Viscosity Measurements of n-Dodecane at Temperatures between 303

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

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Viscosity Measurements of n‑Dodecane at Temperatures between 303 K and 693 K and Pressures up to 10 MPa Song Feng, Zhaohui Liu,* Qincheng Bi, and Hui Pan State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China ABSTRACT: The dynamic viscosities of n-dodecane were measured at temperatures between 303.3 and 693.3 K and pressures up to 10.0 MPa using a dual-capillary viscometer. The accuracy of the dual-capillary viscometer was improved by considering centrifugal effects and the thermal expansion of the capillary. The combined relative standard uncertainty in the dynamic viscosity was calculated to be 0.58−2.92 %. This study provides new data to check the accuracy of Huber’s correlations at supercritical pressures, and a small proportion of the data points at supercritical conditions where few experimental data have been reported. The average absolute deviation between the experimental data and the calculated data from Huber’s correlations is 0.97%, and the maximum absolute deviation is 6.79%. In the near-critical and supercritical region, the accuracy of viscosities of n-dodecane needs more experimental data for verification and to extend the theories.

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INTRODUCTION Viscosity, a transport property, plays an important role in a wide range of engineering disciplines. In the regenerative cooling of aerial vehicles, hydrocarbon fuels are circulated around the channel of the engine compartment as a coolant before being injected into the combustion chamber.1−3 In such applications, the viscosity of hydrocarbon fuels is related to the design of the cooling system and the thermodynamic characteristics determines heat and mass transfer processes of the fuels. However, the prediction of viscosity is difficult because of a large number of components. The viscosity of hydrocarbon fuel was measured with difficulty by experimental systems at high temperatures, because the chemical reactions at temperature beyond 750 K affected both of the components and the flow state of the hydrocarbon fuel.4 Therefore, it is desirable to model the thermodynamic and transport properties of fuels using surrogate materials. Bruno et al.5,6 applied the advanced distillation curve method as a means to analysis the volatility of surrogate mixtures, and predicted the thermophysical properties of the surrogate mixtures and Jet-A fuel at temperatures from 278.15 to 383.15 K. The results showed that the volatility, density, and speed of sound of a simple, three-component surrogate mixture (n-dodecane, n-tetradecane, and 1,2,4-trimethylbenzene with mass fractions of 0.31, 0.38, and 0.31, respectively) agreed with those of Jet-A fuel excellently. Huber et al.7 developed surrogate mixture models to represent the thermophysical properties of two samples of aviation turbine fuel Jet-A at temperatures between 250 and 370 K and pressures up to 35.0 MPa. Most of the surrogate mixtures included n-dodecane. n-Dodecane is often used as a surrogate for two reasons. First, n-dodecane is an important component of hydrocarbon fuels such as Jet-A, JP-4, JP-5, JP-7, JP-8, and RP-1.8−10 Second, n-dodecane has a similar molecular structure to multicomponent © XXXX American Chemical Society

hydrocarbons. Unfortunately, precise viscosity data for n-dodecane above 473.0 K has not been obtained because of a lack of adequate measurements. The residence time, thermal degradation of the sample, and other problems are the key obstacles preventing the measurement of the viscosity of n-dodecane. Huber et al.11 surveyed the literature data and provided correlations for the viscosity of n-dodecane from the triple point (263.6 to 800.0) K, and at pressures up to 200.0 MPa. Their correlations have been validated across a wide range of states and implemented in the REFPROP Version 9.1 software package. However, the data for viscosity of n-dodecane12−25 which are used in the correlations are based on measurements at temperatures below 473.0 K. As shown in Figure 1, the viscosity data for n-dodecane are scarce at temperatures higher than 473.0 K. For the measurement of viscosity, vibrating-wire,26 fallingbody,27 and rotational28 and capillary tube viscometers29 have been constructed. Compared to the above-mentioned viscometers, the dual-capillary viscometer is inexpensive with high accuracy, particularly with longer tubes, and online measurement technology.30 Previously, dual-capillary viscometers were mainly used to measure the viscosities of gas. Berg31 proposed a quartz capillary flow meter (QCFM), and measurements with nitrogen demonstrated that three flow elements could span the flow range from 0.1 to 1000 μmol/s with a standard uncertainty of less than 0.03%. May et al.32 reported the zero-density viscosities of hydrogen, methane, and argon at temperatures from 200 to 400 K using a dual-capillary viscometer. Liu et al.33 measured the viscosity of liquid hydrocarbon at temperatures below 590 K using the dual-capillary viscometer. Yang et al.34,35 measured Received: October 1, 2017 Accepted: January 22, 2018

A

DOI: 10.1021/acs.jced.7b00866 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Article

EXPERIMENTAL SECTION Experimental System. Figure 2 shows the experimental system. A dual-capillary viscometer consists of two test capillaries in series (hereafter called the upstream capillary and the downstream capillary, respectively). The test fluid was pumped by a constant volumetric pump. A filter was attached to the pump to protect the devices from entrained impurity. The fluid first flowed through temperature equilibration coil which was connected before upstream and downstream capillaries to attain thermal equilibrium, and then through upstream and downstream capillaries. Capillaries were made of 316 stainless steel, and the external diameter, internal diameter, and the length of the two test capillaries were 0.76 mm, 0.25 mm and 3100.0 mm, respectively. Test capillaries were coiled nine times to form a helix with 60.0 mm height and 108.6 mm screw diameter. K-type sheathed thermocouples were employed to measure the inlet and outlet temperatures of the test capillaries. The K-type sheathed thermocouples were calibrated by a JOFRA temperature calibrator. The pressure drops of test capillaries were monitored by pressure drop transducers. Each pressure transducer was calibrated by a pressure calibrator supplied by Mesor Ltd. The upstream capillary was housed in a water thermostat at T0 = 298.2 K and the downstream capillary was in an oil thermostat maintained at the test temperature (T = 303.1 < T < 423.1 K) and a molten salt thermostat at T > 423.1 K. Each coiled capillary apparatus was immersed horizontally in a liquid thermostat. The fluid was cooled to below 310.0 K by a condenser, and the fluid pressure was regulated by a back-pressure valve. An electronic balance was placed at the outlet to weight the mass of test fluid. In the experiments, all the information was recorded using a computerized data acquisition system. A molten salt thermostat was constructed, which increased the temperature range of the viscometer to 693 K. Figure 3 shows the thermostat system including a working zone, a mixing zone, and a measuring system. The thermostat had a maximum operating temperature of 773.0 K, and a working volume of 25.0 L. As the working substances are required, water, oil, and a molten salt were added to the thermostat to obtain different temperatures. For stability and safety, the working substances must not produce vapors or harmful products. At temperatures beyond

Figure 1. Experimental data from the literature as well as the experimental data of the present work: ■, Aminabhavi et al.;12−14 □, Aralaguppi et al.;15,16 ●, Knapstad et al.;17,18 ○, Caudwell et al.;19 ▲, Prak et al.;20 △, Ducoulombier et al.;21 ▼, Dymond et al.;22 ▽, Garciá et al.;23 ★, Kashiwagi et al.;24 ☆, Tanaka et al.;25 red ●, present work; bold line, saturation boundary;38 red ○, gas−liquid critical point.38

the viscosity of two kinds of liquid hydrocarbon fuels at temperatures from 303 to 513 K and 303 to 673 K, respectively, using the same method. However, they did not consider centrifugal effects and the thermal expansion of the capillary, which could influence the accuracy of the dual-capillary. In this paper, the viscosities of n-dodecane, determined using a dual-capillary viscometer, are reported at temperatures between 303.3 and 693.3 K and pressures up to 10.0 MPa. The accuracy of the dual-capillary viscometer was improved by considering centrifugal effects and the thermal expansion of the capillary. New viscosity data were obtained experimentally for n-dodecane at supercritical pressures, and a small proportion of the data points at supercritical conditions, which lays the foundation for the design of cooling systems for regenerative cooling of aerial vehicles. In future work, we will measure the viscosity of the pure substances and hydrocarbon fuels at higher temperatures considering the effect of cracking.

Figure 2. Schematic diagram of the experimental system. B



Article

Thus, the viscosity η is given by η=ρ

Δp 1 Q Z

(3)

The upstream capillary was maintained at the constant reference temperature (T0 = 298.2 K), and the downstream capillary was maintained at the test temperature T. In a steady-flow state, the measured fluid flowed through two capillaries in series at a constant mass flow rate and the relationship is ηTmea ηTmea

=

0

mea ρTmea Zup, T0 Δpdown, T mea ρTmea Zdown, T Δpup, T 0

(4)

0

Application of eq 4 to the case where T = T0 permits the evaluation of the structure coefficient from a calibration measurement at that temperature, as follows: Zup, T0

Figure 3. Schematic diagram of the bath thermostat.

Zdown, T0

423.1 K, a binary mixtures of potassium nitrate and sodium nitrite with a mass ratio of 1:1 was used as the working substance. The binary mixtures melt into a liquid at 423.0 K. When the temperature was increased to 700.0 K and maintained for 2 h, the mass loss was less than 0.8%. Thus, the binary mixture did not decompose, and thereby did not produce vapors and harmful products. The temperature of each bath was measured by a K-type sheathed thermocouple. The bath was stirred by a heavyduty fan immersed in the mixing zone, and the working zone and mixing zone were separated by a damping screen. The damping screen increased the disturbance of the flow in the working zone and maintained a uniform temperature. A PID process controller was used to ensure the uniformity and stability of the test temperature within ±0.1 K or better. Experimental Materials. A description of samples is given in Table 1. The samples produced by Aladdin Industrial Corpo-

Zdown, T0 Zdown, T

source

initial mole fraction purity

purification method

toluene n-dodecane

108-88-3 112-40-3

Aladdin Aladdin

0.995 0.995

distillation distillation

ηTmea

3 LT0 rT 4 ⎛ ΔL ⎞⎟ ΔL ≈ ⎜1 + ≈1+3 4 ⎝ L ⎠ L rT0 LT

Zdown, T0 Zdown, T

(6)

increased by 2.2% from

mea ρTmea ⎛ Zup, T0 Zdown, T0 ⎞ Δpdown, T ⎜ ⎟ mea ρTmea ⎜⎝ Zdown, T0 Zdown, T ⎟⎠ Δpup, T0 0

⎤⎛ mea ⎞ ⎛ ρ mea ⎞⎡⎛ Δp mea ⎞ ⎞⎥⎜ Δpdown, T ⎟ Δ L up, T0 ⎟⎛ T ⎟⎢⎜ ⎜ ⎟ = ⎜ mea ⎟ ⎜ mea ⎟⎜1 + 3 mea ⎟ ⎢ ⎝ L ⎠⎥⎦⎜⎝ Δpup, ⎝ ρT0 ⎠⎣⎝ Δpdown, T0 ⎠ T0 ⎠

(7)

where the density ratio can be obtained from ref 38, and the terms in square brackets refer to the calibration experiment with T = T0. Therefore, the ratio of viscosity

ηTmea ηTmea

can be calcu-

0

lated by measuring the pressure drops of the upstream and downstream capillaries. mea To calculate ηmea T , the viscosity of ηT0 must be calculated first. A reference fluid is allowed to flow through the two capillaries at the same mass flow rate to calculate the viscosity of ηmea T0 . According to eq 1, the relationship is ηTmea 0

=

ηTref 0

mea ρTmea Δpup, T 0

ρTref 0

0

ref Δpup, T0

(8)

The temperature dependence viscosity of the test fluid can be calculated as

(1)

where Z is a structure coefficient of the capillary, which is determined from the internal radius and length: 8L πr 4

=

0

4

Z=

=

303 to 693 K. The thermal expansion of the capillary cannot be neglected at high temperatures. By combing the expressions in eq 4 and eq 5, the viscosity ratio of the measured fluid was calculated as

ration are the analytical reagent. The samples were distilled twice before experiments. The initial mole fraction purity of samples is greater than 0.995. Toluene was used as a reference fluid, and its density and viscosity have been reported previously.36,37 n-Dodecane was used as the measured fluid, and its densities have been reported in ref 38. The capillary made of 316 stainless steel was manufactured by Valco Instruments Co. Inc. The thermal expansion of the 316 stainless steel was obtained from ref 39. Experimental Principles. A single-capillary viscometer is based on the Poiseuille’s law which describes the isothermal, slip-free laminar flow of an incompressible Newtonian fluid in a long, straight cylindrical tube. The mass flow rate Q through a capillary with internal radius r and length L depends on the fluid density ρ, fluid viscosity η, and pressure drop Δp: Δp 1 πr Δp Q=ρ =ρ 8L η η Z

(5)

0

According to ref 39, the value of

ηTmea

CASRN

0

mea Δpdown, T

According to eq 2, the structure coefficient for T ≠ T0 can be obtained by considering the integrated thermal expansion of tube length ΔL and tube radius Δr. For a homogeneous material, the relationship on the basis of ref 40 can be simplified to

Table 1. Description of Samples substance

=

mea Δpup, T

ηTmea

(2)

=

⎛ mea ⎞⎡⎛ Δp mea ⎞ ⎛ up, T mea ⎜ ρT ⎟⎢⎜ ηT ⎜ mea ⎟ ⎜ mea 0 ⎟⎟⎜1 0 ⎢ ⎝ ⎝ ρT0 ⎠⎣⎝ Δpdown, T0 ⎠

+3

⎤⎛ mea ⎞ ΔL ⎟⎞⎥⎜ Δpdown, T ⎟ mea ⎟ L ⎠⎥⎦⎜⎝ Δpup, T ⎠ 0

(9) C



Article

In this work, the test fluid in the capillary should remain in the laminar flow state by Poiseuille’s law. According to refs 41−43 the Reynolds number ranged from 20 to 1100. The coiled capillaries lead to secondary flow, which could increase the dissipation of pressure drop. Therefore, the centrifugal effects must be considered. Centrifugal effects are quantified using the Dean number, defined by ⎛ r ⎞ De = Re ⎜ ⎟ ⎝ R curve ⎠

continuously, the centrifugal effect increased remarkably, and the pressure drop of the downstream capillary must be revised. The temperature, pressure, and the mass flow rate are constant in the upstream capillary. De also is constant and less than 10. The pressure drop of the upstream capillary need not be revised. In the downstream capillary, the De increased as the temperature increased. When the De is greater than 10, the pressure drop of the downstream capillary must be revised. Assessment of Uncertainties. The measurement uncertainty of the viscosity was associated with uncertainty in the measured quantities in working eq 7. According to the ref.,46 the combined relative standard uncertainty of viscosity in the measurement is given by

(10)

where Re, Rcurve, and r are the Reynolds number, the radius of curvature of the coil, and the internal radius of the capillary, respectively. The pressure drop of capillaries is corrected through the

(

function fcent De,

r R curve

). Dyke’s theory

44

ur 2(ηTmea ) = ur 2(ηTmea ) + ur 2(ρTmea ) + ur 2(ρTmea ) 0

yields a 12-term

+

polynomial plus a logarithmic term in a variable derived from De by Euler transformation. An iterative solution of the expression yields f VanDyke (De) = fcent (De, 0). Remarkably, Berg45 simplified the expression as ⎡ ⎛ De ⎞ fcent (De) = ⎢1 + 16⎜ ⎟⎥ ⎢⎣ ⎝ De0 ⎠ ⎥⎦

+

2

mea ur (Δpdown, ) T 2

ref ur (Δpdown, ) T0

+

0

2

mea ur (Δpup, ) T0

+

2

ref ur (Δpup, ) T0

⎡ ∂η mea u(T ) ⎤2 ⎥ + 9ur +⎢ T ⎝ L ⎠ ⎢⎣ ∂T η mea ⎥⎦ T ΔL ⎞⎟ 2⎛ ⎜

⎡ ∂η mea u(T ) ⎤2 ⎡ ∂η mea u(p) ⎤2 0 ⎥ +⎢ T ⎥ +⎢ T ⎢⎣ ∂T0 ηTmea ⎥⎦ ⎢⎣ ∂p ηTmea ⎥⎦

4 ⎤−1/16

(11)

(12)

Here, ur(X) is the combined relative standard uncertainty and u(X) is the standard uncertainty of variable X. For the density of n-dodecane, on the basis of equation of state (EOS) of Lemmon and Huber,38 the uncertainty was less than 0.2% at pressures up to 10.0 MPa in liquid phase. The values of ηmea T0 were remeasured at temperature T0 and pressures from 6.0 to 10.0 MPa. The uncertainties of the thermal expansion of capillaries were within 0.2% based on ref 39. In this work, the uncertainties of the main parameters are listed in Table 2. Thus,

The expression describes Dyke’s result at all Dean numbers to within 0.04%. The parameter De0 = 40.58 comes from the analytic solution in the limit of small De. Equation 11 is useful for De < 120 if an error of 1% is tolerable.45 In this study, the Dean numbers are below 80. To verify fcent(De) further, n-dodecane and toluene in the coiled capillaries were measured at different Dean numbers. n-Dodecane and toluene flowed through the coiled capillaries at a constant pressure and temperature. The Dean numbers increased as the mass flow rate increased. The pressure drop of coiled capillaries was measured at a different De. The pressure drop of the straight capillary was calculated by the Poiseuille’s law at the same De. fcent(De) was calculated by the difference between two pressure drops. As shown in Figure 4,

Table 2. Uncertainties of the Main Experimental Parameters parameters

units

standard uncertainty

volume flow rate temperature pressure pressure drop density of n-dodecane thermal expansion of capillary viscosity of n-dodecane at T0

mL/min K MPa kPa kg/m3 μm/(m·K) μPa·s

0.01 mL/min 0.50 K 0.008 MPa 0.19 kPa 1.50 kg/m3 1.78 × 10−6μm/(m·K) 7.6 μPa·s

the combined relative standard uncertainty of dynamic viscosity is calculated to be 0.58−2.92%.

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RESULTS AND DISCUSSION Viscosity of n-Dodecane. The viscosities of n-dodecane were measured from 303.3 to 693.3 K and at pressures from 6.0 to 10.0 MPa. The working pressures are both higher than the critical pressure of n-dodecane, and a small proportion of the data points at supercritical conditions. The data are shown in Table 3, and the variation in the viscosity is presented in Figure 5. To show how the new experimental data completed the reference data, the results of our study as well as the experimental data from ref 12−25 are shown in Figure 1. The list of the considered source of experimental data from ref 12−25 is shown in Table 4. The viscosity decreases with increasing temperature, and the variation becomes smooth at high temperatures. Under working conditions, the pressure has little effect on the viscosity of n-dodecane.

Figure 4. Centrifugal correction, fcent, as a function of De: red □, toluene measured results; red ○,n-dodecane measured results; solid line, Dyke’s analytical theory.44

the measured results disagree with Dyke’s analytical theory44 by as much as 0.9%. Therefore, eq 11 is sufficient when De < 80. Centrifugal effects can be neglected for De < 10. As De increased D



Article

Table 3. Experimental Viscosity Ratios ηT/ηT0 and Viscosities η of n-Dodecane at Temperatures T and Pressures p, with Reference Temperature T0 = 298.2 K.a Also Tabulated: Density Ratios ρT/ρT0 Calculated from EOS of Huber38 T/K

p/MPa

ρT/ρT0

ηT/ηT0

ηT0/μPa·s

ηT/μPa·s

304.9 303.4 303.3 314.5 313.4 313.3 323.4 323.1 323.4 334.1 333.3 333.4 343.3 343.2 343.3 353.4 353.6 353.4 364.0 363.7 363.3 374.0 373.9 373.1 383.0 383.4 383.2 393.1 393.6 393.0 403.0 404.1 403.1 413.2 413.7 412.9 423.0 423.9 423.4 433.2 433.3 432.9 443.5 443.5 443.2 453.3 454.0 453.3 463.4 464.1 463.3 473.4 473.5 473.3 483.4 483.4 483.4

6.02 8.05 10.02 5.99 8.00 10.02 6.05 8.01 10.00 6.01 8.03 10.00 5.99 8.07 10.01 6.08 8.02 9.98 6.04 8.02 10.05 6.00 8.02 10.07 6.01 8.05 10.00 6.08 8.03 10.02 6.04 8.02 10.04 6.08 7.99 10.01 6.08 8.05 10.00 6.01 8.00 10.02 6.02 8.01 10.05 6.02 8.04 9.95 6.04 8.02 10.01 6.04 8.01 10.05 6.05 8.08 10.00

0.993600 0.995102 0.995257 0.984466 0.985705 0.985970 0.976025 0.976628 0.976644 0.965891 0.967098 0.967437 0.957170 0.957861 0.958336 0.947610 0.948172 0.949063 0.937529 0.938735 0.939962 0.927995 0.929179 0.930954 0.919381 0.920261 0.921641 0.909661 0.910638 0.912580 0.900073 0.900683 0.903201 0.890113 0.891526 0.894047 0.880472 0.881716 0.884189 0.870325 0.872613 0.875208 0.859964 0.862630 0.865403 0.849990 0.852262 0.855717 0.839576 0.842173 0.846019 0.829122 0.832670 0.836241 0.818508 0.822555 0.826263

0.922729 0.911945 0.909192 0.776880 0.776523 0.790754 0.667169 0.672635 0.681988 0.585518 0.587052 0.595804 0.517555 0.520243 0.527793 0.461707 0.462253 0.469669 0.412566 0.414818 0.423036 0.373828 0.375601 0.383685 0.342961 0.343199 0.349412 0.313736 0.313602 0.320551 0.288344 0.287547 0.294428 0.265896 0.266368 0.272314 0.245089 0.246259 0.251537 0.228184 0.227485 0.234099 0.211621 0.210583 0.217598 0.195401 0.196152 0.202833 0.182328 0.182924 0.189671 0.171788 0.168693 0.177512 0.157758 0.160676 0.166288

1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9

1348.2 1365.0 1393.7 1135.1 1162.3 1183.6 974.8 1006.8 1020.8 855.5 878.7 891.8 756.2 778.7 790.0 674.6 691.9 703.0 602.8 620.9 633.2 546.2 562.2 574.3 501.1 513.7 523.0 458.4 469.4 479.8 421.3 430.4 440.7 388.5 398.7 407.6 358.1 368.6 376.5 333.4 340.5 350.4 309.2 315.2 325.7 285.5 293.6 303.6 266.4 273.8 283.9 251.0 252.5 265.7 230.5 240.5 248.9

Table 3. continued

E

T/K

p/MPa

ρT/ρT0

ηT/ηT0

ηT0/μPa·s

ηT/μPa·s

493.4 493.4 493.2 503.4 502.7 503.2 514.2 513.0 513.0 523.7 523.0 523.7 533.5 533.3 533.5 542.9 542.9 542.9 552.9 553.2 552.9 563.0 563.0 563.0 573.0 573.0 573.0 582.8 582.9 582.8 593.0 592.9 593.0 603.1 602.9 603.1 613.1 612.9 613.1 623.2 623.0 623.2 633.5 632.8 633.5 643.4 643.0 643.4 653.4 652.9 653.4 663.2 663.1 663.2 673.5 672.9 673.5 683.3 683.2 683.2

6.04 8.01 10.05 6.08 7.96 10.01 6.05 8.04 10.00 5.97 8.00 10.04 6.04 8.04 10.07 6.01 8.06 9.95 6.07 8.01 10.02 6.05 8.02 10.02 6.01 8.02 10.04 6.00 7.96 10.01 6.02 8.07 10.00 6.01 8.02 10.01 6.04 8.02 10.01 6.05 7.99 10.05 6.02 8.04 10.01 6.02 8.05 10.01 6.07 8.02 10.04 6.01 8.01 10.05 6.04 8.01 9.98 6.01 8.05 10.04

0.807721 0.812186 0.816472 0.796746 0.802417 0.806374 0.784679 0.791450 0.796357 0.773838 0.780616 0.785277 0.762424 0.769289 0.774994 0.751263 0.758548 0.764990 0.739116 0.746809 0.754228 0.726555 0.735442 0.743188 0.713794 0.723637 0.732121 0.700980 0.711725 0.721107 0.687259 0.699453 0.709495 0.673298 0.686942 0.697830 0.659071 0.674204 0.686112 0.644283 0.661081 0.674142 0.628722 0.648104 0.661773 0.613334 0.634355 0.649750 0.597306 0.620792 0.637487 0.581159 0.606591 0.625357 0.563704 0.592749 0.612510 0.546636 0.578015 0.600340

0.147834 0.150120 0.156200 0.137225 0.138629 0.146780 0.127301 0.129943 0.138295 0.118883 0.121726 0.130211 0.112107 0.115981 0.121928 0.104442 0.107095 0.112373 0.0971186 0.100615 0.105559 0.0915064 0.0944682 0.100815 0.0848676 0.0899920 0.0960048 0.0793238 0.0845136 0.0881881 0.0755595 0.0801710 0.0843800 0.0717268 0.0753608 0.0792357 0.0663199 0.0706841 0.0745591 0.0621450 0.0670096 0.0705505 0.0580384 0.0623330 0.0662747 0.0540004 0.0586585 0.0630011 0.0501677 0.0550508 0.0581240 0.0466772 0.0517771 0.0549840 0.0434604 0.0485703 0.0524452 0.0403805 0.0457643 0.0498397

1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9 1461.1 1496.8 1532.9

216.0 224.7 233.8 200.5 207.5 219.7 186.0 194.5 207.0 173.7 182.2 194.9 163.8 173.6 182.5 152.6 160.3 168.2 141.9 150.6 158.0 133.7 141.4 150.9 124.0 134.7 143.7 115.9 126.5 132.0 110.4 120.0 126.3 104.8 112.8 118.6 96.9 105.8 111.6 90.8 100.3 105.6 84.8 93.3 99.2 78.9 87.8 94.3 73.3 82.4 87.0 68.2 77.5 82.3 63.5 72.7 78.5 59.0 68.5 74.6



Article

Table 3. continued T/K

p/MPa

ρT/ρT0

ηT/ηT0

ηT0/μPa·s

693.2 693.2 693.2

6.01 8.02 10.01

0.528969 0.563561 0.587745

0.0375060 0.0429583 0.0475681

1461.1 1496.8 1532.9

ηT/μPa·s 54.8 64.3 71.2

a

Standard uncertainties u are u(T) = 0.5 K, and u(p) = 8.0 kPa, and the expanded uncertainty U is U(η) = 17.8 μPa·s (0.95 level of confidence).

Figure 6. Deviations of the experimental viscosities of n-dodecane from those predicted by the correlations of Huber et al.,11 where the subscripts “exp” represent experimental value and the subscripts “ref” represent a reference correlation in Huber:11 ■, Aminabhavi et al.;12−14 □, Aralaguppi et al.;15,16 ●, Knapstad et al.;17,18 ○, Caudwell et al.;19▲, Prak et al.;20 △, Ducoulombier et al.;21 ▼, Dymond et al.;22 ▽, Garciá et al.;23 ★, Kashiwagi et al.;24 ☆, Tanaka et al.;25 red ◀, present work at p = 6.0 MPa; red ▶, present work at p = 8.0 MPa; red ★,present work at p = 10.0 MPa. Figure 5. Viscosity variations of n-dodecane versus temperature at ●, p = 6.0 MPa; red △, p = 8.0 MPa; blue ▽, p = 10.0 MPa.

and MAD are within 0.65% and 1.84% at temperatures in the range of 303.3 to 503.4 K and pressures up to 10.0 MPa. Our results agree well with the majority of the experimental data reported in the literature; however, the data of Aminabhavi et al.13,14 at temperatures below 400 K show marked deviations. In the literature, the experimental data are scarce at temperatures above 503.0 K. Figure 7 compares the new data with the

Table 4. List of Available Experimental Viscosity Data of n-Dodecane author

T/K

p/MPa

methoda

uncertaintyb

Aminabhavi12−14 Aralaguppi15,16 Knapstad17,18 Caudwell19 Prak20 Ducoulombier21 Dymond22 Garciá 23 Kashiwagi24 Tanaka25

298−318 298−308 288−425 298−473 293−373 293−373 293−373 278−318 298−345 298−348

0.1 0.1 0.1 0.1−200 0.1 0.1−100 0.1−100 0.1 0.1 0.1

CAP CAP OSD VIW ROT GFB GFB GFB VIC VIC

0.20% 0.20% 0.30% 2% 1.09% nul 2% nul 2% 2%

a

Notation: CAP, Capillary; OSD, Oscillating Disk; VIW, Vibrating Wire; ROT, Rotational; GFB, Guided Falling Body; VIC, Vibrating Crystal. bnul: no uncertainty given in source reference.

Figure 6 compares experimental data with the correlations of Huber et al.11 at temperatures ranging from (303.3 to 503.4) K. In the calculation, the percentage deviation (PD), the average absolute deviation (AAD), and the maximum absolute deviation (MAD) of the results were defined: ⎛η − η ⎞ ref exp ⎟100 PD % = ⎜⎜ ⎟ η ⎝ ⎠ exp

Figure 7. Deviations of the experimental viscosities of n-dodecane from those predicted by the correlations of Huber et al.,11 where the subscripts “exp” represent experimental value and the subscripts “ref” represent a reference correlation in Huber:11 ■, present work at p = 6.0 MPa; red ●, present work at p = 8.0 MPa; blue ▲, present work at p = 10.0 MPa.

(13)

n

AAD =

∑i = 1 abs(PDi ) n

MAD = max(abs(PDi ))

(14) (15)

correlations of Huber et al.11 at temperatures ranging from 513.0 to 693.3 K. The results show that AAD was small 1.42%, and MAD was within 6.79%. At temperatures beyond 640.0 K, the deviations of compared results increased with increasing

where the subscripts “exp” and “ref” represent experimental value and a reference correlation in Huber,11 respectively. The new data agree well with the correlations of Huber et al.11 The AAD F



Article

temperatures. The following reasons may result in this phenomenon. First, Huber et al.11 represent the viscosity η of a pure fluid as the sum of a dilute gas contribution and a residual term. The experimental viscosity data were required to calculate the dilute region and residual contribution. However, the experimental data are scarce at temperatures beyond 473 K, and we could hardly find the experimental data in the supercritical region. In the supercritical region, the correlations have been fitted to little experimental data, which may lead deviations of the correlations increased. Huber et al.11 estimates that the uncertainty of dilute-gas correlation is 3% and there were no experimental data validating the correlations in the supercritical region. Second, the EOS was used to calculate the density ratio. Huber et al.11 found the deviations of correlations of viscosities increased when the EOS exceeded the recommended limit. In the supercritical region, the EOS was not validated by the experimental data, which may lead to the deviations of the viscosities of n-dodecane being increased. Third, as the Dean number increased, the additional dissipation of the secondary flow might lead to an increase in the deviation. Fourth, in the supercritical region, the density fluctuations that can occur, and the effects on other thermodynamic and transport properties that might exhibit a systematic deviation with temperature increasing. At high pressures, the flow was more stable, and the deviation decreased. In the near-critical and supercritical region, the accuracy of viscosities of n-dodecane needs more experimental data for verification and to extend theories. Correlation of the Viscosity. The viscosity of n-dodecane was measured by the dual-capillary viscometer at temperatures from 303.3 to 673.3 K and supercritical pressures ranging from 6.0 to 10.0 MPa. The experimental viscosity ratios ηT/ηT0 were fitted as a function of density ratios ρT/ρT0 according to the following equation: r=

ηT ηT

0

⎡ ⎛ ⎞1.78 ⎤ ⎛ ρ ⎞14.5 ⎢ ⎜ ρT ⎟ ⎥ T = exp⎢a⎜ ⎟ + b⎜⎜ ⎟⎟ − (a + b)⎥ ρ ρ ⎝ T0 ⎠ ⎣ ⎝ T0 ⎠ ⎦

Figure 8. Relative deviations 100·(r cal − r exp )/r exp between experimental viscosity ratios rexp and values rcal calculated from eq 16: ■, p = 6.0 MPa; blue ▲, p = 8.0 MPa; red ●, p = 10.0 MPa.

method was found to work well and should permit future measurements at higher temperatures.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +86-029-82665287. ORCID

Zhaohui Liu: 0000-0003-2904-9230 Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 21306147), the National Science Foundation for Postdoctoral Scientists of China (Grant No. 2013M532044) and the Fundamental Research Funds for the Central Universities. Notes

(16)

The authors declare no competing financial interest.

■

where a and b are regression coefficients. The regression coefficients are a = 3.61076 and b = 0.82583. Deviations of the experimental viscosity ratios from eq 16 are plotted in Figure 8. The AAD and MAD are within 1.09%, and 3.46%, respectively. From these deviations, we can also conclude that the fitted formula can accurately predict the experimental viscosities of n-dodecane.

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CONCLUSIONS The experimental temperature range of the dual-capillary viscometer was increased to 693 K using a molten-salt thermostat. Considering centrifugal effects and the thermal expansion of the capillary, the viscosities of n-dodecane were measured in a dual-capillary viscometer at temperatures between 303.3 and 693.3 K and pressures between 6.0 and 10.0 MPa. The combined relative standard uncertainty of dynamic viscosity measurement was 0.58−2.92 %. The new data confirm the accuracy of Huber’s correlations11 for the viscosities of n-dodecane at temperatures from 303.3 to 503.4 K. The results show MAD and AAD are 1.84% and 0.65%, respectively. At temperatures beyond 503.4 K, MAD increased to 6.79%, and AAD increased to 1.42%, indicating that the improvement in the correlation11 is necessary and more experimental data are required in the supercritical region. The experimental G



Article

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Viscosity Measurements of n-Dodecane at Temperatures between 303

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