g Std. Dev.

0.09309 1.789 0 0.4758 – 9.436 2.7%

5.204E05 10.024 0.01669 0.3440 8.438 1.712 2.6%

0.02846 3.677 0.7694 0.3226 1.7715 5.657 2.9%

0.010015 5.341 0.5316 0.2922 1.9435 5.051 3.5%

B R0

e Std. Dev.

An attempt was made to fit the Batchinsky [19] free volume form to the viscosities of A16 ELV.

ð9Þ

1  R0 VV1R

V VR

However, the fit was only satisfactory when viscosities greater than 1 mPa s were excluded. 4.3. Thermodynamic scaling A thermodynamic scaling rule that has been found to be accurate for many organic liquids is l ¼ FðTV g Þ, where 3g can be related to the exponent of the repulsive intermolecular potential [10]. The Ashurst–Hoover scaling parameter can therefore be written as

u¼



T TR



V VR

g





BF u1 u  u1

d ln l dð1=uÞ

 ð11Þ

12 ð12Þ

This /-Stickel function has been plotted for the measured viscosities as the points for B20 in Fig. 9. Pressure increases to the right, temperature to the left. There is a local maximum in Sð1=uÞ which cannot be represented by the Vogel-like relation. A more general form of lðuÞ is 0.7

S (1/φ)

0.5

B20

0.3 0.2 0.1

0

100

313K

353K

423K

B&L Model

1st branch

2nd branch

1

200

313K

353K

423K

B&L Model

Doolile Model

Yasutomi Model

300

400

500

Pressure / MPa Fig. 10. Viscosity of B20 comparing models.

suggested by the scaling function introduced by Bair and Laesecke [21]. The parameters are given in Table 13.

l ¼ A exp½Buq þ C uQ ; 0 < q < 1; 1 < Q

ð13Þ

This relation can reproduce both the ascending and descending portions of the Stickel plot as shown by the curves representing the first branch, l ¼ A expðBuq Þ, and the second branch, l ¼ A expðC uQ Þ, in Fig. 9. This model has been plotted along with the Yasutomi and Doolittle models for B20 in Fig. 10 for comparison with the measured viscosities. The exceptionally low value of g for ED95, 1.71, is similar to the value of g = 1.45 for pure ethanol found in [22]. In the case of 16A ELV, no improvement was found by including the second powerlaw in the brackets of the above equation. That is to say, C ¼ 0. The resulting relation was introduced by Pensado and coworkers [23].

l ¼ A exp½Buq ; 0 < q < 1

ð14Þ

The /-Stickel plot for 16A ELV is shown in Fig. 11. Bair and Laesecke [21] discovered a normalization of the scaling parameter, bV ¼ ðV molecule =VÞg =T, which mapped the viscosity of three widely different liquids onto a master Stickel curve represented by a combination of two exponential power law terms, Eq. (12). They found that a maximum in the derivative bV  Stickel function occurred at a value of the volume-normalized Ashurst– Hoover scaling parameter 1  104 6 bV 6 7  104 K 1 . We may approximate V molecule  V 1 [21]. Then the new scaling parameter may be found from the old parameter.

0.6

0.4

B20

1.0

ð10Þ

However, this form, which was developed for supercooled liquids such as lubricating oils, results in the Stickel function for 1=u, Sð1=uÞ, decreasing in a linear fashion with 1=u.

Sð1=uÞ ¼

10.0

0.1

Here temperature is absolute and T R ¼ 313:15 K. Furthermore, an accurate scaling function for viscosities less than the dynamic crossover can be obtained from a Vogel-like form introduced by Bair and Casalini [20].

l ¼ l1 exp

Viscosity / mPa·s

l¼

100.0

1 b VV1R

10

1/φ Fig. 9. /-Stickel plot for B20, increasing pressure to the right, increasing temperature to the left.

bV 

Rg0 uT R

ð15Þ

The bV  Stickel function for all samples has been plotted in Fig. 12 where the maximum can be seen at bV  4  104 K 1 for the three liquids showing such behavior.

118

S. Bair / Fuel 135 (2014) 112–119

behavior illustrated in Fig. 12 explains the poor fit of the Doolittle model to the data for all samples except 16A ELV which lies entirely on the first branch. The measurements for the other materials lie across a transition in the response of viscosity to temperature and pressure. Lubricating oils will usually lie entirely to the right of peak in Fig. 12.

10

S(1/φ )

16A ELV

1

5. Conclusions

313K 353K 423K Pensado Model

0.1 0.1

1

10

1/φ Fig. 11. /-Stickel plot of the Pensado et al. [22] type response of 16A ELV, increasing pressure to the right, increasing temperature to the left.

0.016

[d(lnμ )/dβV]-1/2 / K1/2

0.014

β V-Sckel Plot 16A ELV ED95

0.012

B20

0.01

B100 Master 1st Term

0.008

(1) The relative volumes and the viscosities of four Diesel fuels have been measured experimentally to pressures up to 350 MPa and temperatures to 160 °C. (2) The Tait and Murnaghan equations of state represented the pressure–volume–temperature response equally well. (3) The improved Yasutomi model for supercooling liquids, which accurately represents the temperature and pressure dependence of the viscosity of lubricating oils, did not fit the data well except for B100. (4) Surprisingly, the Doolittle free-volume equation was accurate only for 16A ELV. (5) The reason for the correlation difficulties may be illuminated by the behavior of the thermodynamic scaling of the viscosities. The Stickel analysis of the normalized Ashurst–Hoover parameter indicates that, for all liquids but 16A ELV, the measured viscosities lie across the transition in the response of viscosity to temperature and pressure. Consequently, the Bair and Laesecke model successfully fits all data.

0.006

Acknowledgement 0.004

This work was funded by a research contract with Cummins Engine Company.

0.002 0 0.000001

0.00001

0.0001

0.001

βV / K-1 Fig. 12. bV-Stickel plot for all samples, increasing pressure to the right, increasing temperature to the left.

Table 14 Parameters of the Bair and Laesecke model with fixed q and Q. Sample

16A ELV

ED95

B20

B100

A=mPa s B C q Q g Std. Dev.

0.01963 3.344 0.02505 0.24 1.78 9.593 3.3%

1.225E06 13.608 0.1963 0.24 1.78 1.754 4.1%

0.009851 4.717 0.7923 0.24 1.78 5.651 4.7%

0.005357 5.821 0.6765 0.24 1.78 5.051 3.5%

SðbV Þ ¼

 1 d ln l 2 dbV

ð16Þ

The master curve from [21] for R134a and dimethyl pentane is plotted as the broken curve in Fig. 12. It was found in [21] that the viscosities of R134a, dimethyl pentane and squalane could all be represented with fair accuracy by q = 0.24 and Q = 1.78. These ‘‘universal’’ values when fixed in the regression of Eq. (13) with 1=bV substituted for u resulted in standard deviations of 3.3%, 4.1%, 4.7% and 3.5% for 16A, ED95, B20 and B100, respectively. The parameters are listed in Table 14. Clearly, there is a consistency among the viscosities of these fuels and published results for model pure liquids in [21]. The

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