J.C.S. Faraday I, 1980,76,2531-2541Determination of the Viscosity of Molten KN03with an Oscillating-cup ViscometerBY YOSHIYUKI ABE, OTOYA KOSUGIYAMA, HIROYUKI MIYAJIMAAND A m NAGASHIMA"Faculty of Engineering, Keio University,3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223, JapanReceived 29th August, 1979An oscillating-cup viscometer for studying high temperature melts was built and absolute measure-ments on the viscosity of molten KN03 along the saturation line have been performed.For an accurate numerical evaluation of the viscosity, an assessment on existing rigorous workingequations was made.The temperature range of the present measurements covered up to 973 K where no earlier experi-mental data are available. In the temperature range where earlier experimental data are available,the measured viscosity of the present study was in good agreement with those of recent studies.A correlation based on the present results and those of other selected studies was made and theequation is believed to be valid over the whole temperature range where molten KN03 is stable.Compared with a current recommendation for the viscosity of KN03 proposed by Janz ef al., thepresent equation gives more satisfactory results for the viscosity of molten KN03.Although a number of experimental studies on the viscosity of high temperaturemelts exist, few satisfactory studies have been performed because of experimentaldifficulties mainly due to elevated melting points and severe conditions of chemicalhandling.Hence, even if experimental data obtained by different authors are avail-able, a sizeable disagreement is commonly found.Since precise information onthe thermophysical properties of high temperature melts is urgently required inpower technology and nuclear power technology in particular, the present situationmust be improved.is based on theidea of establishing " calibration-quality " data for moderately high and high tem-perature physical property measurements; KN03 and NaCl were selected as thestandard salts. For the viscosity of KN03, several consistent data sets obtained bydifferent experimental methods are available and the equation recommended byJanz et aL2 gives a reliable viscosity value up to 750 K. However, taking into accountthe fact that the melting point of KN03 is 607.4 K and comparing with other physicalproperties such as the density and the electric conductivity whose values are establishedup to 880 K with even smaller ~ncertainties,~ further precise knowledge of theviscosity in the extended high temperature range is still required.On the other hand,the situation as regards the viscosity of NaCl is also unsatisfactory [a recommendationappearing in ref. (3) is definitely doubtful1.tThe present study is an attempt to construct a new apparatus and to performprecise measurements on a series of high temperature melts over a wide temperaturerange. For this purpose the oscillating-cup method was applied and the viscosityThe Molten Salts Standards Program initiated by Janz in 1974t We will publish new results of the viscosity of NaCI.2532532 VISCOSITY OF KNOjcalculation was carried out on an absolute base using a working equation which wasselected after an assessment of existing rigorous working equations.The viscosity measurements on molten KN03, for the first step, were performedin the temperature range 623-973 K with an estimated accuracy of 1.1 %.A newcorrelation for the viscosity of KN03 was formulated by evaluating the resultsobtained in the present study and those of some other earlier studies judged by theauthors to be reliable.EXPERIMENTALPRINCIPLEFor the viscosity measurements on high temperature melts which have relatively lowviscosities, two major types of experimental method are generally applicable, that is, thethe capillary method and the oscillation method.While the capillary method has mostlybeen used as a relative method in a number of studies on high temperature melts, difficultiesdue to the corrosion and the machine working mean that the temperature range of this methodis limited to w 1100 K as the highest temperature.The oscillation method, since it enjoys two different variations, can be applied in the mostpreferable way. One method has a suspended pendulum, generally a disc, cylinder orsphere, immersed in the fluid to be measured and the other method has the fluid containedin a suspended spherical or cylindrical hollow crucible. Using either method, the viscositycan be determined from the geometrical dimensions of the suspended system and the charac-teristics of its torsional oscillation.The second method, the so-called oscillating-cup viscometer, consists essentially ofan oscillating system with a thin suspension wire and a hollow cylindrical cup which containsfluid.Once a torsional oscillation is given to the system, the executed oscillation is graduallydamped with a constant period and a constant decrement and these parameters are charac-terised by the viscosity and density of the fluid, the moment of inertia of the system andthe dimensions of the cylindrical cup. In other words, the viscosity can be evaluated fromthese measurable parameters.The following features enabled us to believe that the oscillating-cup method would bethe most suitable one for the viscosity measurements on high temperature melts : (1) Onlya small amount of specimen is required for measurements (for instance, 14.666 g in the presentstudy).(2) The desired temperature condition can be easily obtained since the specimenis contained in the small cup. (3) Suitable material for the cup can be successfully contactedwith corrosive fluids and its cylindrical shape allows precise machine working.WORKING EQUATIONIn contrast with the inherent advantages of the oscillating-cup method over others, itsmathematical complexity has restrained its extensive application. Most earlier measure-ments, therefore, were performed on a relative base. Meanwhile, several efforts have beenmade to derive a rigorous solution for this viscometry and the solutions thus obtained canbe classified into two types.The first ~ n e , ~ - ~ which involves the Bessel functions of complexarguments, has not been able to be employed practically for viscosity calculations. Onlymanageable approximations from this rigorous solution allowed the viscosity calculationon the absolute base; however, the applicability of such approximations is dependent onthe characteristic dimensions of the cup and the kinematic viscosity of fluid under investiga-tion. Further details about such restrictions are mentioned else~here.~ The secondsolution lo was derived by introducing the Laplace transform so as to facilitate the compli-cated evaluation.In the present study the following assessment on these two rigorous solutions was madeand the numerical evaluation of the measured viscosity was carried out using a selectedworking equation.First, we briefly show how the rigorous solutions were derived in theprevious analysesY . ABE, 0. KOSUGIYAMA, H . MIYASIMA AND A. NAGASHIMA 2533The motion of the oscillating system is described by the following basic equationd2a dadt2 dtI -+L -+Ka = 0where I is the moment of inertia of the system, a is the azimuthal angle of the oscillation, Kis the torsional constant of the suspension wire and L(da/dt) denotes the damping torque onthe system. To obtain an expression for L, one has to solve the Navier-Stokes equationtogether with the equation of continuity under the following assumptions : (1) There is nomotion in the axial direction. (2) There are no body forces except gravity.(3) There is noslip at the wall of the cup. (4) The velocity is small enough to neglect the non-linear terms.A final form for L is I ,m h r J2( m r) a) tanh (Znh)+2m4r2 c = 2nV[ - Jl(mr) n = l lik:where Z i = k i - m2 and m2 = (p/qT)(S- 2ni), (i = , / T ) r is the inner radius of the cup, h theheight of the fluid in the cup, p the density, q the viscosity, kn the roots of Jl(knr) = 0 and J1and J2 denote the Bessel functions of order 1 and 2, respectively.Consequently, the rigorous working equation for the viscosity evaluation is expressed inthe formwhere A = 6/2n, 6 represents " actual " logarithmic decrement and Tand To are the periodsof oscillation for the presence and absence of fluid, respectively. The left-hand side of eqn (3)is a modified form of that appearing in ref.(6) where some errors are found in the calculation.As we have mentioned, no attempt has so far been made to solve eqn (3) directly since theBessel functions of complex arguments could not be generated. We have employed sub-routines, however, which could generate the Bessel functions of complex arguments with anaccuracy of 0.001 % and, therefore, make it possible to solve eqn (3) without any approxima-tion.From eqn (3) one can obtain two viscosity values independently by solving either theimaginary or real part. In the imaginary part the term T/To-1, which is nearly zero, isdominant. On the other hand, the value 6 is dominant in the real part, so that the realpart is considered to give a more accurate viscosity value.Alternatively, another approach to obtain an expression for L was developed by Kestinand Newell.lo They facilitated the complicated evaluation by introducing the Laplacetransform and a series of dimensionless space coordinates.Their analysis took account ofan intrinsic damping torque due to internal friction of the suspension wire in addition to thatdue to the fluid contained in the cup. The rigorous working equation they derived iswhere(S+Ao)2+ 1 + D(S) = 0, (4)I' is the moment of inertia of the fluid in the cup, a0 is the logarithmic decrement due to thesuspension wire, ,Un are the roots of J1(pn) = 0 and qo and to are the dimensionless spacecoordinates h(2np/qTo)* and r(2rp/qTo)*, respectively.Eqn (4) also gives two viscosityvalues for the same reason as eqn (3). However, the real part is considered to give a lessaccurate viscosity value in this case, since the relationship between the real and imaginaryparts held in eqn (3) reverses in eqn (4)2554 VISCOSITY OF KN03Prior to the numerical evaluation of the measured viscosity, convergence manners andestimated accuracies for these four equations, the imaginary and real parts of both eqn (3)and (4) were examined. Table 1 lists the viscosity values calculated from each equationwith summation terms up to 10, 30, 50 and 100; estimated accuracy for each case is alsogiven. The accuracy was estimated by evaluating all the uncertainties of the values appearingin the equation. The considerable difference between the period dominant equations,imaginary part of eqn (3) and real part of eqn (4), and the decrement dominant equations,real part of eqn (3) and imaginary part of eqn (4), indicates that an uncertainty of the periodsTand To due to conducted thermal effect on the suspension wire is possibly greater than weestimated.However, for the decrement dominant equations, their accuracy estimationsare believed to be reasonable since even such an uncertainty scarcely affects them.TABLE 1 .-COMPARISON OF THE RIGOROUS EQUATIONS~50 100n = l c c equation accuracy( %) n = 1 n= 1 n= 1estimatedreal part of eqn (3) * 1.1 2.104 2.104 2.104 2.104imaginary part of eqn (3) & 5.1 2.71 5 2.71 5 2.715 2.715real part of eqn (4) & 3.5 3.781 3.065 2.924 2.819imaginary part of eqn (4) & 1.1 2.123 2.098 2.099 2.099a First experimental value at 673 K was tested for these calculations.The slight difference between the values obtained from two decrement dominantequations may be attributed to the difference in evaluation methods for 6.In eqn (3) the" actual " decrement S is regarded as a value which is corrected by subtracting the intrinsicdecrement do from the measured one. On the other hand, both intrinsic and measureddecrements are individually taken into account in eqn (4).Although table 1 indicates that eqn (3) is converging with a smaller sum, eqn (4) canconverge more rapidly in the iteration process and takes less time. For these reasons theimaginary part of eqn (4) was selected as the present working equation, but both decrementdominant equations can still be regarded to give identical viscosity values within the accuracyof the present measurements.Note that the viscosity calculated from an approximation deduced by Roscoe,' whichhas been employed by most of authors who made absolute measurements by the oscillating-cup method, was always 0.6-1.5 % lower than the rigorous one, so that the approximationcannot be validly applied in the present situation.APPARATUSA schematic diagram of the main part of the apparatus is shown in fig.1. The oscillatingsystem, which consists of a Pt92-W8 suspension wire l1 of 0.2 mm diameter (2), a reflectionmirror with its holder (3), an inertial disk (4), a molybdenum connecting rod ( 5 ) and acylindrical cup (6), is suspended in a closed vessel.The vessel can be evacuated to w 10-3Paso that disturbances due to any remaining gas can be eliminated during the run. An initialtorsional oscillation is fed to the suspension system using an oscillation initiator (1) locatedon the top of the vessel.The specimen contained in the cup is heated with an electric furnace which has fourteenheating elements made of Sic and the temperature of the cup can thus ultimately reach1800 K. The temperature is measured by means of five Pt-Ptl3Rh thermocouples (10)arranged around the cup. Each thermocouple was previously calibrated at the meltingpoints of six extra pure metals, uiz., tin (231.968"C), zinc (419.58"C), aluminium (660.46"C)Y . ABE, 0.KOSUGIYAMA, H . MIYASIMA AND A . NAGASHIMA 2535VacuumFIG. 1.-High temperature oscillating-cup viscometer. (1) oscillation initiator ; (2) Pt92-W8suspension wire ; (3) mirror ; (4) inertial disc ; (5) molybdenum connecting rod ; (6) cylindricalcup ; (7) molybdenum radiation shields ; (8) AI2O3 pipe ; (9) Sic heating elements ; (10) Pt-Ptl3Rhthermocouples.RG. 2.-Cylindrical cup2536 VISCOSITY OF KN03silver (961.93"C), gold (1064.43"C) and palladium (1 554°C). Electromotive forces cot-responding to these melting points were determined by applying the wire method l2 and itsestimated uncertainty is believed to be < 0.5 K even at the highest temperature.The cylindrical cup is made of stainless steel and is shown in detail in fig. 2 with majorcharacteristic dimensions.In any kind of oscillation viscometry, experimental error is mostly governed by an un-certainty due to the decrement measurement.In the present study the measurements ofthe period and the logarithmic decrement were perfolmed using the optical measuringsystem shown in fig. 3. The motion of the torsional oscillation can be detected as themotion of a beam which is a reflection of a He-Ne laser from the mirror fixed to the sus-pension system. The laser and the detecting devices are placed w 2 m from the viscometer.Slit A has to be properly placed so that it is positioned at the centre of the oscillation andslit B is placed beside slit A approximately 15 cm from it. When the moving reflectioncomes to a photomultiplier through the slit A or B, a signal to trigger a digital counter isgenerated.Mirr Slit A MirrorHe-Ne Laser nSlit B 3 Photomuttiptier %: Counter (period)Counter (logarithmicPrinterFIG.3.-Optical measuring system.The period is determined from a successive passing of the reflection at the slit A. Thelogarithmic decrement is also optically determined from an increment of successive timeintervals of the reflection passing through between slits A and B,13 that is,where ti is the time interval at an arbitrary moment and ti+n is that of nth period later. Theseprocedures are also graphically shown in fig. 4. The optical measuring system thus allowedprecise measurements for the decrement and the period, with an accuracy of k0.3 and& 0.003 %, respectively.PROCEDUREBefore the viscosity measurements were made, the moment of inertia, the period and thedecrement were measured under conditions exclusive of the fluid.The moment of inertiaat 20°C was determined using two different brass rings whose moments of inertia wereaccurately known. The period and the decrement were measured in the temperature rangefrom 20°C to the highest temperature where the viscosity measurements would be performed.While the period was fairly well represented as a linear function of temperature, the decrementwas found to be constant throughout the whole temperature rangeY. ABE, 0. KOSUGIYAMA, H. MIYAJIMA AND A . NAGASHIMA 2537The specimen, KNO, of purity 99.99 %, was first dried at 200°C in vacuo for 20 h andplaced in the cup in a nitrogen atmosphere.With the aid of a nut and a copper gasket, thiswas finally sealed in a vacuum chamber. Consequently, the measured viscosity turns out tobe a value along the saturation line.AFIG. 4.-Damped oscillation and time measurements.RESULTS AND DISCUSSIONThe viscosity measurements on molten KN03 were then performed in the tem-perature range 623-973 K at intervals of z 50 K. At each temperature the measure-ment was repeated several times and a reproducibility of k0.5 % was attained.The accuracy was estimated to be & 1.1 %. The density was calculated from anequation recommended in ref. (3). Table 2 iists the experimental results and eachnominal viscosity corrected to the nominal temperature.At higher temperatures a problem due to decomposition may arise.In theatmosphere the decomposition of KN03 is said to begin near 920 K.I4 At least atthe highest temperature of this study, 973 K, it is likely that decomposition has takenplace to some extent. The equilibrium constant at the highest temperature for thefollowing reactionis calculated to be 1.43 x lO-I.l4 Furthermore, taking account of the volume of themelt and of the space above the melt inside the cup, 9.14 and 4.67 cm3, respectively,the mole fraction of KNOz produced is estimated to be 0.025 and therefore the partialpressure of the evolved oxygen in the closed cup will be z 30 bar when vaporizationof the melt is not considered. The decomposition is considered to have two effectson the measured viscosity. First, the effect due to the evolved oxygen, from a roughestimation of the effect on the viscosity of KN03, is shown to influence the measuredviscosity values by 0.6 % at the most, since the logarithmic decrement is approximatelyproportional to the square root of the product of the viscosity and the density of thesubstance.When the vaporization of KN03 is taken into account, the estimatedinfluence will be smaller. Secondly, the effect due to the produced KN02 is estimatedto be of the order of 0.5 % by assuming that the viscosity for the mixture of KN03and KN02 is approximately expressed by a linear interpolation of their viscosityvalues extrapolated to the referred temperature.KN03 = KN02+902 (62538 VISCOSITY OF KNOSFrom these estimations and there being no anomaly in the measured viscosity,we conjectured that the influence of the decomposition on the present measurementsshould have been negligibly small or comparable to the experimental error at the most.In the case of unassociated liquids, it is empirically known that the ArrheniusTABLE 2.-EXPERIMENTAL RESULTStemperature Y mom./K T/s 6/10-2 /10-3Pa s Tnom./K /10-3Pa s622.8623.3624.0672.8672.1672.3671.8723.7723.5723.3723.7724.3773-5772.9773.3773.7773.1813.18 12.7812.4812.3812.3873.1872.9873.2873.5923.5923.1923.1923.4923.2973.8973.8973.5973.1973.36.955 566.956 276.955 846.955 966.956 156.956 316.956 356.958 826.958 616.958 496.958 756.958 786.960 766.960 486.960 526.960 496.960 506.963 746.963 586.963 516.963 526.963 616.968 776.968 566.968 576.968 486.972 906.972 686.972 786.972 636.972 726.976 046.976 396.976 046.976 116.976 311.81491.81291.80831.69621.69581.69571.70001.60191.60001.60021.59951.59891.51 531.51511.51261 S1051.51261.45091 .a 6 71.45021.44961.44871.36961.36801.37061.371 51.31831.32281.31851.32131.31861.27551.28071.27571.27851.27582.7402.7382.7102.0992.0972.0972.1151.6971.6911.6921.6891.6851.3981.3981.3911.3851.3911.2091.1991.2071.2061.2040.9990.9951.0011.0020.8790.8880.8800.8850.8800.7860.7950.7870.7920.787623 .O673 .O723.0773.0813.0873.0923.0973.02.7372.7432.7262.0972.0882.0902.1011.7021.6941.6941.6941.6941.4011.3981.3931.3891.3921.2091.1981.2051.2031.2011 .0000.9951.0021.0030.8800.8880.8800.8860.8800.7870.7960.7880.7920.78Y .ABE, 0. KOSUGIYAMA, H. MIYAJIMA A N D A . NAGASHIMA 2539equation is valid to describe qualitatively their temperature dependence of theviscosity. A least-squares fit of the present results to the Arrhenius equation yieldswith q in Pa s and R in J K-l mol-l. Although this equation represents themeasured viscosity (see fig. 5), a slight systematic deviation has been observed athigher temperatures.At lower temperatures some reliable studies by other authors are available.Taking these selected earlier experimental data into account in addition to the presentq = 8.541 x exp (17 928/RT), (7)-0.51 I I I I I1.0 1.2 1.4 1.6 1.8103 KITFIG.5.-Natural logarithm of measured viscosity (9 in Pas) as a function of inverse temperature.experimental data, an attempt at establishing a better correlation was made. Apolynomial expression with an inverse power series was found to be quite adequateand the resultant correlation has the form6q = aiTmi, (8)i = lwith the coefficientsQ, = - 1.298 806 x lo4,a3 = -9.978 276 x lolo,a2 = 5.845 601 x lo7,a4 = 8.421 488 xa5 = -3.492 264 x 10l6, a6 = 5.770 989 xThe measured viscosity values are fairly well represented with a standard deviationof 0.69 %.While eqn (7) is a simpler form with only two disposable parameters,to describe the viscosity over a wide temperature range with sufficient accuracy eqn(8), which is also based on other selected earlier experimental data, is more adequate.Fig. 6 shows the deviation plots of all the available data which have been reportedso far in reference to eqn (8) and table 3 lists the details of these experiments and thestandard deviations from eqn (8). The last column of table 3 also lists standarddeviations but from the value in ref. (2) which, as we have already mentioned, giv2540 VISCOSITY OF KN03the viscosity up to 750 K. The calculation of the standard deviations, therefore,excludes the experimental data beyond this temperature limit.When theseeliminated data are also taken into account, the standard deviations turn out to bethe values appearing in the parentheses of the last column. Within the temperaturerange the values in ref. (2) certainly give the viscosity with nearly the same standard4.0-4.0 0-6.0 * 1. I I 1 1 I600 700 800 900 1000temperature /I<FIG. 6.-Deviation from qcalc [eqn (8)]. 0, ref. (15) ; 0, ref. (16) ; A, ref. (17) ; +, ref. (18) ;A, ref. (20); 0, ref. (21); V, ref. (22); V, ref. (23); c), ref. (24); A, ref. (25); 0, ref. (26);H, ref. (27) ; 0, ref. (28) ; a, ref. (29) ; *, present study.TABLE 3 .-SUMMARY OF PREVIOUS VISCOSITY MEASUREMENTS ON KNOjs.d. fromref. temperature stated s.d. from valuesno. method range/K accuracy( %) eqn (8)( %) in ref.(2)( %)151617181920212223242526272829capilIarycapillaryoscillating-discoscillating-ballcounterbalancedoscillating-balloscillating-cupcapillarycapillarycapillaryoscillating-balloscillating- balloscillating-balloscillating-cupcapiHarysphere606-686620.2-779.2622-680621-815620-700630.2-716.2632-764621-764616-765613.2-743.2615-764615-671625.6-685.2635-78761 3-7431- +22f l21.81 -221.2- + 1.51.91.53 .O0.81.20.50.61.80.71.63.60.62.50.72.01.6 (2.6)2.90.7 (2.8)0.90.4 (1.3)0.7 (0.8)1.9 (1.8)0.61.2 (1.6)3.11 .o2.0 (4.4)0.Y. ABE, 0. KOSUGIYAMA, H. MIYAJIMA AND A . NAGASHIMA 2541deviation as eqn (8), however, it is obvious from the last column of table 3 that theextrapolation of the values to higher temperatures is no longer valid.From thisstandpoint eqn (8) of the present study is expected to be more adequate as the viscosityequation of KN03 for calibration use. Eqn (8) can represent the viscosity valueswith an uncertainty comparable to the experimental errors and is believed to bevalid over the whole temperature range where molten KN03 is stable.We thank Prof. J. Kestin of Brown University for his helpful suggestions on theapparatus and Drs. K. Furukawa and H. Ohno of Japan Atomic Energy ResearchInstitute for their kind advice on the chemical preparation of the molten salts.G. J. Janz, personal communication.G. J. Janz, R. P.T. Tomkins, C. B. Allen, J: R. Downey Jr, G. L. Gardner, U. Krebs andS. K. Singer, J. Phys. Chem. Ref. Data, 1975, 4, 871.G. J. Janz, F. W. Dampier, G. R. Lakshminarayanan, P. K. Lorenz and R. P. T. Tomkins,Nat. Stand. 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