A new technique for measuring the Fickian diffusion coefficient in binary liquid solutions

Experimental Thermal and Fluid Science 18 (1998) 33±47

A new technique for measuring the Fickian di€usion coecient in binary liquid solutions Massimo Capobianchi

a,* ,

Thomas F. Irvine Jr. b, Narinder K. Tutu a, George Alanson Greene a

a

b

Department of Advanced Technology, Brookhaven National Laboratory, Upton, NY 11973-5000, USA Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, NY 11974-2300, USA Received 8 July 1997; received in revised form 23 July 1997; accepted 10 February 1998

Abstract The present study details the development of a new technique for measuring the Fickian di€usion coecient in binary liquid solutions, and reports the coecients obtained using this new technique for two electrolytic systems. The new method, called the decaying pulse technique, takes advantage of the behavior of a semi-in®nite system exposed to a transient concentration pulse. The method permits simple, direct, and absolute determination of the di€usion coecient, and requires measurement of only time and distance. It is applicable to many di€erent types of ¯uid pairs, and requires no knowledge of solution properties. The decaying pulse technique was used to measure the average di€usion coecient of potassium chloride in water and sodium chloride in water, at 18.5, 25.0, and 30.0°C. The current experimental results were compared to those from other published investigations, and were generally found to agree within the predicted uncertainty of the current measurements ‹7.6%. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Di€usion coecient; Decaying pulse technique; Di€usion mass transfer; Mass transport property measurement

1. Introduction The di€usion coecient is a transport property which characterizes the rate of propagation of a concentration disturbance within a mixture of two or more constituents. Knowledge of this property is important in the prediction of di€usional mass transfer phenomena, since the di€usion coecient is required in the solution of the governing equations. While data are available in several handbooks for many ¯uid combinations, many cases exist where the data are incomplete or non-existent. Furthermore, care must be exercised when using reported measured data since these values may be dependent upon the experimental technique used. Hence, it is useful to de®ne the di€erent types of di€usion coecients which may be encountered, and to describe their physical signi®cance.

*

Corresponding author. Address: Department of Mechanical Engineering, Gonzaga University, 502 E. Boone Ave., Spokane, WA 99258-0026, USA. Tel.: 1 509 323-3541; fax: 1 509 324 5871; e-mail: [email protected]. 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 8 ) 1 0 0 0 6 - 7

Consider a stationary mixture of two components at constant temperature and pressure. If the concentration is non-uniform, mass transfer occurs which tends to drive the mixture to a homogenous state. This mass transfer is di€usional in nature and is proportional to the existing concentration gradients. Termed ordinary di€usion, it is governed by Fick's ®rst law: ~ ˆ ÿDrc; m

…1†

~ is the mass ¯ux vector, D is the di€erential difwhere m fusion coecient, and c is the concentration. The di€erential di€usion coecient, as de®ned in Eq. (1), is a fundamental transport property and is the constant of proportionality between the mass ¯ux and the driving concentration gradient. It generally varies with concentration, temperature, and pressure for any given ¯uid pair. Between concentration limits c1 and c2 , it is convenient to de®ne a mean value, called the integral di€usion coecient, which is valid within that concentration range. The integral di€usion coecient, Dint: , is calculated from the distribution of the di€erential di€usion coef®cient as follows:

34

Dint:

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

1 ˆ c2 ÿ c1

Zc2 D…c† dc:

…2†

c1

Many of the existing experimental techniques measure neither the di€erential nor the integral di€usion coecient. They measure some complicated average value which is dependent upon all the concentration pro®les existing within the apparatus during the experiment as well as the ¯uid combination under test. These coecients will be termed average di€usion coecients, Dave: , and will generally di€er from both the integral and the di€erential di€usion coecients. However, if a suciently narrow concentration range is selected for the experiment, the di€erential di€usion coecient may be considered constant. Then the average and the integral di€usion coecients will both approach the differential di€usion coecient, so that the measured data point may be considered the di€erential di€usion coecient at the mean concentration of the experiment. Many experimental methods have been developed for measuring di€usion coecients and literature describing the theory and the apparatus of the techniques is generally available. Woolf et.al. [1] o€er a comprehensive review of some of the more common methods. Many of these techniques, along with a few unique techniques, were used by previous investigators to measure di€usion coecients for the KCl±H2 O and NaCl±H2 O systems. A review of these studies may be found in Capobianchi [2]. A new method, called the decaying pulse technique, exploits the transient behavior of a semi-in®nite system exposed to a concentration pulse of limited duration at the boundary. Consider two semi-in®nite tubes each ®lled with a homogenous solution of di€erent concentration, as shown in Fig. 1. A concentration pulse is introduced into the upper tube by bringing its free surface into contact with the free surface of the other tube. Di€usion is allowed to proceed across the common interface until the end of the pulse (time t ˆ tp ) when they are suddenly separated. Thereafter, the concentration gradient within the upper tube decays from that existing at the end of the pulse. If the concentration is monitored at a selected location

Fig. 1. Schematic of the apparatus for the decaying pulse technique, and a di€erential control volume.

along the length of the tube, a peak will be observed 1. The concentration begins at its initial value, monotonically increases to a maximum some time after the end of the pulse, and ®nally decreases to a steady state value as time approaches in®nity. For a given pulse duration, tp , the time required for the peak to occur, tmax , is unique and depends upon the distance from the free end, xm , and the di€usion coecient, D. If tp , tmax , and xm are measured, then D may be determined from the solution of the governing equation. This is the essence of the decaying pulse technique. The decaying pulse technique has several advantages over other methods. The method is absolute and requires no calibration so that measured data are as accurate as the apparatus permits. The coecient is determined by measuring only time (tp and tmax ) and distance (xm ), and no ¯uid properties need be known except for the predetermined concentrations, cu0 and cl0 . The technique is relatively simple in both theory and apparatus, and run times with D  10ÿ9 m2 /s are of the order of 10 h. The thermal analog to this procedure was ®rst used by Kim and Irvine [3] to measure the thermal di€usivity and the heat capacity of carboxymethyl cellulose polyethylene oxide solutions. A probe was inserted into the center of a tank ®lled with solution and a square thermal pulse was generated by the probe. The temperature response of the solution was measured by two thermocouples located 0.7 and 1.2 mm from the probe. Two measurements were required since the heat capacity of the solution was also unknown. By measuring the time required for the peak temperature to occur at each thermocouple, the authors were able to evaluate the thermal conductivity, the thermal di€usivity, and the volumetric heat capacity of the ¯uid from the numerical solution of the governing equations. A similar technique was reported by Kosky [4] for measuring the thermal di€usivity of a diamond sheet. Rather than using a square pulse, a laser was used to apply a sinusoidal pulse to the edge of the sample. Temperature changes were measured at a given position along the sheet with an infrared microscope. From the solution of the governing equation and the measured time required for the pulse to reach the monitoring location, the thermal di€usivity was determined. The purpose of the current investigation was to develop the theory and the apparatus for the decaying pulse technique, and to use it to measure the di€usion coecient of potassium chloride in water and sodium chloride in water at 18.5, 25.0, and 30.0°C. These systems have been extensively studied at 25°C and provide a convenient check of the apparatus and of the technique at that temperature. However, few investigations exist near 18.5°C, and none were found at 30.0°C. The

1 It is assumed that the concentration in the lower tube is initially greater than that in the upper tube. If the reverse is true, a minimum will be observed.

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

data from this study were compared to published values where available.

0 6 t‡ 6 1;

t‡ > 1;

c‡ …x‡ P 0; 0† ˆ 0; 2. Analysis

c‡ …x‡ < 0; 0† ˆ 1;

Consider the two semi-in®nite tubes shown in Fig. 1. Both tubes are of uniform and equal cross-section and each is ®lled with a solution of known but di€erent concentration. The free surfaces of the tubes are brought together at the start of the experiment (time t ˆ 0), and di€usive mass transfer is allowed to continue across the common interface at x ˆ 0 until time t ˆ tp . The two tubes are then separated so that the boundary at x ˆ 0 becomes impermeable to the mass ¯ux. For the binary system being considered, the following assumptions are made: 1. All properties are independent of concentration, including the di€erential di€usion coecient. 2. The denser solution is in the lower tube and the temperature is uniform throughout, so that no natural convective motion exists. In addition, no other source of mechanical mixing is present. 3. The tubes are selected of sucient length that the concentrations at their ends, x ˆ ‹L, remain unchanged from their initial values during the duration of the experiment. Mathematically, this implies L ® 1. The mass balance on the di€erential control volume shown in Fig. 1 is then: o2 c oc ˆ ox2 ot with initial and boundary conditions:

D

0 6 t 6 tp ;

…3†

t > tp ;

oc …0; t† ˆ 0; ox c…x < 0; 0† ˆ cl0 ; c…1; t† ˆ cu0 ; c…1; t† ˆ cu0 ; c…ÿ1; t† ˆ cl0 ; c…x P 0; 0† ˆ cu0 ;

…4†

Introducing the above de®nitions and simplifying gives: oc‡ o2 c ‡ ˆ ; ot‡ ox‡2

oc‡ …0; t‡ † ˆ 0; ox‡ c‡ …1; t‡ † ˆ 0;

…6†

…7†

c‡ …1; t‡ † ˆ 0; c‡ …ÿ1; t‡ † ˆ 1; c‡ …ÿ1; t‡ † ˆ 1: Notice that no parameters remain. The dimensionless concentration pro®le depends only on the dimensionless time, t‡ , and the dimensionless position coordinate, x‡ . Furthermore, the di€usion coecient, D, appears exclusively in the de®nition of x‡ . Solution of Eq. (6) is performed in two steps (see Capobianchi [2] or Jost [5] for solution details). First, Eq. (6) is solved for the pulse period, 0 6 t‡ 6 1, using the initial and boundary conditions given in Eq. (7). The resulting complementary error function is obtained from similarity considerations: for c‡ ˆ c‡ (x‡ , 0 6 t‡ 6 1):  ‡  1 x ‡ p : …8† c ˆ erfc 2 2 t‡ Next, Eq. (6) is solved using the concentration pro®le existing at the end of the pulse period as the initial condition: for c‡ ˆ c‡ (x‡ , t‡ ˆ 1):  ‡ 1 x : …9† c‡ ˆ erfc 2 2 With the boundary conditions for the decay period (i.e. t‡ > 1) given in Eq. (7), the solution of Eq. (6) becomes: for c‡ ˆ c‡ (x‡ P 0, t‡ > 1): Z1 1 ‡ erfcjv=2j exp c ˆ p 4 p…t‡ ÿ 1† vˆÿ1

c…ÿ1; t† ˆ cl0 ; where cu0 is the initial concentration in the upper tube and cl0 is the initial concentration in the lower tube. The resulting equation with initial/boundary conditions shows the concentration to be dependent on position, x, and time, t, with the initial concentrations, cu0 and cl0 , the di€usion coecient, D, and the pulse duration time, tp , as parameters. To reduce the number of parameters, the governing equation and initial/boundary conditions are non-dimensionalized using the following de®nitions: c ÿ cu0 x t c‡ ˆ t‡ ˆ : ; x‡ ˆ p ; …5† cl0 ÿ cu0 tp Dtp

35

2

ÿ…x‡ ÿ v† 4…t‡ ÿ 1†

! dv:

…10† While the solution is analytical, the resulting integral in Eq. (10) must be evaluated numerically. The concentration pro®les for the upper tube (i.e. x‡ P 0) may then be plotted at selected times as shown in Fig. 2. Examination of these curves reveals behavior consistent with expectations. During the pulse period, 0 6 t‡ 6 1, the concentration at the interface is ®xed at c‡ ˆ 0.5, a consequence of the assumption of constant properties. After the pulse, the concentration pro®le at t‡ ˆ 1 decays with 0 slope at x‡ ˆ 0, indicating that the interface is impermeable to the mass ¯ux. The concentration decreases with increasing x‡ for all times, asymptotically approaching the upper tube initial concentration, c‡ ˆ 0. The concentration history at any selected x‡ location may also be plotted as shown in Fig. 3. At each location away from the interface, the concentration begins at its initial value, c‡ ˆ 0, increases to a maximum value, and ®nally decreases for large times.

36

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

Fig. 2. Concentration pro®les for the upper tube (i.e. x‡ P 0) at selected times: t‡ ˆ 0, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00, 3.00, and 4.00.

Also, the peak concentration decreases with increasing distance from the interface. More importantly, each location has only one peak, the time of which is unique and depends only on x‡ . Hence, it is possible to calcu‡ late the x‡ location for any given t‡ max , where tmax is the dimensionless time at peak concentration. This is of critical importance. Since D occurs only in the de®ni-

tion of x‡ , knowledge of x‡ for a given set of experimental conditions (i.e. location, x ˆ xm , and pulse duration, tp ) allows the determination of D from its de®nition, Eq. (5). ‡ The evaluation of t‡ max as a function of x is performed by numerically searching for the time at which the peak concentration passes at discrete values of x‡ . The

Fig. 3. Concentration histories at x‡ ˆ 0, 0.50, 1.00, 1.50, 2.00, 2.50, and 3.00 for 0 6 t‡ 6 10.

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

37

‡ Fig. 4. t‡ max at discrete values of x , from a numerical analysis.

results (see Fig. 4) reveal a smooth curve increasing with x‡ . It should be emphasized that this solution is subject to the assumptions previously listed. A least squares curve may be ®t to the calculated results shown in Fig. 4 once an appropriate model is determined. The recommended functional form, suggested from an order of magnitude analysis on a related system is ‡ 1=2 ‡ ÿ1=2 ‡ ÿ3=2 ‡ ÿ5=2 x‡ ˆ a0 ‡ a1
tmax ‡ a2
tmax ‡ a3
tmax ‡ a4
tmax ;

…11† where a0 , a1 , a2 , a3 , and a4 are the least squares ®t parameters. The domain of the calculated results was broken into 5 subdomains, and Eq. (11) was ®t to the computed data of each. The results of the least squares ®t are given in Table 1. These polynomials ®t the calculated numerical results within 0.0063%. The least squares polynomials provide convenient access to the numerical solution and permit a simple exper-

imental procedure for measuring the di€usion coecient: 1. Using an apparatus consistent with the schematic shown in Fig. 1, the investigator selects a location to monitor the concentration. The distance from that location to the tube interface is measured and recorded as xm . 2. The free ends of the tubes are brought together for a period of time, tp , after which the tubes are separated. The pulse duration, tp , is measured. 3. Throughout the duration of the experiment, the concentration is monitored at location x ˆ xm by observing some property which changes monotonically with concentration. At the conclusion of the experiment, this history is reviewed and the time at which the monitored property is maximum, tmax , is extracted. ‡ t‡ max is then calculated from the de®nition of t , Eq. (5): tmax ‡ tmax ˆ : …12† tp

Table 1 ‡ Parametres for least squares ®t of Eq. (11) to numerical t‡ max vs. x data Fit par.

1.072241 6 t‡ max 6 1.114184

1.114184 < t‡ max 6 1.177403

1.177403 6 t‡ max < 1.361544

‡ 1.361544 < t‡ max < 1.634899 1.634899 6 tmax 6 1.993997

a0 a1 a2 a3 a4

)121.536792 102.772178 )103.411537 210.790780 )88.2362346

)34.7553452 28.7554724 )28.1963293 63.6691895 )29.0271407

380.982990 )101.326630 )414.780503 187.140068 )51.5321010

37.2517840 )7.69627361 )45.0191936 24.7669733 )8.71496377

27.3527155 )4.88962209 )35.0652950 21.4487782 )8.32901274

38

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

4. The appropriate least squares ®t polynomial parameters are selected from Table 1 and the x‡ corresponding to the calculated t‡ max is evaluated using Eq. (11). 5. Finally, the di€usion coecient is determined from the de®nition of the dimensionless variable x‡ . Solving Eq. (5) for D with x ˆ xm gives: 2

Dˆ

…xm =x‡ † : tp

…13†

Note that nowhere is the knowledge of the concentration or any other solution property required. Nor does one need to know how the monitored property varies with concentration. All that is required is that the chosen property vary monotonically with concentration within the concentration range of interest. No calibration is required and the value of the initial concentrations are needed only to ascribe the measured di€usion coecient to the proper concentration. 3. Experiment An apparatus for performing measurements using the decaying pulse technique must meet several design requirements. First, the device must contain two tubes, and these must be arranged such that their free ends may be joined at the start of the experiment and separated at the end of the pulse. The mechanism which provides this function should do so with minimal disturbance to the ¯uids within the tubes. In addition, since the di€usion coecients are temperature dependent, provisions must be made for maintaining isothermal conditions in the tubes. Finally, a scheme for monitoring the concentration dependent ¯uid property must be incorporated. For the binary systems studied, sodium chloride in water and potassium chloride in water, changes in concentration were tracked by monitoring the temporal changes in the local index of refraction of the ¯uid at the location of interest (i.e. at x ˆ xm ). The index of refraction was selected because of its sensitivity to concentration and because of the simplicity of the monitoring technique. Glass tubes of square cross-section were selected for the apparatus. A laser beam was passed horizontally through the upper tube at the monitoring location (i.e. at x ˆ xm ), and forced to egress from the wall adjacent to the entering surface. The beam then traversed approximately 10 m, ending as a spot of light on a Cartesian grid. Local changes in the index of refraction of the solution caused the exit angle of the beam to change, thereby displacing the laser spot on the grid. Throughout the duration of the experiment, the location of the spot and the time from the experiment start were simultaneously recorded with a video camera. The spot would begin to move across the grid during the pulse period, continue in the same direction during the beginning portion of the decay period, and then move in the reverse direction for the remainder of the experiment. tmax was found by reviewing the recorded data

and determining the time at which the maximum spot displacement occurred. As the beam crosses the ¯uid under test at the monitoring location, it encounters a concentration gradient. Since the beam is of ®nite diameter (1.5 mm), the monitoring occurs over an area rather than at a discrete position. However, the spot movement history is developed from the experimental data by tracking the geometric center of the spot. These data are treated as if generated by an in®nitely thin laser beam. Therefore, a numerical investigation was performed to ascertain the consequences of this monitoring scheme (see Capobianchi [2]). The time at peak concentration obtained by monitoring over an area equivalent to the laser beam diameter was calculated to be negligibly di€erent (less than 0.05%) from that which would be measured if the concentration were monitored with an in®nitely thin beam. Fig. 5 shows a schematic diagram of the experimental set-up, and the front view cross-section of the test unit is shown in Fig. 6. The latter ®gure also shows the path of the laser beam through the test section and the surrounding water jacket. An optical rail was placed on a vibration dampening table and supported a laser with polarizing lens on one end, and the test unit on the opposite end. The laser beam, whose intensity was controlled by the polarizing lens, pierced the upper tube of the test unit at the monitoring location, passing through the ¯uid and exiting the tube, ®nally ending on the grid. To increase the path length from the tube to the grid, the beam was re¯ected o€ mirror no. 1 ®rst, then o€ mirror no. 2, and ®nally ended on the grid approximately 10 m from the exit point of the tube as shown in Fig. 5. A video camera with integral timer continuously monitored the beam position and the time, sending its signal to a high resolution video cassette recorder (VCR). During the experiment, isothermal conditions were maintained within ‹0.1°C by a pair of constant temperature baths ®lled with distilled water. The bath labeled ``bath no. 2'' in Fig. 5 maintained the temperature at the test value by use of an integral heater and controls, and provided no refrigeration. A built-in pump supplied this water via insulated tubing to water jackets which surrounded each tube of the test unit. The water jacket of the lower tube was circular in cross-section, its purpose being simply to maintain the lower tube isothermal. The upper tube's water jacket was rectangular in crosssection and served two functions: it maintained the temperature of the ¯uid in the upper tube constant, and provided corrective optics for the traversing laser beam as described by Capobianchi [2]. The other water bath, ``bath no. 1'' in Fig. 5, supplied refrigerated distilled water to a heat exchanger within bath no. 2, producing a thermal load and allowing test temperatures below ambient. The experiment temperature was measured by calibrated thermocouples at six locations: the inlet and the outlet of the upper tube jacket, the inlet and the outlet of the lower tube jacket, beneath the upper tube (during the decay period), and within bath no. 2. The thermocouples were previously calibrated with a precision plat-

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

39

Fig. 5. Schematic of the experimental set-up for the decaying pulse technique. Laser beam trajectory shown in plan view.

Fig. 6. Front view cross-section of the test unit. The inset in the lower left hand corner depicts a top view cross-section of the upper tube and jacket. Also shown in the inset is the trajectory of the laser beam.

40

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

inum resistance thermometer. An ice bath provided the reference junction and the voltages were remotely recorded throughout the experiment at 15 s intervals by a voltmeter/scanner/PC data acquisition system. Three precision mercury thermometers provided con®rmation of the thermocouple readings. These were located at the supply tube, at the return tube, and in bath no. 2. An additional precision mercury thermometer was placed in the room to measure the ambient temperature. To establish and break the concentration pulse, the test unit was designed with a rotating mechanism as shown in Fig. 6. Two circular stainless steel plates were stacked together, separated by a thin layer of vacuum grease. The grease acted as a sealant and also lubricated the sliding surfaces during rotation. The plates were held together by a center compression spring installed about a shoulder bolt and by two perimeter rings, one above the upper plate and the other below the lower plate, which were pushed together by eight compression springs equally spaced about their circumference. The plates had a set of three matching holes: a center hole through which the shoulder bolt was inserted to act as a pivot so that the lower plate could rotate while the upper plate was held stationary, and two 1 ´ 1 cm square holes located on opposite sides of the center hole at a common radius. On the upper plate, a 1 ´ 1 cm square glass tube, 15.24 cm long, was attached to one of the square holes. An identical tube was joined to the lower plate at the opposing square hole. The tube lengths (15.24 cm) were selected to approximate the semi-in®nite tube boundary condition. With measured di€usion coecients of the order of 10ÿ9 m2 / s, the dimensionless concentration predicted by the in®nite tube solution at the end of the tube is vanishingly small (2 ´ 10ÿ63 ) throughout the full 12 h duration of the experiment. Even with a di€usion coecient of 10ÿ8 m2 /s, an order of magnitude higher than those measured, the dimensionless concentration is always less than 4 ´ 10ÿ7 at the tube end. Hence, the semi-in®nite tube assumption is valid for the tubes selected. By rotating the lower plate assembly 180° about the shoulder bolt, the free ends of the tubes could be brought together to start the pulse. By a reverse rotation of 45°, the tubes could be separated to begin the decay portion of the experiment. Additional holes in the lower plate were provided which aligned with a mating hole in the upper plate at these critical positions. Two di€erent structures were used to support the test unit. Originally, the unit was suspended from a support post as shown in Fig. 6. The opposite end of the support post was attached to a translation stage which permitted the entire test unit to be moved vertically by means of an integral micrometer. A peripheral groove was cut about the outer wall of the upper ring as shown in Fig. 6. The groove was designed to be 0.5 mm smaller than the laser beam diameter and to have its center coincide with the interface of the two tubes. Prior to the start of the experiment, the laser beam was centered within the groove by translating the apparatus vertically until an equal

amount of the laser beam's edge was visible on either side of the exterior of the groove. The apparatus was then translated to the monitoring position, nominally xm ˆ 5 mm, the traversed distance being measured with an electronic indicator and con®rmed by the micrometer reading. This scheme had the advantage of allowing the monitoring position to be chosen for each experiment, but it consequently contributed a source of random error. In addition, the support structure was insuciently rigid, permitting the possibility of disturbing the position of the apparatus during rotation of the lower tube, thereby adding an additional uncertainty in xm . To reduce this random error and to resolve the support rigidity problem, a revised set-up was designed and installed. The test unit was fastened via a redesigned upper ring to a heavy aluminum channel which was permanently clamped to the optical rail. This ®xed the vertical location of the unit so that any uncertainty in xm might produce a systematic error but no random error. This design provided a more stable platform when rotating the lower half of the test unit, and also allowed ®ner control when setting the initial incidence angle between the laser beam and the tube. With the revised set-up, xm was determined by performing di€usion experiments with a binary liquid pair whose di€usion coecient was ``known'', rather than measuring the monitoring position directly. As reported by Woolf et al. [1], the integral di€usion coecients between potassium chloride solutions and pure water at 25°C are used for calibrating porous diaphragm di€usion cells, a common device employed in the measurement of integral di€usion coecients. To use these data, however, the relationship between the average and the integral di€usion coecients had to be investigated. Capobianchi [2] compared, via numerical analysis, the average di€usion coecients obtained using the decaying pulse technique to previously reported di€erential and integral di€usion coecients. The analysis was performed for both the NaCl±H2 O and for the KCl± H2 O systems at 25°C. The average di€usion coecients were found to di€er from the corresponding di€erential di€usion coecients by less than 1% for the concentration ranges used in the present experiments. Also, for experiments run at 25°C between pure water and 0.700 M KCl (5.072 wt.% KCl), the average di€usion coecients are essentially the same as the integral di€usion coecients. In fact, they di€er by less than 0.16%, which is well within the present apparatus ‹7.6% measurement uncertainty (Capobianchi [2]). Hence, experiments were run at 25°C with initial concentrations of 0.700 M KCl in the lower tube and pure water in the upper tube. Rather than performing the measurement as previously described, the published value was used to extract the monitoring position, xm , from Eq. (5). Using this technique, xm was determined with an uncertainty of 2.2%, yielding a maximum systematic uncertainty in the measured di€usion coecients of 4.4%. It should be noted that subsequent experiments which use the calibrated value of xm yield results which are relative to the calibration system data. It is however important to realize that

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

the decaying pulse technique is inherently an absolute technique rather than a relative method. The fact that xm was obtained by calibration rather than by direct measurement is a consequence of the diculties encountered in measuring xm directly with the equipment available. A more re®ned apparatus combined with a better measurement method would preclude the need to calibrate, and permit direct measurement of xm . Experiments began by ®rst starting the voltmeter/ scanner/PC data acquisition system. A tape was inserted into the VCR and recording started. With the video camera focused onto the spot, the video camera's timer was reset then triggered on. Hence the time since triggering was continuously recorded onto the tape. The lower tube was rotated slowly to the pulse position, with the rotation rate not exceeding 1 deg/s. With such a slow rotation rate, no mechanical mixing occurs at the interface, as was demonstrated (Capobianchi [2]) by ¯ow visualization experiments conducted on a mock-up of the test unit which was geometrically identical to the real unit. The mock-up was built completely of Lucite sheet, which is a clear cast acrylic material. Tests were run with either of two ¯uid pair combinations: a 0.5 wt.% NaCl solution in the upper tube with a 6.0 wt.% NaCl solution (the more dense solution) in the lower tube, or with pure water in both tubes. In each case, ®ve drops of red food coloring were added to the ¯uid in the lower tube, and mixed thoroughly. The tubes were ®lled and left undisturbed to allow the ¯uids to come to rest. The lower tube was then rotated to the pulse position at a constant rate of rotation. The ¯uid's response to the rotation in the vicinity of the interface was recorded on video tape. Two average rotation speeds were examined: 30 deg/s and 2 deg/s. A total of three experiments were performed for each of the ¯uid pair combinations using an average rotation rate of 2 deg/s; the results showed no visible disturbances, with the interface remaining sharp and unperturbed during the mating of the tubes. With a 30 deg/ s average rotation rate, disturbances were noted at the interface for each of the ¯uid pair combinations used. Hence, the adopted technique included always running experiments with the denser solution installed in the lower tube, and maintaining the average rotation rates at or below 1 deg/s, which is one half of the rate at which no disturbances were observed for either ¯uid pair combination. In addition, temperature gradients which may cause buoyancy induced mechanical mixing were precluded by circulating constant temperature water through the jackets surrounding the tubes. Also, temperature variations caused by the release/absorption of heat of dilution as the concentration changed within the tubes were calculated by Capobianchi [2] to be less than 0.004°C. Since this value was calculated assuming insulated tube walls, it is conservatively high and yet is still two orders of magnitude less that the temperature control of the apparatus, ‹0.1°C. A verbal announcement was made at the start of rotation, at the point of initial tube engagement, and at the

41

instant the two tubes were fully mated and rotation stopped (the beginning of the pulse). These announcements were recorded on the video tape by the video camera's microphone. Since the elapsed time was also continuously recorded on the video tape, the verbal announcements marked the times at which these events occurred. With typical pulse durations of approximately 4 h, the resulting random uncertainty in the knowledge of the pulse duration, estimated to be ‹30 s, only accounts for approximately 0.2% of the total random uncertainty in the experiment (‹7.6%). The experiment was left undisturbed except for regular checks to ascertain that all was operating normally, and that the video camera was properly focused. After 4 h, the lower tube was slowly rotated back to the rest position. Once again, verbal announcements were used to mark the time at the start of rotation, the instant the two tubes were totally separated (the end of the pulse), and when rotation ended. Nothing else remained to be done except for exchanging the VCR tape with a new one after 6 h. In this manner, the spot position could be monitored for 12 h. At the conclusion of the experiment, the spot displacement versus time data which were recorded on the two VCR tapes were analyzed to obtain the average di€usion coecient. To ®nd tmax , a least squares ®t was performed on the near peak data of the spot displacement history. tmax was then determined by ®nding the maximum of the function ®t to the data. The following model was used for the curve ®ts and is recommended:   K1 ÿK2 ‡ K0 ; d ˆ p exp …14† t=tp t=tp where d is the spot displacement at time t, and K0 , K1 , and K2 are the least squares ®t parameters. Note that the form of the model is extracted from the decay period solution, Eq. (10). The coecient in the ®rst term of Eq. (14) is similar to the coecient of the integral in Eq. (10), while the remainder of the expression is similar to the decaying exponential term in the argument of the integral. Other models for ®tting the peak data were studied. However, Eq. (14) was found to be superior (see Capobianchi [2] for details). Di€erentiating Eq. (14) and equating to zero gives the time at which the function extremum occurs, tmax : …15† tmax ˆ 2tp K2 : Hence, with tmax known, the di€usion coecient was determined as described in the ``Analysis'' section. Figs. 7 and 8 show the spot displacement history for an entire experiment and for the near peak data, respectively, for run number 27 (see Table 2 for run parameters). 4. Results and discussion The apparatus described above was used to measure the average di€usion coecients of sodium chloride in water and potassium chloride in water. A total of 32 experiments were conducted. The ®rst 9 runs were per-

42

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

Fig. 7. Spot displacement history for the full duration of run no. 27. Squares denote experimental data points. The solid line is the least squares ®t to the exact solution, Eqs. (8) and (10).

Fig. 8. Spot displacement history in the near peak region of run no. 27. Squares denote experimental data points. The solid line is the least squares ®t of the peak model, Eq. (14), to the experimental data.

formed on the NaCl±H2 O system using the original setup with a physical measurement of xm . The remaining 23 runs utilized the revised set-up, and consisted of measurements on both the NaCl±H2 O and the KCl±H2 O systems. The speci®c parameters for each run and the measured average di€usion coecients are tabulated in Table 2, where each table entry corresponds to one experiment. The values of xm for runs 10 through 32 are the results of calibration and were evaluated as described earlier utilizing the ``known'' value of the di€usion coecient reported by Woolf et al. [1]. With xm determined, the revised set-up was used to measure the di€usion coecients for the NaCl±H2 O system at nominal temperatures of 18.5, 25.0, and 30.0°C. Similar measure-

ments were made on the KCl±H2 O system, except that those at 25°C were used either for calibration (runs 13 through 18, inclusive), or as a check of calibration of xm (runs 21 and 32). The measurements performed in the current study supplement the existing di€usion coecient data for the NaCl±H2 O and for the KCl±H2 O systems at 25°C, and provide new data at 18.5 and at 30.0°C for which little or no information is currently available. A review of existing literature reveals that studies were performed for both systems using a variety of experimental techniques. Most of the reported values are at 25°C, and consist of di€erential, integral, and average di€usion coecients. Detailed description of the techniques and apparatus used may be found in either the references provided, and/or in any of several books, such as that by Woolf et al. [1], and will not be repeated here. The cited investigations are presented for comparison with the present study. Hence, the given references all have concentration ranges consistent with those used in the current experiments. A review of the published NaCl± H2 O data was performed by Kukulka et al. [6]. A comprehensive review of many aqueous electrolyte systems, including the systems studied in the present investigation, is presented by Lobo [7]. These literature data are plotted, along with the experimental results from the current study 2, in Figs. 9 and 10. The di€erential di€usion coecients reported in the literature are presented in these ®gures at concentrations of 2.000 wt.% NaCl for the sodium chloride-water system, and 2.536 wt.% KCl for the KCl± H2 O system. These literature data were interpolated where necessary. The integral and the average di€usion coecients presented are the literature values reported for the concentration limits of the individual studies. To determine the temperature dependence of the differential di€usion coecient between 0°C and 50°C, a second order polynomial was ®t to the existing published di€erential di€usion coecient data. The result of the ®t are shown as the heavy dashed lines in Figs. 9 and 10. The lighter dashed lines are the least squares ®t curve displaced by ‹10%. In performing the ®t, investigators were allowed one average value at each reported temperature, thereby precluding arti®cially weighing any particular investigator's data more than the data from other investigators at the same temperature. Hence, when an investigator reported several data at a given temperature, these were averaged and the resulting value was used in the ®t. As is shown in Fig. 9, most of the existing experimental studies for the NaCl±H2 O system are at 25.0°C with few studies existing at other temperatures. At 25.0°C

2 Runs 1, 2, 7, and 9 were excluded from the plots since these experiments were executed between concentration limits di€erent from the remaining runs.

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

43

Table 2 Run parameters for performed experiments, and the corresponding measured average di€usion coecients Run no.

1a 2a 3a 4a 5a 6a 7a 8a 9a 10 b 11 b 12 b 13 c 14 c 15 c 16 c 17 c 18 c 19 b 20 b 21 d 22 b 23 b 24 b 25 b 26 b 27 b 28 b 29 b 30 b 31 b 32 d a b c d

T (°C)

24.9 24.6 24.7 24.9 25.2 24.9 18.5 18.5 18.5 25.0 25.0 25.1 25.1 24.8 24.9 25.0 24.8 25.0 18.6 18.6 25.0 18.6 30.1 30.3 29.9 30.2 30.1 30.1 18.6 18.6 18.5 25.1

Solute

NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl NaCl KCl KCl KCl KCl KCl KCl NaCl NaCl KCl NaCl NaCl NaCl NaCl KCl KCl KCl KCl KCl KCl KCl

Initial concentration (wt.% solute) c10

cu0

5.995 2.991 3.966 4.012 4.008 4.006 2.994 4.003 5.998 3.993 3.994 3.994 5.077 5.072 5.084 5.083 5.076 5.062 4.000 4.000 5.076 4.000 4.000 4.000 4.000 5.081 5.071 5.068 5.071 5.076 5.071 5.076

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

xm (mm)

tp (s)

Dave ´ 109 (m2 /s)

5.057 5.057 5.057 5.057 5.057 5.057 5.057 5.057 5.057 5.358 5.358 5.358 5.358 5.358 5.358 5.358 5.358 5.358 5.358 5.372 5.372 5.372 5.372 5.377 5.377 5.372 5.371 5.370 5.370 5.370 5.370 5.370

14560 14640 14660 14540 14660 14700 14880 14460 14720 14580 14820 14910 14860 14600 14570 14730 14690 15000 14490 14530 14640 14720 14480 14570 14600 14620 14600 14620 14650 14470 14520 14530

1.40 1.51 1.39 1.44 1.43 1.38 1.27 1.20 1.20 1.54 1.43 1.43 1.84 1.72 1.81 1.89 1.81 1.84 1.23 1.26 1.81 1.28 1.68 1.87 1.75 2.14 2.11 2.19 1.70 1.69 1.66 1.93

Run performed with the original set-up. xm determined by direct measurement. Run performed with the revised set-up. xm determined by calibration. Calibration run, performed with revised set-up. Calibration check run, performed with revised set-up.

and 18.5°C, the results from this study agree with those from the other investigations within the maximum predicted uncertainty as determined by Capobianchi [2]. At 30.0°C, the experimental values from this study appear high when compared to the least squares ®t curve. However, these data are still within the predicted uncertainty. The total random uncertainty was ‹7.6%, with the largest contribution due to the errors in determining the location of the center of the spot on the grid, which in turn produced an uncertainty in tmax . In addition, errors in measuring xm produced a maximum systematic uncertainty of 4.4%. If the systematic uncertainty is near its maximum predicted value, then the total uncertainty may range from +12.0% to )3.2%. Hence, errors near +10% may occur if the systematic and the random errors are in the same direction. Therefore, the measured values generally fall within the predicted overall uncertainty (for additional details on the uncertainty analysis, see Capobianchi [2]).

Care must be exercised when using the least squares ®t curve for comparative purposes. Since there are few data 3 outside of those at 25°C, the least squares ®t line is in¯uenced primarily by the data of Vitagliano [8] at temperatures other than 25°C, with only the studies by Clack, [9,10] adding additional data at 18.5°C. Vitagliano [8] included no uncertainty estimates and only one set of data for each temperature, so that it is dicult to judge the accuracy of his measurements. In addition, the uncertainty in temperature in Clack's studies may be large, since the temperature, which was controlled via a room thermostat, is only claimed to be ``in the neighborhood of 18.5°C'' by the author. Also, the variation of the di€erential di€usion coecient with concentration from the earlier Clack [10] study has a shape inconsis3

The least squares line is based upon the reported di€erential di€usion coecient data only.

44

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

Fig. 9. Measured values for the NaCl±H2 O system ((a) 4 data points; (b) 7 data points; (c) 3 data points). Dashed line is a least squares ®t polynomial through the published di€erential di€usion coecient data. D, Dint: , and Dave: indicate the type of di€usion coecient measured in the experimental study.

Fig. 10. Measured values for the KCl±H2 O system ((a) 3 data points; (b) 8 data points; (c) 3 data points). Dashed line is a least squares ®t polynomial through the published di€erential di€usion coecient data. D, Dint: , and Dave: indicate the type of di€usion coecient measured in the experimental study.

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

tent with that of his later study and from the shape of Vitagliano's [8] data. Therefore, the lack of data outside 25°C and the uncertainty in the existing data pose diculties in establishing the accuracy with which the least squares ®t curve actually represents the temperature dependence of the di€erential di€usion coecient. Similar results were obtained for the KCl±H2 O system, shown in Fig. 10, and the same arguments apply. However, in addition to the studies by Clack [9,10], Nienow et al. [11] reported di€erential coecients at 20°C and Harned and Blake [12] provided data at 4°C. At temperatures above 25°C, Vitagliano and Caramazza [13] were the only source of di€erential coecients found for the given concentration. Therefore, the least squares curve in this temperature region is controlled only by their data. However, if the resulting least squares ®t curve in fact represents the true variation of the differential di€usion coecient with temperature, then it should pass through the di€erential data of Gosting [14] and of Harned and Nuttall [15], providing that these too are correct. These data were used by Woolf and Tilley [16] in deriving the integral di€usion coecients used in calibration, and subsequently reported by Woolf et al. [1]. But the least squares curve predicts a di€erential coecient of 1.816 ´ 10ÿ9 m2 /s at 25°C, whereas Gosting [14] reports 1.841 ´ 10ÿ9 m2 /s, and Harned and Nuttall [15] report 1.843 ´ 10ÿ9 m2 /s. Hence, the di€erential values are higher than the least squares ®t prediction by 1.4% and 1.5%, respectively. This may make the values measured in this study higher than they should be, although the exact magnitude is dicult to predict since the integral coecient was used in calibrating. It is also possible that the least squares ®t curve is lower than it should be. Again, the di€erence between the least squares ®t curve and the true temperature variation is dicult to quantify. Therefore, conclusions regarding the accuracy of any given data set based upon comparison to the least squares ®t curve are dicult to justify. 5. Conclusions The ®rst iteration in the development of a new experimental method for measuring the di€usion coecient has been achieved. The decaying pulse technique has several advantages: 1. The method is direct and conceptually simple, requiring measurements of time (tp and tmax ) and distance (xm ) only. From these data, the di€usion coecient can be readily obtained. Since no other solution property or data are required, and since any property which changes monotonically with concentration may be monitored, the technique is applicable to a wide variety of ¯uid pairs. One need not monitor changes in the index of refraction as was done in this study. Rather, the property to be monitored should be selected according to the particular ¯uid pair being tested. And the exact dependence of the selected property on concentration is unnecessary; one needs only to know that it varies monotonically within the concentration range of interest.

45

2. The decaying pulse technique is absolute, in principle requiring no calibration. Hence, no ¯uid pairs with ``known'' di€usion coecient are needed. 3. The ability to obtain results from runs of short duration has important practical consequences. Run parameters such as temperature must be controlled throughout the full duration of the experiment. The equipment needs to be motionless to avoid any mechanical mixing of the ¯uids. Longer experiments are more susceptible to disturbances, so that there are practical advantages in maintaining run times as short as possible. Many of earlier methods for measuring di€usion coecients in liquids, such as the diaphragm cell and the capillary techniques, require run times of the order of days (Woolf et al. [1]). The decaying pulse technique requires measuring the time at which the peak concentration occurs. Once the peak concentration has passed and sucient data have been collected to delimit and de®ne the peak region, the experiment may be terminated. In the present study, experiment durations of 12 h were used, but as may be seen in Fig. 7, this may be reduced to approximately 7±8 h for the purposes of resolving the peak data. Further decrease is possible with an improved technique for monitoring concentration. Other methods exist, and each has its own advantages and disadvantages. The decaying pulse technique features a simple, direct, ¯exible, and absolute method of measuring the di€usion coecient in binary liquid solutions. 6. Recommendations and future research needs Based on the work performed in this study, the accuracy of the measurements must be improved, and the sensitivity must be enhanced so that di€erential di€usion coecients may be measured. More speci®cally, the primary areas requiring re®nement are in reducing the uncertainties in xm and in x‡ , and in increasing the sensitivity of the monitoring technique to allow measurements driven by small initial concentration di€erences, thereby yielding di€erential di€usion coecients. Accomplishment of both objectives is tied into the monitoring technique chosen.

Nomenclature A a0 a1 a2 a3 a4

cross-sectional area (m2 ) least squares ®t parameter, less) least squares ®t parameter, less) least squares ®t parameter, less) least squares ®t parameter, less) least squares ®t parameter, less)

Eq. (11) (dimensionEq. (11) (dimensionEq. (11) (dimensionEq. (11) (dimensionEq. (11) (dimension-

46

c c1 c2 cl0 cu0 D Dave: Dint: d K0 K1 K2 L m !x m T t tmax tp x xm

M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47

concentration (kgsolute /m3 ) concentration limit in Eq. (2) (kgsolute /m3 ) concentration limit in Eq. (2) (kgsolute /m3 ) initial concentration in the lower tube (kgsolute /m3 ) initial concentration in the upper tube (kgsolute /m3 ) di€erential di€usion coecient (m2 /s) average di€usion coecient (m2 /s) integral di€usion coecient (m2 /s) spot displacement (m) least squares ®t parameter, Eq. (14) (m) least squares ®t parameter, Eq. (14) (m) least squares ®t parameter, Eq. (14) (dimensionless) tube length (m) mass ¯ux in x direction (kgsolute /(m2 s)) mass ¯ux vector (kgsolute /(m2 s)) temperature (°C) time (s) time at peak concentration (s) pulse duration (s) coordinate direction (m) distance from tube interface to monitoring location (m)

Dummy variable of integration in Eq. (10) (dimensionless)

Superscripts +

[8] [9] [10] [11] [12] [13]

[14] [15] [16]

Greek m

[7]

[17] [18]

Denotes dimensionless variable [19]

References [1] L.A. Woolf, R. Mills, D.G. Leaist, C. Erkey, A. Akgerman, A.J. Easteal, D.G. Miller, J.G. Albright, S.F.Y. Li, W.A. Wakeham, Di€usion coecients, in: W.A. Wakeham, A. Nagashima, J.V. Sengers (Eds.), Measurement of the Transport Properties of Fluids, Experimental Thermodynamics, vol. III, Blackwell Scienti®c Publication, Boston, 1991, pp. 227, 449. [2] M. Capobianchi, A new experimental technique for measuring the di€usion coecient in binary liquid solutions, Ph.D. Dissertation, Department of Mechanical Engineering, State University of New York at Stony Brook, Stony Brook, New York, 1996. [3] S.C. Kim, T.F. Irvine Jr., A transient technique to determine the thermophysical properties of liquids, in: Proceedings of the Second World Conference on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics, Dubrovnik, Yugoslavia, Elsevier, New York, 1991, pp. 1546. [4] P.G. Kosky, A method of measurement of thermal conductivity: Application to free-standing diamond sheet, Rev. Sci. Instrum. 64 (4) (1993) 1071. [5] W. Jost, Di€usion in Solids, Liquids, Gasses, 3rd Printing with Addendum, Academic Press, New York, 1960, pp. 16. [6] D.J. Kukulka, B. Gebhart, J.C. Mollendorf, Thermodynamic and transport properties of pure and saline water, in: J.P. Hartnett,

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[23]

[24] [25]

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M. Capobianchi et al. / Experimental Thermal and Fluid Science 18 (1998) 33±47 [26] D.R. Caldwell, Thermal and Fickian di€usion of sodium chloride in a solution of oceanic concentration, Deep-Sea Res. 20 (1973) 1029. [27] L.A. Woolf, D.G. Miller, L.J. Gosting, Isothermal di€usion measurement on the system H2 O±Glycine±KCl at 25°; Tests of the Onsager reciprocal relation, J. Amer. Chem. Soc. 84 (1962) 317. [28] J. Lielmezs, H. Aleman, G.M. Musbally, External transverse magnetic ®eld e€ect on electrolyte di€usion in KCl±H2 O solution, Zeitschrift f ur Physikalische Chemie Neue Folge 90 (1974) 8.

47

[29] R.H. Stokes, Integral di€usion coecients of potassium chloride solutions for calibration of diaphragm cells, J. Amer. Chem. Soc. 73 (1951) 3527. [30] J.W. McBain, C.R. Dawson, The di€usion of potassium chloride in aqueous solution, Proc. Roy. Soc. London Ser. A 148 (1935) 32.