Viscosities of Aqueous LiNO3 Solutions at Temperatures from 298 to

Azerbaijan State Oil Academy, Baku 370601, Azerbaijan. The viscosities of four (0.265, 0.493, 1.074, and 1.540 mol‚kg-1) aqueous LiNO3 solutions wer...
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Ind. Eng. Chem. Res. 2005, 44, 416-425

Viscosities of Aqueous LiNO3 Solutions at Temperatures from 298 to 573 K and at Pressures up to 30 MPa I. M. Abdulagatov* Institute for Geothermal Problems of the Dagestan Scientific Center of the Russian Academy of Sciences, Shamilya Street 39-A, 367003 Makhachkala, Dagestan, Russia

N. D. Azizov Azerbaijan State Oil Academy, Baku 370601, Azerbaijan

The viscosities of four (0.265, 0.493, 1.074, and 1.540 mol‚kg-1) aqueous LiNO3 solutions were measured in the liquid phase with a capillary flow technique. Measurements were made at four isobars (0.1, 10, 20, and 30 MPa). The range of temperatures was from 298 to 573 K. The total uncertainties of viscosity, pressure, temperature, and concentration measurements were estimated to be less than 1.5%, 0.05%, 15 mK, and 0.014%, respectively. The reliability and accuracy of the experimental method was confirmed with measurements on pure water for five selected isobars (1, 10, 20, 40, and 50 MPa) and at temperatures between 294.5 and 597.6 K. The experimental and calculated values from the International Association for the Properties of Water and Steam formulation for the viscosity of pure water show excellent agreement within their experimental uncertainty (AAD ) 0.27%). The temperature, pressure, and concentration dependences of the relative viscosity (η/η0, where η0 is the viscosity of pure water) were studied. The behavior of the concentration dependence of the relative viscosity of aqueous LiNO3 solutions was discussed in light of the modern theory of transport phenomena in electrolyte solutions. The values of the viscosity A and B coefficients of the Jones-Dole equation for the relative viscosity (η/η0) of aqueous LiNO3 solutions as a function of temperature were studied. The derived values of the viscosity A and B coefficients were compared with the results predicted by the Falkenhagen-Dole theory of electrolyte solutions and calculated with the ionic B coefficient data. Different theoretical models for the viscosity of electrolyte solutions were stringently tested with new accurate measurements on LiNO3(aq). The predictive capability of the various models was studied. The measured values of the viscosity at atmospheric pressure were directly compared with the data reported in the literature by other authors. 1. Introduction The viscosity of aqueous electrolyte solutions is of importance in many industrial and scientific applications because the long-range electrostatic interactions (Coulombic forces between ions) presented make it difficul to describe ionic solutions.1-8 Long-range electrostatic interactions govern thermodynamic and transport properties of ionic electrolyte solutions. The viscosity of electrolyte solutions is usually studied to obtain information on ion-solvent interactions.1-9 Knowledge of the pressure, temperature, and composition dependences of the viscosities of aqueous salt solutions is essential to understanding a variety of problems in a number of technological and engineering applications such as environmental applications, treatment of wastewater, and flue and vent gases often requiring modeling of electrolyte solutions, geothermal power, hydrothermal formation of minerals, for understanding various geologic processes, the mass- and heat-transfer phenomena in the hydrothermal environments, and design calculation.10 However, the lack of reliable thermodynamic and transport property data over temperature, pressure, and * To whom correspondence should be addressed. Present address: Physical and Chemical Properties Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305. Tel.: (303) 497-4027. Fax: (303) 497-5224. E-mail: [email protected].

concentration ranges makes it necessary to estimate the missing properties by extrapolating high-temperature, high-pressure, and high-concentration results. Because the transport properties of aqueous salt solutions undergo dramatic changes as temperature T and composition m are increased,11-14 it is impossible to obtain them by using an extrapolating technique (extrapolations are not accurate). So far, the theory (calculation of the thermodynamic and transport properties of fluids from knowledge of the intermolecular interactions) can give inaccurate or even often physically incorrect results when applied to strong electrolyte solutions. Better predictive models should be developed. A database of transport properties in high-temperature and high-pressure aqueous electrolyte solutions is needed to support the advancement of theoretical work. For engineering uses, reliable methods for the prediction and estimation of the viscosities of solutions over wide ranges of concentration, temperature, and pressure would be extremely valuable. Therefore, experimental data for the viscosities of aqueous systems at high temperatures and high pressures are needed to test theoretical models and improve their predictive capability. However, measurements of the viscosities of aqueous salt solutions have so far been limited to rather narrow ranges of temperature, pressure, and concentration with less than satisfactory accuracy.

10.1021/ie0494014 CCC: $30.25 © 2005 American Chemical Society Published on Web 12/08/2004

Ind. Eng. Chem. Res., Vol. 44, No. 2, 2005 417

In this paper we report new high-precision viscosity measurements for aqueous LiNO3 solutions over wide ranges of temperature, pressure, and composition using a capillary flow technique, which allows the performance of accurate measurements at high temperatures and high pressures.11,12 The research presented in this paper is part of a continuing series of measurements of transport properties of aqueous solutions at high temperatures and high pressures.11-17 A literature survey revealed that all previously reported viscosity data for aqueous LiNO3 solutions were performed at atmospheric pressure or in the range of a slightly higher saturation curve (0.1-2 MPa). There are no viscosity data for aqueous LiNO3 solutions as a function of pressure. Thus, the main objective of this work is to provide reliable experimental viscosity data for aqueous LiNO3 solutions at high temperatures (up to 573 K) and high pressures (up to 30 MPa). The present results considerably expand the temperature, pressure, and concentration ranges in which the viscosity data for aqueous LiNO3 solutions are available. Another objective of the present study was to determine the viscosity A and B coefficients in the Jones-Dole equation for aqueous LiNO3 solutions and to compare them with the results predicted from various theoretical models. The measured values of the viscosity of LiNO3(aq) were used for stringent testing of various theoretical models for the viscosity of electrolyte solutions. 2. Brief Literature Review of the Experimental Viscosity Data for H2O + LiNO3 Solutions Puchkov and Sergeev18 reported viscosity data for nine H2O + LiNO3 solutions between 0.57 and 26.15 mol‚kg-1, at temperatures between 298 and 548 K, and at pressures slightly above the saturation curve (0.1-2 MPa). Measurements were made with a glass capillary viscometer. They claimed the uncertainty in the viscosity measurements was 0.5-1.0%. Klochko and Grigor’ev19 reported viscosities of H2O + LiNO3 solutions in the temperature range from 298 to 373 K and at concentrations between 0.94 and 71.64 wt %. No uncertainty of the measured values of the viscosity was given in the paper. Polovnikova and Karev20 reported the viscosities of H2O + LiNO3 solutions at high concentrations from 17.871 to 68.614 mol‚kg-1 and at temperatures between 352 and 421 K using a glass viscometer. The uncertainty of the measured values of the viscosity is 2.5%. Popevic´ and Nedeljkovic´21 reported of the viscosity of H2O + LiNO3 solutions in the temperature range from 293 to 343 K for compositions between 11.2 and 47.5 wt %. The temperature dependence of the viscosity was expressed by the relation

(

η ) A exp

B T - T0

)

(1)

where adjustable parameters A, B, and T0 were determined for each composition using measured values of the viscosity. Applebey22 reported the relative viscosities of the H2O + LiNO3 solutions at three temperatures (273.15, 291.15, and 298.16 K) using an Ostwald type viscometer. The concentration range was from 0.0174 to 5.849 m. No uncertainty was given in the paper. The measured values of the viscosity of H2O + LiNO3 solutions at 291.15 K were fitted by the equation

1/η ) 1 - 0.00818c1/2 - 0.08458c

(2)

where c is the molarity. Campbell and Friesen23 reported viscosity data for H2O + LiNO3 solutions at temperatures of 298 and 308 K and concentrations up to 1.005 mol‚L-1. Campbell et al.24-26 reported the viscosities for H2O + LiNO3 solutions at temperatures of 298, 308, and 383 K for compositions up to 14.4 mol‚L-1. Aseyev27 represented available experimental viscosity data from the literature for H2O + LiNO3 solutions by following the correlation equation

log η ) log η0 + (1.225 - 0.00108t)x

(3)

where η0 (mPa‚s) is the viscosity of pure water, x is the composition (mass fraction), and t is the temperature (°C). Moulik and Rakshit28 expressed available relative viscosity data for aqueous salt solutions at 298.15 K by the correlating equation

η/η0 ) 1 + Bc + (B2)nc1+l

(4)

where the constant l is within the range 1 e l e 2. For H2O + LiNO3 solutions, the parameters B, n, and l are 0.1034, 1, and 1.5, respectively. 3. Experimental Apparatus and Procedures The apparatus and procedures used for the viscosity measurements of the H2O + LiNO3 solutions have been described in detail in previous papers11,12,15,17,29 and were used without modification. Only essential information will be given here. The measurements were made using a capillary flow method that gives an uncertainty of 1.5% for the viscosity. The main parts of the apparatus consisted of a working capillary, a high-temperature and high-pressure autoclave, movable and unmovable cylinders, electrical heaters, and a solid red copper block. All parts of the experimental installation that have contact with the sample were made from stainless steel (1X18H10T9, 1:18:9 chrome-nickeltitanium). The capillary together with the extension tube was located in the high-temperature and highpressure autoclave. The capillary tube was filled with mercury. When the movable cylinder was moved vertically at constant speed, the fluid flowed through the capillary. Mercury flowed from the movable cylinder to the unmovable (fixed) cylinder and acted as a piston. Both cylinders were supplied with two viewing windows, which were made of Plexiglas. The autoclave was placed in a solid red copper block. Two electrical heaters were wound around the surface of the copper block. The temperature of the sample was measured with a 10 Ω platinum resistance thermometer (PRT-10, R100/R0 ) 1.39245, and the resistance R0 at 0 °C was 9.9980 Ω). The thermometer (the sensitive elements of the PRT) was located very close to the viscometer. The thermometer was calibrated at the All Russian Scientific Research Institute for Physical and Technical Measurements (ARSRIPTM, Moscow, Russia). The resistance of the thermometer was measured by a compensation method using automatic potentiometer P-362/2 (accuracy rating of 0.002) and standard resistance (accuracy of 0.01%). The standard resistance was thermostated with an oil bath with an accuracy of (0.03 K. Measurements were performed in two directions of the working current in the thermometer. The difference between readings of the thermometers in both directions of the working current was 0.005%. This confirmed that

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there was very small noxious emf in the circuit of the thermometer. This scheme of the temperature measurements was provided to detect the sample temperature with a precision of (0.015 K. To create and measure the pressure, the autoclave was connected to a dead-weight pressure gauge (MP-600) by means of separating vessel. The pressure of the sample (solution) was measured with a deadweight pressure gauge MP-600. The absolute sample pressure was calculated as

Table 1. Experimental Viscosities, Pressures, and Temperatures of Pure Water η, mPa‚s 294.46 K P, MPa

this work

1 10 40

0.9696 0.9669 0.9605

(

)

Fc Fc F η ) Aτ 1 (1 + R∆t)3 - B , Fc FHg τ A)

nVC gπR4∆H0F0,Hg , B) (5) 8LVC 8πL

where R ) 0.150 91 mm is the inner radius of the capillary, L ) 540.324 ( 0.005 mm is the capillary tube length, τ is the time of flow, R is the linear expansion coefficient of the capillary material, ∆t is the temperature difference between the experimental and room temperatures, n ) 1.12 is a constant, F(P,T) is the density of the fluid under study at the experimental conditions (P and T), VC ) 1.2182 cm3 is the volume of the unmovable (measuring) cylinder, FC is the density of the fluid under study at room temperature and experimental pressure, ∆H0 ) (H1 - H2)/ln(H1/H2), where H1 and H2 are the mercury levels at the beginning and ending of the fluid flow, respectively, at room and atmospheric pressures, FHg is the density of mercury at room temperature and experimental pressure, and F0,Hg is the density of mercury at room temperature and atmospheric pressure. The values of the parameters A and B can also be determined by means of a calibration technique. In this work, the capillary was calibrated from the known viscosity of a standard fluid (water) with well-established viscosity information (IAPWS31 formulation) at pressures of 0.1, 10, and 30 MPa and selected temperatures of 296.54, 323.65, 348.15, 420.75, 493.69, and 577.86 K. The viscosities of pure water at these conditions were (IAPWS31) as follows: P ) 0.1 MPa (0.9237, 0.5423, and 0.377); P ) 10 MPa (0.9211, 0.5441, 0.3803, 0.1881, 0.1231, and 0.084 49), and P ) 30 MPa (0.9175, 0.5482, 0.3856, 0.1930, 0.1279, and 0.091 49). The volume of the unmovable cylinder VC was determined using a weighing technique.11 This working equation (5)

IAPWS31

this work

0.9700 0.9669 0.9600

0.5910 0.5924 0.5986

359.40 K

IAPWS31

this work

IAPWS31

0.5915 0.5928 0.5980

0.3289 0.3314 0.3380

0.3287 0.3311 0.3390

η/mPa‚s

P ) Pm + Pb + ∆hm + ∆hr + ∆hl where Pm is the pressure reading by a dead-weight pressure gauge (MP-600 and MP-60), Pb is the barometric pressure, ∆hm is the pressure created due to the difference in oil levels in the manometer and separated vessel, ∆hr is the pressure created due to the difference in mercury levels in the separated vessel and viewing window, and ∆hl is the pressure of the sample column between the viscometer center and the mercury level in the viewing window. The maximum uncertainty in pressure Pm read by the dead-weight pressure gauge was 0.05 bar. The uncertainty of the barometric pressure Pb was 0.0013 bar. The uncertainties of the ∆hm, ∆hr, and ∆hl measurements were 5 mm or 0.0065 bar. Therefore, the total uncertainty in the pressure measurements was 0.016%, and the maximum uncertainty was 0.05%. The final working equations for this method is

318.60 K

395.73 K P/MPa 10 20 40 50

this work

448.31 K

IAPWS31

this work

IAPWS31

0.2375

0.2370

0.1630

0.1630

0.2290

0.2294

0.1575

0.1568

486.79 K this work

IAPWS31

0.1279 0.1293

0.1273 0.1297

0.1369

0.1360

η/mPa‚s 527.16 K P/MPa 10 20 40 50

567.60 K

this work

IAPWS31

this work

IAPWS31

0.1055 0.1090

0.1059 0.1086

0.0884 0.0923

0.0887 0.0922

0.1151

0.1150

0.1009

0.1000

597.56 K this work

IAPWS31

0.0804 0.0876 0.0905

0.0807 0.0879 0.0908

takes into account the acceleration of a fluid at the inlet and outlet and the variation of the geometrical sizes of the capillary with T and P. The mercury and sample densities at the experimental conditions were varied with temperature and pressure. The time of fluid flowing through the capillary τ was measured with a stopwatch with an uncertainty of less than 0.1 s (0.5%). All values of τ are averages of at least 5-10 measurements. At a temperature of 573 K, the minimal value of τ was 40 s. The viscosity was obtained from the measured quantities R4, ∆H0, L, VC, τ, FHg, FC, T, P, and m [see working equation (5)]. The accuracy of the viscosity measurements was assessed by analyzing the sensitivity of eq 5 to the experimental uncertainties of the measured quantities. At the maximum measured temperature (573 K), the values of the root-mean-square deviations in the viscosity measurements were δη ) 2 × 10-5 g‚cm-1‚c-1. On the basis of the detailed analysis of all sources of uncertainties likely to affect the determination of viscosity with the present apparatus, the combined maximum relative uncertainty δη/η in measuring the viscosity was 1.5%.11 The experimental uncertainty in the concentration is estimated to be 0.014%. The Reynolds (Re) number occurring during all measurements was less than the critical values (Rec ) 300). As one can see from eq 5, to calculate the dynamic viscosity from measured quantities, the values of the density of the solution under study at room temperature and experimental pressure, FC, and the density at the experimental conditions, F(P,T), are needed. For this purpose, we used the density data measured in our previous paper30 for aqueous LiNO3 solutions at high temperatures (up to 573 K) and high pressures (up to 40 MPa). As a check of the method and procedure of the measurements, the viscosities of pure water were measured from 294.46 to 597.56 K at pressures up to 50 MPa. Table 1 provides the present experimental

Ind. Eng. Chem. Res., Vol. 44, No. 2, 2005 419 Table 2. Experimental Dynamic Viscosities, Pressures, Temperatures, and Concentrations of H2O + LiNO3 Solutions T, K

0.1 MPa

10 MPa

20 MPa

30 MPa

T, K

0.1 MPa

10 MPa

20 MPa

30 MPa

mol‚kg-1

298.18 307.85 323.36 335.45 348.55 356.55 374.47 389.65 400.25 412.77 423.55 433.78

0.9105 0.7460 0.5612 0.4623 0.3900 0.3520

0.9110 0.7475 0.5637 0.4650 0.3920 0.3541 0.2927 0.2531 0.2308 0.2080 0.1920 0.1788

0.9111 0.7515 0.5665 0.4680 0.3944 0.3567 0.2951 0.2556 0.2333 0.2105 0.1945 0.1813

m ) 0.265 0.9118 0.7525 0.5690 0.4700 0.3969 0.3595 0.2976 0.2582 0.2359 0.2131 0.1971 0.1838

450.76 458.85 477.49 486.55 497.05 511.99 527.74 539.89 553.45 561.98 573.25

0.1603 0.1528 0.1378 0.1315 0.1256 0.1181 0.1106 0.1051 0.0989 0.0952 0.0901

0.1628 0.1553 0.1404 0.1341 0.1281 0.1203 0.1126 0.1073 0.1018 0.0983 0.0937

0.1652 0.1576 0.1429 0.1365 0.1304 0.1222 0.1152 0.1102 0.1047 0.1014 0.0969

298.11 308.95 323.56 337.78 347.76 353.57 372.46 384.05 397.26 413.56 424.05

0.9305 0.7411 0.5721 0.4589 0.4039 0.3741

0.9308 0.7423 0.5749 0.4618 0.4063 0.3772 0.3078 0.2744 0.2440 0.2140 0.1977

0.9309 0.7441 0.5777 0.4643 0.4089 0.3800 0.3103 0.2769 0.2464 0.2164 0.2002

m ) 0.493 mol‚kg-1 0.9309 433.74 0.7460 447.75 0.5801 459.84 0.4670 475.25 0.4113 489.96 0.3825 499.66 0.3128 512.05 0.2792 524.53 0.2489 535.75 0.2190 555.25 0.2028 572.87

0.1844 0.1687 0.1571 0.1443 0.1341 0.1284 0.1220 0.1162 0.1110 0.1021 0.0940

0.1870 0.1713 0.1596 0.1468 0.1365 0.1307 0.1244 0.1186 0.1136 0.1051 0.0974

0.1895 0.1737 0.1621 0.1493 0.1390 0.1332 0.1267 0.1211 0.1163 0.1082 0.1010

298.35 307.65 324.04 336.55 348.07 362.35 375.25 387.63 398.55 407.85 420.77

0.9846 0.8088 0.6053 0.4990 0.4340 0.3678

0.9848 0.8108 0.6080 0.5018 0.4362 0.3700 0.3227 0.2860 0.2605 0.2410 0.2190

0.9849 0.8125 0.6108 0.5045 0.4386 0.3725 0.3253 0.2885 0.2631 0.2438 0.2215

m ) 1.074 mol‚kg-1 0.9853 437.85 0.8140 446.36 0.6133 461.76 0.5071 475.75 0.4412 485.86 0.3750 497.05 0.3278 513.28 0.2910 523.77 0.2655 534.88 0.2461 552.65 0.2241 573.85

0.1950 0.1848 0.1691 0.1566 0.1488 0.1418 0.1328 0.1273 0.1218 0.1128 0.1025

0.1971 0.1870 0.1714 0.1591 0.1514 0.1443 0.1352 0.1299 0.1245 0.1158 0.1057

0.1996 0.1894 0.1739 0.1616 0.1539 0.1468 0.1377 0.1325 0.1271 0.1189 0.1092

298.45 306.75 323.56 334.66 347.35 358.73 371.75 384.85 397.87 409.66 422.75

1.0399 0.8753 0.6519 0.5482 0.4658 0.4059

1.0408 0.8765 0.6548 0.5500 0.4672 0.4083 0.3576 0.3164 0.2837 0.2588 0.2364

1.0410 0.8778 0.6572 0.5523 0.4698 0.4111 0.3600 0.3190 0.2863 0.2614 0.2389

m ) 1.540 mol‚kg-1 1.0413 437.75 0.8788 450.97 0.6597 461.88 0.5553 474.56 0.4722 484.76 0.4138 498.36 0.3626 510.07 0.3218 524.87 0.2888 539.97 0.2638 555.09 0.2414 571.76

0.2148 0.1982 0.1866 0.1753 0.1670 0.1573 0.1506 0.1419 0.1337 0.1255 0.1168

0.2172 0.2006 0.1890 0.1779 0.1695 0.1599 0.1532 0.1444 0.1364 0.1284 0.1199

0.2197 0.2032 0.1916 0.1804 0.1721 0.1624 0.1557 0.1470 0.1389 0.1309 0.1223

viscosity data for pure water measured using the same experimental apparatus. These data were compared with values calculated from the IAPWS31 formulation. The agreement between IAPWS31 calculations and the present results along the isobars at 1, 10, 20, 40, and 50 MPa is excellent. Deviation statistics for the present viscosity data for pure water and values calculated with IAPWS31 formulation are as follows: AAD ) 0.27%, bias ) -0.05%, std dev ) 0.32%, std err ) 0.06%, and max dev ) 0.59% (N ) 25). The maximum deviation of 0.59% is found at the maximum temperature of 597.56 K and at the maximum pressure of 50 MPa. No systematic trend of the deviations was found for pure water. This excellent agreement between the present data and IAPWS31 calculations for pure water confirms the reliability and high accuracy of the measurements for H2O + LiNO3 solutions and correct operation of the present instrument. This generally good agreement provides some confidence in the experimental values of Table 2 for H2O + LiNO3 solutions. Chemically pure LiNO3 and distilled water were used to prepare the solutions. The solutions at the desired

concentration were prepared by a gravimetric method, and the concentration was checked by means of a pycnometer using a density at 20 °C with reference data. 4. Results and Discussion The dynamic viscosity, η, measurements for aqueous LiNO3 solutions were made in the temperature range from 298 to 573 K at pressures up to 30 MPa for four compositions (0.265, 0.493, 1.074, and 1.540 mol‚kg-1). All experimental viscosity data were obtained as a function of temperature at four isobars (0.1, 10, 20, and 30 MPa). The experimental temperature, viscosity, pressure, and composition values for the aqueous LiNO3 solutions are presented in Table 2. Some selected experimental results for H2O + LiNO3 solutions as an example of the present results are shown in Figures 1-3 in the η-m, η-T, and η-P projections together with values calculated from the IAPWS31 formulation for pure water (m ) 0). To check the reproducibility of the experimental data, the measurements of the viscosity were performed 5-10 times at the same selected tem-

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Figure 2. Measured values of the viscosity of H2O + LiNO3 solutions as a function of the temperature at a selected constant concentration of 0.57 mol‚kg-1 at pressures slightly above saturation (0.1-2 MPa) together with data reported by other authors. The solid curve is a guide to the eye.

Figure 1. Measured values of the viscosity of H2O + LiNO3 solutions as a function of the concentration along the three selected temperatures 298.15 K (a) and 323.15 and 348.15 K (b) together with values reported by other authors from the literature at atmospheric pressure. The solid curve is a guide to the eye.

perature and pressure. The scatter of the experimental data was within of (0.2-0.4%. Figures 1-3 illustrate how the viscosity of the solution changes with the temperature, pressure, and concentration. These figures also contain data reported by other authors and calculated with correlation equation (3) by Aseyev.27 As Figure 1a shows, along isotherm 298.15 K and at compositions between 0 and 22 mol‚kg-1, the viscosity of the solutions changes by a factor of 13. The temperature dependence of the viscosity of the solution for a selected composition of 0.57 mol‚kg-1 near saturation (0.1-2 MPa) is illustrated in Figure 2 together with data reported by other authors. As Figure 2 shows, the viscosity of the solution decreases considerably (by a factor of 10) with temperature changes between 298 and 543 K. As one can see from Figure 3, the viscosity of the solution along a constant composition (1.54 mol‚kg-1) and selected temperatures increases slightly linearly as the pressure is increased. The viscosity is little affected (up to 2% at pressure changes between 0.1 and 30 MPa) by pressure along the isotherms (see Figure 3). For solutions, the relative viscosity, η/η0 (where η and η0 are the viscosity of the solution and pure water at the same P and T, respectively), is often used to compare them with theoretical models. Figures 4-6 demonstrate the concentration, temperature, and pressure dependences of the relative viscosity η/η0 for H2O + LiNO3 solutions. As Figure 4 shows, the relative viscosity η/η0 for H2O + LiNO3 solutions increases monotonically with the con-

Figure 3. Measured values of the viscosity of H2O + LiNO3 solutions as a function of the pressure along various isotherms at a selected concentration of 1.54 mol‚kg-1. The solid curve is a guide to the eye.

Figure 4. Measured values of the relative viscosity (η/η0) of H2O + LiNO3 solutions as a function of the concentration along the various isotherms at atmospheric pressure. The solid curve is a guide to the eye.

centration along fixed temperatures. The temperature dependence of η/η0 at a selected pressure of 10 MPa and various concentrations is shown in Figure 5. At low concentrations, the relative viscosity η/η0 is little affected by the temperature, while at high concentrations, the temperature significantly (up to 13%) influences η/η0 (see Figure 5). Between temperatures 320 and 440 K

Ind. Eng. Chem. Res., Vol. 44, No. 2, 2005 421

Figure 5. Measured values of the relative viscosity (η/η0) of H2O + LiNO3 solutions as a function of the temperature along the various constant compositions for selected isobar 10 MPa: O, 1.540 mol‚kg-1; b, 1.074 mol‚kg-1; 4, 0.493 mol‚kg-1; 2, 0.265 mol‚kg-1. The solid curve is a guide to the eye.

Figure 6. Measured values of the relative viscosity (η/η0) of H2O + LiNO3 solutions as a function of the pressure along the various constant compositions for isotherm 298.15 K at atmospheric pressure. The solid curve is a guide to the eye.

on the η/η0-T dependence curve, a slight, but notable, maximum (approximately at 400 K) is observed. At the same temperature range, the maximum in thermal conductivity of LiNO3(aq)13 and other aqueous solutions is found (see, for example, work by Abdulagatov and Magomedov33-35). Probably this is the same effect that is occurring in aqueous solutions at high temperatures (around 400 K). As Figure 6 shows, the relative viscosity η/η0 of the solution is almost independent of pressure (between 0.1 and 30 MPa, the values of η/η0 change within 0.70-0.98%). The present results for the viscosity of H2O + LiNO3 solutions at atmospheric pressure were directly compared with experimental values reported in the literature. Figures 1a,b and 2 contain the values of viscosity reported by Puchkov and Sergeev,18 Popevic´ and Nedeljkovic´21 Applebey,22 Campbell and Friesen,23 and Campbell et al.24-26 together with the present results for selected isotherms (298.15, 323.15, and 348.15 K) and a selected isopleth (0.57 mol‚kg-1) at atmospheric pressure. These figures include also the values of viscosity for H2O + LiNO3 solutions calculated with the correlation equation (3) reported by Aseyev.27 As one can see from these figures, the agreement between various data sets is good, except a few data points reported by Applebey22 at 298.15 K and at concentrations of around

5 mol‚kg-1. Figures 1a,b and 2 illustrate that our data are well consistent with literature values at atmospheric pressure and various temperatures. Good agreement with AAD ) 1.13% is found between the present measurements and the data reported by Puchkov and Sergeev.18 The deviation is very close to their experimental uncertainty of 1.0% and the uncertainty of the present results (1.5%). The data reported by Campbell et al.24-26 agree with the present viscosity values within 0.54%. The deviation between the present data and the values reported by Applebey22 at low concentrations is within 0.05-0.4%. Differences between our measurements and the values calculated with correlation equation (3) within 1.5-2% are slightly higher than their experimental uncertainty. This excellent agreement between the present and published data at atmospheric pressure and low temperatures also confirms the reliability of the present measurements and its consistency with literature data. The measured viscosities were used to calculate viscosity A and B coefficients of the Jones-Dole equation for various isotherms. The values of viscosity A and B coefficients of an electrolyte provide information on the interaction between dissolved ions and molecules of a solvent. Therefore, viscosity A and B coefficients of electrolyte solutions are useful tools in the study of structural interactions (ion-ion, ion-solvent, and solvent-solvent) in solutions. The experimental and calculated values of the viscosity A and B coefficients of a series electrolyte solutions have been tabulated by various authors.1,2,36-43 We examine the viscosity A and B coefficient values of aqueous LiNO3 solutions as a function of temperature. Falkenhagen-Onsager-Fuoss44,45 and Debye-Hu¨ckel-Onsager46,47 theories predict a square-root concentration, η/η0 ∝ xc, dependence of the viscosity of ionic solutions at infinite dilution (c f 0). This theory correctly explains the rise of viscosity with the concentration in the limit of very low ion concentrations (c < 0.05 mol‚L-1). This model was based on macroscopic consideration. Therefore, this model is inadequate when intermolecular correlation becomes important. Jones and Dole48 proposed an empirical extension of the Falkenhagen44 model to high concentrations as

η ) 1 + Axc + Bc η0

(6)

for the viscosity of electrolyte solutions. In eq 6, η and η0 are the viscosities of an electrolyte solution and pure solvent (water), respectively, A is an always positive constant, and c is the electrolyte molarity concentration (mol‚L-1). The viscosity A coefficient is a related to the long-range Coulombic force interactions and the mobilities of solute ions, and B is the result of interactions between the solvent and ions and ion size. The sign of the B coefficient depends on the degree of solvent structuring introduced by the ions. A positive value of the B coefficient is associated with structure-making (ordering) ions, while a negative value of the B coefficient is associated with structure-breaking (disordering) ions. This equation is valid only for concentrations below 0.1 mol‚L-1. Equation 6 provides a better description of the experimental viscosity data. Falkenhagen and Dole49 gave a theoretical derivation of the A coefficient. Its general form is

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Ind. Eng. Chem. Res., Vol. 44, No. 2, 2005

A)

A* )

A* f(λ∞+,λ∞-,z+,z-) 1/2 η0(0T)

Fe2NA1/2(1 + x2)

f)

12π(*k)1/2 z2(λ∞+ + λ∞-)

,

4(2 + x2)(λ∞+ λ∞-)

[

1-

4(λ∞+ - λ∞-)2 (1 + x2)2(λ∞+ + λ∞-)2

(7)

]

(8)

where A* ) 1.113 × 10-5 °C2 (m‚K‚mol-3)1/2, f(λ∞+, λ∞-,z+,z-) is the function of the equivalent conductances λ∞( at infinite dilution of the ions, and z( is the charges. The value of parameter A depends also on the viscosity of the solvent η0, its relative permittivity (dielectric constant) 0, and temperature T. For example, at temperature 298.15 K and pressure 0.1 MPa, the viscosity and dielectric constant of pure solvent (water) are η0 ) 0.8901 × 10-3 Pa‚s and 0 ) 78.41, respectively,50 and λ∞+ and λ∞- for LiNO3 are 38.6 and 71.46, respectively.38 Therefore, the value of the A coefficient calculated with eq 7 is 0.007 228. Falkenhagen51 reported the values of the A coefficients for some electrolyte solutions at several temperatures. Figure 7 shows the temperature dependence of η0-1(0T)-1/2 for pure water calculated with IAPWS31,50 formulations for the viscosity and relative permittivity for various pressures. As one can see from this figure, the values of the complex η0-1(0T)-1/2 almost linearly increase with temperature and are little affected by pressure. The present experimental data for the relative viscosity η/η0 at atmospheric pressure and for various temperatures together with the data reported by other authors at low concentrations were used to calculate A and B coefficients in the Jones-Dole48 equation (6). The B coefficient can be estimated from the experimental viscosity data as

B)

(η/η0) - 1 - Ac1/2 c

Figure 8. Jones-Dole plot, [(η/η0) - 1]/c1/2 vs c1/2, for H2O + LiNO3 solutions for isotherm 298.15 K at atmospheric pressure. Table 3. Viscosity A and B Coefficients of Aqueous LiNO3 as a Function of the Temperature

(9)

using the theoretical value of A or as the slope of the dependence [(η/η0) - 1]/c1/2 on c1/2 (the Jones-Dole plot; see Figure 8)

[(η/η0) - 1]/c1/2 ) A + Bc1/2

Figure 7. Temperature dependence of the complex η0-1(0T)-1/2 in eq 7 for the pure water at various pressures calculated with IAPWS.31,50

(10)

The present viscosity data for H2O + LiNO3 solutions are presented in Figure 8 together with the data reported by other authors at low concentrations in the Jones-Dole plot, [(η/η0) - 1]/c1/2 vs c1/2, for the selected temperature of 298.15 K. The coefficients A and B, the intercept and the slope, respectively, of a the JonesDole plot, were calculated using least-squares analysis of the present data and the data reported in other studies from the literature for various temperatures. As Figure 8 shows, the data lie on the straight line with negligible scatter in the concentration range c < 0.112 mol‚L-1, but this concentration range might change with temperature. The A and B coefficients were calculated with only the data points in the linear region. The results are summarized in Table 3 and presented in Figures 9 and 10 as a function of temperature together with values reported by other authors and calculated from theory. As one can see from this table and Figures 9 and 10, the agreement between A and B coefficients

T, K 298.15 323.15 348.15 353.15 373.15

B, dm3‚mol-1 (ionic B A, B, A, coefficients dm3/2‚mol-1/2 dm3/2‚mol-1/2 dm3‚mol-1 (experiment) data9,38,52) (theory, eq 7) (experiment) 0.103 0.117 0.130 0.133 0.146

0.104, 0.101 0.131

0.0072

0.0084

0.0071 0.0075 0.0080 0.0081 0.0085

derived in the present study and calculated with theory and ionic B coefficient data is good. The viscosity A coefficient almost linearly increases with temperature like the complex η0-1(0T)-1/2 (see Figure 7). Therefore, the values of η0-1(0T)-1/2 are the leading contribution to the temperature behavior of the viscosity A coefficient. Cox and Wolfenden52 assumed the additivity of the ionic B coefficients. This additivity is B ) ∑νiBi, where the summation extends over all of the ions present and the ionic B coefficients Bi. The values of Bi are constant at a given T for given ions in a specific solvent and describe solely the ion-solvent interactions. The present results for the B coefficient satisfactorily agree with the data estimated from the additivity principle (see Table 3 and Figure 10). For electrolyte solutions such as LiNO3(aq), the viscosity B coefficient is positive. More typical large positive values of the B coefficient are found with ions that are strongly hydrated, for example, at 298.15 K;

Ind. Eng. Chem. Res., Vol. 44, No. 2, 2005 423 Table 4. Values of Viscosity A, B, and D Coefficients of Equation 11 for Aqueous LiNO3 as a Function of the Temperature

Figure 9. Experimental viscosity A coefficient of the H2O + LiNO3 solutions as a function of the temperature together with values calculated from theory (eqs 7 and 8). The solid curve is a guide to the eye.

Figure 10. Experimental viscosity B coefficient of the H2O + LiNO3 solutions as a function of the temperature together with values calculated from ionic B-coefficient data reported by various authors from the literature: b, this work (experiment); O, Jenkins and Marcus;9 0, Cox and Wolfenden;52 ×, Robinson and Stokes.38 The solid curve is a guide to the eye.

the B coefficient value for Li+ is 0.1495 (structureordering ion). Negative values of the B coefficients are found for the ions that exert a “structure-breaking” effect on water (for example, NO3- structure-disordering ion). The values of the B coefficients for NO3- ions become less negative or even change to positive as the temperature is raised.9 There are other types of electrolyte solutions (for example, H2O + KNO3, H2O + KBr, H2O + KCl, H2O + KI, H2O + RbCl, and H2O + CsCl) for which the viscosity decreases with concentration at low electrolyte concentrations, reaching a minimum value and then increasing monotonically for higher concentrations. For these types of electrolyte solutions, the B coefficient is negative. Kaminsky,36,37 Desnoyers and Perron,40 Feakins and Lawrence,53 Desnoyers et al.,54 and Robertson and Tyrrell55 added a quadratic term

η/η0 ) 1 + Axc + Bc + Dc2

(11)

to extend the Jones-Dole equation for more concentrated electrolyte solutions (c > 0.1-0.2 m). The new Dc2 term of eq 11 includes all solute-solvent and solute-solute structural interactions that were not accounted for by the Axc and Bc terms at high concentrations such as40 high terms of the long-range

T, K

A, dm3/2‚mol-1/2 (exp. this work)

B, dm3‚mol-1 (exp. this work)

D, dm3/2‚mol-1/2 (exp. this work)

298.15 323.15 348.15 353.15 373.15

0.0071 0.0073 0.0079 0.0081 0.0084

0.104 0.118 0.131 0.131 0.133

0.00163 0.00167 0.00169 0.00395 0.01305

Coulombic forces, high terms of the hydrodynamic effect, and interactions arising from changes in solute-solvent interactions with concentration. The range of the concentration in the present study overlaps the range where the Dc2 term is essential. The values of the A, B, and D coefficients calculated with the present viscosity data together with data reported by other authors at low concentrations are presented in Table 4 for various temperatures. As one can see from this table, the values of the D coefficient are sharply increasing with temperature at T > 348 K. This is means that at high temperatures the effect of solute-solvent (ion-solvent) and solute-solute (ion-ion) long-range Coulombic interactions on experimental values of the viscosity of the solution becomes more essential (on the order of 0.008). Lencka et al.8 developed the model for multicomponent, concentrated electrolyte solutions. They developed the method for calculating the viscosity of aqueous systems ranging from dilute to very concentrated. Chandra and Bagchi5,6 developed a new microscopic model for the ionic contribution to the viscosity of electrolyte solutions on the basis of mode coupling theory. They presented a microscopic study of the concentration dependence of the viscosity of an electrolyte solution. This theory predicts a stronger increase of the viscosity with concentration than the classical theory of Falkenhagen.44 According to this theory at finite concentration, the viscosity increases nonlinearly with the square root of the concentration, in contrast to the Falkenhagen44 equation. Jiang and Sandler3 developed a new statistical mechanics-based model for the viscosity of electrolyte solutions. This model based on the combination of liquidstate theory and absolute rate theory

η/η0 ) (1 + axc + bc) exp(fEX/RT)

(12)

where fEX is the excess contribution of the activation Helmholtz energy of the solution with the pure solvent. Jiang and Sandler3 provided the analytical expression for hard-sphere and electrostatic contributions fEX ) fHS + fEL. The parameters a and b in eq 12 have physical meaning. As is the case for the parameter B in the Jones-Dole equation (6), the b value indicates the degree of order or disorder introduced by ions into the solvent structure. The value of the parameter a was fix at 1.6.3 The analytical expression for fEX (for details, see work by Jiang and Sandler3) contain three physical meaning fitting parameters (σ+ 1 , 1, and b), which defined the diameter cations and dielectric constant of an electrolyte solution,3 + σ+ ) σ+ P + σ1 c

and

 ) 0/(1 + 1c)

+ where σ+ P ) 1.2 Å (Li ) and σP ) 3.78 Å (NO3 ). The + values of parameters σ1 , 1, and b for LiNO3(aq) derived from the present experimental viscosity data at

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viscosity (η/η0) of aqueous LiNO3 solutions as a function of temperature are studied. The predictive capability and validity of the various theoretical models are tested. Acknowledgment I.M.A. thanks the Physical and Chemical Properties Division at the National Institute of Standards and Technology for the opportunity to work as a Guest Researcher at NIST during the course of this research. The authors also thank Dr. D. Diller for his assistance in improving the manuscript. Literature Cited Figure 11. Comparison of concentration dependences of the present experimental viscosity results and the data reported by other authors for H2O + LiNO3 solutions with values calculated from various theoretical models at atmospheric pressure and at a temperature of 298.15 K. Table 5. Optimized Values of Adjustable Parameters of Equation 12 for Aqueous LiNO3 at Atmospheric Pressure as a Function of the Temperature T, K

σ+ 1

1

b

298.15 323.15 348.15 353.15 373.15

0.0325 0.0328 0.0330 0.0331 0.0333

0.0268 0.0271 0.0273 0.0275 0.0280

0.389 0.392 0.410 0.418 0.422

0.1 MPa for various temperatures are presented in Table 5. Figure 11 demonstrates the predictive capability of the various models for the viscosity of electrolyte solutions in the wide concentration range. As one can see from Figure 11, this statistical mechanics-based model by Jiang and Sandler3 reproduces the present viscosity data for LiNO3(aq) solutions as well as multiparametric empirical correlations (AAD is comparable with empirical correlations), which were directly fitted to the experimental data. Conclusion Dynamic viscosities of four (0.265, 0.493, 1.074, and 1.540 mol‚kg-1) of the aqueous LiNO3 solutions have been measured in the liquid phase with a capillary flow technique. Measurements were made at four isobars (0.1, 10, 20, and 30 MPa). The range of temperatures was from 298 to 573 K. The total uncertainties of the viscosity, pressure, temperature, and concentration measurements were estimated to be less than 1.5%, 0.05%, 15 mK, and 0.014%, respectively. The reliability and accuracy of the experimental method were confirmed with measurements on pure water for five isobars (1, 10, 20, 40, and 50 MPa) and at temperatures between 295 and 578 K. The experimental and calculated values of the viscosity for pure water from the IAPWS31 formulation show excellent agreement within their experimental uncertainties (AAD ) 0.27%). The measured viscosity values of solutions at atmospheric pressure were compared with the data reported in the literature by other authors. Good agreement (deviations within (0.5-1.2%) is found between the present measurements and the data sets reported by other authors in the literature. The temperature, pressure, and concentration dependences of the relative viscosity (η/η0) are studied. The values of the viscosity A and B coefficients of the Jones-Dole equation for the relative

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Received for review July 8, 2004 Revised manuscript received September 30, 2004 Accepted October 26, 2004 IE0494014