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Electrochemical mass transfer studies in annuli
Electmchimica Acta. 1965. Vol. 10. pp. 1093 to 1106.
Psrgaunn Press Ltd.
Printed in Northern Ireland
ELECTROCHEMICAL MASS TRANSFER IN ANNULI* T. IL Department
Ross and A. A.
STUDIES
WRAGG
of Chemical Engineering, Faculty of Technology, University of Manchester, England
Abstract-Rates of mass transfer in conditions of transport control and in the mass transfer entry region have been determined by measuring limiting currents for the deposition of copper from acidified solutions of copper sulphate on to copper cathodes of different lengths. The cathodes formed part of the inner wall of an annular flow cell and conditions were such that in both streamline and turbulent flow the hydrodynamic conditions were fully developed at the mass-transfer section. The variation of the mass transfer coefficient with electrode length has been clearly demonstrated, and results for streamline flow have been compared with a form of the L&&que equation suitably modified to account for the annular geometry. The data have been successfully correlated by the equation? Sh = 1.76(Re. SC . dcIL))“” for an annulus radius ratio of 05. For amnrli of radius ratios 0.25 and O-125 the constant in the correlatingequation was found to increase in the manner expected from velocity-profile considerations. In turbulent flow with an annulus diameter ratio of 05, experimental data have been correlated by the equation St = 0.276 Sc-a12Re-o’m(d,L)1’s. R&urn&-Des mesures de courants-hmites d%lectrodeposition de Cu en solution acide CuSO, sur cathodes Cu de differentes longueurs ont permis de determiner des vitesses de transfert massique dans les conditions oh elles se manifestent r@latrices. Le coefficient de transfer? massique apparait dependre nettement de la longueur de l’electrode et l’on compare les r&ultats avec les previsions issues de l’equation de LRveque, convenablement modifiee pour tenir compte dune geometric ici annulaire. L’equation Sh = 1,76(Re . SC . dJL)1’8 en rend convenablement compte pour un armeau de rapport radial 0,5. La constante numerique s’accroit de facon previsible pour des rapports 0,25 et 0,125. En regime turbulent, avec un rapport 0,5, l’equation devient St = 0,276 Sc-*/sRe-o:4B(d,L)11s. ZusanunenfassnnR-Stofftransportgeschwindigkeiten unter Bedingungen der Transportkontrolle im Anlaufbereich des Stofftiberganges werden durch Grenzstrommessungen bei der Kupferabscheidung aus sauren Kupfersulfatliisungen auf Kupferkathoden verschiedener Abmessungen ermittelt. Die Kathoden waren Teil der inneren Wand einer ringfbrmigen Striimungszelle. Die Redingungen waren derart gewahlt worden, dass sowohl voll ausgebildete laminare wie such turbulente hydrodynamische Striimungen am 01% des Stofftibergang-Messgebietes enielt werden konnten. Die Aenderung der Stoffiibergangskoefbzienten mit der AnstrLimlinge konnte deutlich gezeigt werden. Die Ergebnisse im laminaren Gebiet konnten fiir ein Radienverhaltnis von 0,5 mit einer ftir die Geometrie des Ringspaltes modifizierten Form der Ltveque-Gleichung Sh = 1,76(Re . SC . dJL)‘ls erfolgreich korreliert werden. Ftir Radienverhlltnisse von 0,25 und 0,125 vergriissert sich die Konstante der Gleichung in einer aus Striimungsprotil-Retrachtungen zu erwartenden Weise. * Manuscript received 1 October 1964 t See Notation list at end of paper 4
1093
T. K. Ross and A. A. WRAGG
1094
Im turbulenten Gleichung
Gebiet mit einem Radienverhlltnis
von 0,s lassen sich die Ergebnisse durch die
St = 0,276 Sc-*18Re-o,4~(d%Y L)l/* darstellen. INTRODUCTION
RECENT paperGs in which ionic mass-transfer experiments are described give an incomplete account of the principles of mass transfer from flowing solutions, especially with regard to hydrodynamic factors. For instance, the experimental arrangements are such that there is uncertainty as to the degree of hydrodynamic development at the mass-transfer section in the flow cell,l and this important factor is not adequately considered in the analysis and correlation of the results. Further, despite the fact that in many of the experiments hydrodynamic conditions are fully, or almost fully, developed, the theoretical relationship with which the experimental results are compared1 applies only in conditions of developing velocity distributions. The theoretical expression with which the results are found to agree is in fact that for which the hydrodynamic and mass-transfer boundary layers develop from the same point, a condition tihich did not apply in the experiments described. Moreover the results from outside the laminar region were included in the overall correlation and the authors2 appear to misunderstand the conditions of the Ledque solution,3 which in fact predicts that I& varies as Uz” in fully developed hydrodynamic conditions. The claim that the experimental dependence of KL on Vi” indicates fully developed conditions is thus mistaken. The only other work on solid-liquid mass transfer in an annulus is that of Lin et a1.,4 who worked with four different electrochemical reactions. Hydrodynamic conditions were fully developed and these workers reported their mass transfer measurements to be in agreement with the Ltv&que solution in streamline flow, and with the Chilton-Colburn equation in turbulent flow. The experimental work was however restricted to one annulus geometry and one electrode length. The case of the circular tube has been treated by Van Shaw, Reiss and Hanratty,6 who also used an electrochemical technique and concentrated on the turbulent flow region. Further useful work on mass transfer in circular ducts involving the dissolution of tubes of varying lengths, cast from substances such as benzoic acid, has been contributed by Linton and Sherwood,6 and more reliably by Meyerink and Friedlander,’ who investigated fully developed streamline and turbulent conditions. In view of the limitations of earlier investigations the present work was undertaken to provide a more comprehensive treatment. Rates of mass transfer from a flowing electrolyte to the core of an annulus in fully developed streamline and turbulent flow were measured, the goemetry of the annular flow cell, the length of the working electrode, the solution flow rate, and the electrolyte concentration being varied at a single temperature. THEORETICAL
DISCUSSION
Streamline Jlow
The simple solution laminar flow past a flat and applied to the case profile terms. For mass
developed by L&&que3 for the heat-transfer coefficient in plate can be conveniently expressed in mass-transfer terms of tubes or annuli by suitable modification of the velocitytransfer in laminar, unidirectional, fully developed flow we
Electrochemical
mass transfer studies in annuli
1095
can write, in the notation given at the end of the paper,
(1) where the x and y directions are parallel and perpendicular to the direction of flow respectively. In the region near the wall the variation of U, with y is approximately linear, and the equation yields the following solution for the mean mass-transfer coefficient between surface and fluid KL = 0.807 D & “’ ( DL ) '
(2)
where @is the velocity gradient at the wall. For a tube, the velocity at any point is given by
u,= 2U,(l-
(f)“),
(3)
so that by differentiation (4) and the LevCque solution for a tube can be written as Sh=y=
1*614(Re.Sc.g)liq
where Re = dump and SC = -k P PD.
(5)
In an annulus the velocity distribution is given by the equation
u, = 2u,
(6)
where a = rl/rz = annulus radius ratio, and for the inner wall it can be shown that
where
(‘3)
T. K. Ross and A.
1096
A. WRAGG
and d, = annulus equivalent diameter
=
4 x cross sectional area wetted perimeter
= d2 - dl. Thus for mass transfer to the inner wall of an annulus the L&Cque solution gives Sh = K;de - 1.614 [&)]lj3( Re . SC . :)““, where
Re
=
(9)
dJd -. P
Lin et aZ.4 compared their experimental laminar flow data with a form of (5) in which the equivalent diameter of the annulus was substituted for the tube-diameter term, and excellent agreement was obtained. However, this procedure has been criticized by Friend and Metzner,s who point out that the correct form of the Levv&que solution with which this experimental data should be compared is (9), which, for rl/rz = O-5 (the radius ratio used by Lin et al.) becomes Sh=
1.939(Re.Sc.%r3.
(10)
It is clear therefore that the results in question are nearly 17 per cent lower than the more correct theoretical solution, and cannot be considered to agree with the theory as well as the authors make it appear. Turbulent$ow
The basic differential equation of diffusion and convection in the system may be written
The superscript bar indicates time-averaged values, and the subscript f fluctuating velocity and concentration components. Following the procedure of Linton and Sherwood6 and Van Shaw, Reiss and Hanratty,5 three of the terms on the right hand side of the equation are assumed to be negligible and the equation simplifies to
(12) similar to (1). Assuming an approximately linear velocity distribution in the region close to the wall, the solution for the mass transfer coefficient is ,
(13)
Electrochemical
mass transfer studies in ammli
1097
where @’is the velocity gradient at the wall in turbulent flow. The velocity near the wall can be conveniently expressed in terms of the friction factor, f, (14) whence (15) Using the Blasius formula for the friction factor and substituting for r in (13) gives for the tube St = K$ = 0.276
~~-213
Re-0.42
(@)1/3.
m
In applying this procedure to the annulus it is necessary to use the appropriate equation for the friction factor, and this involves making the distinction between the friction factors at the inner and outer walls of the annulus, fiand f2respectively. This question has recently been discussed by Knudsen,g who presents a method by which the friction factor at both walls of the annulus may be determined for different values of Re and r,/r,. This treatment, based on the work of Rothfus et aLlo who fist defined two distinct annulus friction factors, yields the following expressions, both of which approximate the exact values of the core friction factor within 5 per cent in the Reynolds number range 4 x 18 and 106 with the present system. (17) and
fl=
O-079 Re- 025(~j@“”
($
:;)),
where il. = rmsX/r2 and
Insertion of the appropriate values of a and il for a radius ratio of two, and substituting in (15) and then (13) yields St = z
= 0.25 Re-w40
~~-213
(d,/L)l/3
(1%
m
and St = z
= O-2995 Re-0.42 Sc-2/3(d8/L)1’3. m EXPERIMENTAL
Apparatus Rates of mass transfer were determined by measuring limiting electrolysis current for the cathodic deposition of copper from solutions of cupric sulphate containing excess sulphuric acid. The vertical annular flow cell consisted of an outer tube of
T. K. Ross and A. A. WRAOO
1098
TABLE 1. SUMMARY
OF ANNULUS
Annulus number Diameter of tube, da, cm Length of tube, cm Diameter of core, d,, cm Equivalent diameter, de, cm Entrance length, cm Diameter ratio, da/d1 Value of [$(a)1118in equation (9) Electrode lengths, cm
AND ELECTRODE OIZQME~
1 2.54 152 1.21 1.27 125 2
1.201 o-05, 0.10 0.05, 1.0 3.81, 7.62 15.24
2 5.08 214 2.54 2.54 165 2
1.201 1-o 3-81 15.24
3
4
2.54 152 0.635 1905 115
5.08 214 0.635 4.445 125
4
1.289 1.0 3-81 15.24
8
1.418 1.0 3.81 15.24
copper,
flanged at both ends to Perspex inlet and outlet sections which served to hold the Perspex core concentrically within the copper tube. The copper cathode sections on which the mass-transfer process took place formed part of the core of the annulus, and were situated at such a distance from the cell entrance as to ensure fully developed flow conditions. (The annulus hydrodynamic entry length problem has received attention from Heaton et al.,ll and the results of this treatment were used in ascertaining the required entry length in the present work.) Twenty equivalent diameters were allowed as an exit length. The outer tube acted as anode in the electrolysis cell, and the electrolyte flowed vertically upwards. Four annulus assemblies were used and the geometries are summarized in Table 1, which also gives details of the electrode lengths used with each annulus. The cell assembly and electrical circuit is illustrated in Fig. 1. The source of variable voltage was a transistorized power supply capable of a current throughput of 5 A.
Detail
of wow-tight --
fitting
F-10. 1 Diagram of apparatus, including details of amndus assembly.
Electrochemicalmass transfer studiesin annuli
1099
Any one of a series of 8 ammeters covering a range of O-10 ,uA to O-10 A could be selected to measure the electrolysis current. The potential of the cathode was measured relative to that of the anode by means of an electronic millivoltmeter. (Since the area of the anode was large compared with that of the cathode, it constituted a convenient reference electrode for the purpose of limiting current measurements.) A decade resistance box was also incorporated in the circuit. The rate of solution flow was measured by the use of one of four specially calibrated Rotameters equipped with stainless steel floats. A thermometer was situated in the electrolyte reservoir and cooling of the electrolyte was effected by a film of cold water on the outside of the reservoir surface. During the course of the work a variety of centrifugal pumps, fabricated in both plastic and stainless steel, were used. Procedure
The most extensive series of experiments was performed with Annulus 1, which permitted seven different electrode lengths to be studied. Prior to each experimental run the electrode surface was treated with increasingly fine grades of emery cloth, degreased with trichlorethylene and washed with alcohol before being secured in the core assembly and introduced into the annulus. After being made up to 1.5 M in H,SOI and 0.01 or 0.05 M in CuSO,, the electrolyte was treated for about 90 min with oxygen-free nitrogen and its temperature adjusted to 18°C. A low current density was passed through the cell for a few minutes to achieve a uniform fresh deposit of copper, and then current/potential curves were obtained for different flow rates by increasing the electrical potential across the cell in steps of about 40 mV until hydrogen evolution commenced. The steady state current corresponding to each value of potential was noted. Readings were taken in laminar flow with all seven electrodes and in turbulent flow with the 15.24-cm, 3*81-cm and l-cm electrodes, and were repeated three times for each electrode at each flow rate. Samples of solution were withdrawn at frequent intervals for analysis. Confirmatory experiments in streamline and turbulent flow were performed with Annulus 2 in which the diameter ratio was also 0.5. The effects of varying diameter ratio were studied by employing Annuli 3 and 4 with an electrolyte concentration of 0.01 M in streamline flow. In the case of Annulus 4, for which the ratio of entry length to equivalent diameter is low, Reynolds numbers were restricted to a maximum of 1,300. Secondary experiments involving low flow rates and widely varying electrolyte concentrations were performed in order to examine natural convection effects. By this means results obtained in conditions where natural convection effects were known to be significant could be excluded from the final streamline flow correlation. Tables of results, a more detailed account of the apparatus and experimental procedure, and a fuller exposition of the theory, are given elsewhere.12 RESULTS Typical current/potential curves from which values of the limiting current may conveniently be estimated are shown in Fig. 2. Since electrical migration effects are rendered negligible by the presence of a large excess of indifferent electrolyte, and since at the limiting rate of deposition the concentration of copper ions is reduced virtually
1100
T. K. Ross and A. A. WRAGG
mV
AE,
FIG. 2. Current/potential curves obtained using Anmdus 1 (Table 1).
to zero at the electrode surface, the mass-transfer coefficient due to diffusion and convection, KL, is related to the limiting current by the equation KL=-.
k .zFAC,
Figure 3 shows the manner of variation of the observed mass-transfer coefficient with Reynolds number; the change in gradient between the regions of streamline and turbulent flow is clear. A plot of the average value of the mass-transfer coefficient obtained for varying electrode lengths for three different values of Re is given in Fig. 4. This affords an excellent illustration of the variation of the rate of mass transfer with electrode length in the mass-transfer entrance region. The mass-transfer coefficient
/
100 / 50 /'$
2:r,AF
Annulus I : S, q2450 L =3*8lcm:&,=OOlM 50-
20 KS
I
I
IO'
104
Re Fw. 3. Variation of average mass-trans~u”,~lient
with Reynolds Number, employing
Electrochemical
1101
mass transfer studies in annuli
can be seen to tend toward an infinite value at the leading edge of the electrode, (corresponding to zero mass-transfer boundary-layer thickness), and to decrease very rapidIy over the first few cm. These results were obtaitied with Annulus 1.
Variation af average mass
transfer coefficient with electrodelength Sc=2450 0: =l27cm
0
I
I I
I
2
I
3
II 4
5
I 6
IIll
7
8
9
Electrodelength,L.
FIG. 4. Variation
KJ
I II
I
I2
I
I3
I I I
I4
L
I5
cm
of the average mass-transfer coeficient (d, = 1.27 cm, SC = 2450).
with electrode
length
Laminar flow Further calculations based on the experimental data were performed in the manner suggested by the LMque solution as expressed by (9). Using the viscosity and density data of &e&erg, Tobias and Wilke13 and values of diffusivity based on the data of Gordon and Cole,%he relevant dimensionless groupswere evaluated. Figure 5 shows a plot of all the laminar flow results obtained with Annuli 1 and 2. The appropriate annulus L&&que solution corresponding to these conditions (lo), is represented by the upper broken line, while the Iower broken line represents the circular tube equation (5). All the results fall between these two lines and are well represented by the equation Sh = 1*76(Re. SC. c&/L)~~.
(22)
Experimental values of Re . SC . d,/L have varied between 1oQand IOB,so that a thorough examination of the mass-transfer entrance region has been made. Comparison of the present results with those of Lin et aL4shows that they lie closer to the annulus L%que equation, falling an average of about 9 per cent below the line compared with the 17 per cent discrepancy of the data of Lin et al. It is suggested that this may be due to the fact that the desired conditions of transport control are realized
T. K. Ross and A. A. WRAGG
1102
more closely with the electrolytic system employed in the present work. In experiments using circular pipes, Linton and Sherwood6 reported results in agreement with the LBv&que solution, and it would be expected that for an annulus the results would be above, and not (as Lin et al. report) on the same line. The close agreement with the prediction of the LBv&que solution is not unexpected, since the assumptions involved are not unrealistic in the system examined.
106 Re.
IO'
SC d. /L
FIG. 5. Results obtained in streamline flow when r = 05
The data obtained with the annuli of different geometries were treated in a similar way, and the results are best represented by Sh = 1*80(Re . SC . d&)1/3
(23)
for rl/ra = 0.25, and Sh = 2*03(Re . SC . de/Q113 (24) for rl/ra = 0.125. Comparison of (22), (23) and (24) indicates that as the value of rl/ra decreases, the correlating line on the plot of Sh against Re . SC . d,/L lies higher, but in each case is some 9 to 13 per cent lower than the appropriate form of (9). This fact demonstrates the inadequacy of correlations for annular flow mass transfer, such as that of Lin et ~1.~ which merely substitute the annulus equivalent diameter term into the equation applicable to a tube. Turbulentpow Calculations based on the experimental data for turbulent flow were performed in the manner suggested by (19) and (20). The results obtained with Annulus 2 are presented in Fig. 6 in the form of a plot of (KJU,> Sc213against Re, the solid line representing the solution for a tube, (16). Coincidence with this equation is obtained except for the case of the 15.24cm electrode, for which there is a marked deviation as the Reynolds number increases due to the rapid onset of fully developed mass transfer conditions. The empirical Chilton-Colbum equation jo = St . Sc213= O-023 Re-“‘2,
(25)
Electrochemical mass transfer studies in annuli
1103
with which the turbulent flow data of Lin et al4 agreed, is shown as a broken line in the diagram. The results obtained with both Annuli 1 and 2 are given in Fig. 7, where (KJ iJ,J . Sc2J3 (L/d,)“3 is plotted against Re. The results are compared with (16) for the
Reynolds FIG.
6.
number
Results obtained in turbulent flow with Annulus 2.
case of a tube with de substituted ford, and also with the approximations to the exact annulus solution represented by (19) and (20). It can be seen there that is good agreement with the tube equation, whereas the results lie some 9 per cent below the theoretical annulus line. This finding is similar to that of Van Shaw, Reiss and Hanratty,6
Ix
2r
FIG.
7.
Results obtained in turbulent flow with Annuli 1 and 2.
whose results for the circular tube experiments lay, on average, 7 per cent below the theoretical line. The turbulent flow results obtained with the two Annuli of radius ratio 1 to 2, are therefore well represented by the equation St = !$ = 0.276 Sc-2/sRe-o42(d,/L)l/s m
T. K. Ross and A. A. WRAOO
1104
Mass transfer entry length in laminar flow Although only relatively short electrodes have been used in the present study, it is possible to estimate the magnitude of the mass-transfer entrance length from the fact that close agreement with a theoretical analysis is obtained. The solution of Graetz,15 which is not restricted by the assumption of a linear velocity gradient, provides a more extensive solution than that of LCv&que, and while the two solutions coincide when
IO 7-
Gmetzsolution
3-
100
I _ 1000
Re. Scd/x
Fm. 8. Representation of solutions of Lkdque and Graetz for a tube.
Re . SC d/x > 200, the Graetz solution predicts that the total Sherwood number (Sh,) approaches a constant value at low values of Re . SC d/x. This corresponds to the fully developed mass-transfer condition. The situation is somewhat complicated in the present system by the anode mass-transfer boundary conditions, but if the simpler case of the round tube is considered (Fig. 8), the mass-transfer conditions are fully developed when Re . SC d/x is less than about 8. For a typical electrolytic process with SC = 2,000 and Re = 500, x > 125,000d (27) for fully developed conditions. The magnitude of the entry length for mass transfer is thus far greater than the corresponding length in most heat-transfer situations, because of the high value of the Schmidt number, which represents the ratio between momentum and mass diffusivities. CONCLUSIONS
When inspecting the results of experiments designed to measure mass transfer in circular or annular ducts, it is important to be clear about the different relationships which apply to various flow regimes. As the expressions presented in the theoretical portions of this and other paperP make clear, the mass transfer coefficient (KL) may vary as the average liquid velocity (Cl,,,) to powers of 0*33,0.50, 0.58 or 0.60 as the flow is developed streamline, developing streamline, developed turbulent or developing turbulent respectively. If, therefore, a range of results obtained under all these conditions is correlated it is not surprising that an approximate dependence on Uk” is reported.1,2 Such treatments do not however contribute to an understanding of the relationships between mass transfer and various hydrodynamic conditions. The present work has confirmed the dependence of KL on lJL33 and U,o’58for fully developed streamline and turbulent flow conditions respectively.
Electrochemicalmass transfer studies in annuli
AckrwwZe&ments-We us (A. A. W.), and to
1105
wish to acknowledge the receipt of a D.S.I.R. studentshipawarded to one of thank Dr. B. Gay for helpful discussionsduring the research.
NOTATION Electrode surface area, cm2 A Annulus radius ratio, r1/r2 a Concentration of transferred species, mole/ems (Subscripts b and s indicate C bulk and surface values respectively) Tube diameter, cm d 4 4 Annulus inner and outer diameters, cm Annulus equivalent diameter, d, - d,, cm 4 Diffusion coefficient, cm2/s Friction factor Faraday i i Electrolysis current, A Limiting current, A iL j-factor for mass transfer jD Average mass transfer coefficient, cm/s KL Electrode length, cm L Hydrodynamic entrance length, cm LS Radial distance from conduit axis, cm r Inner and outer annulus radii, cm h/r2 Radial distance of point of maximum velocity from annulus axis, cm rmax Reynolds number, dU,p/p Re Schmidt number, ,u/p D SC Sherwood number, K,d/D Sh Stanton number, KL/Um St Average liquid velocity in x direction, cm/s u?n Point liquid velocity in x direction, cm/s G Velocity in y direction, cm/s V Distance in direction of flow, cm X Distance perpendicular to direction of flow, cm Y Z Number of faradays in electrodeposition of one mole of ionic species. Velocity gradient at conduit wall, s-l Ratio of radial distance of point of maximum velocity to outer radius (rmax/r2) Viscosity, g/s.cm Kinematic viscosity, cm2/s Density, g/cm3 REFERENCES 1. J. C. B.u.~N and A. J. ARVfA, Electrochim. Acta 9, 17 (1964). 2. J. C. BAZAN and A. J. ARVfA, Electrochim. Acta 9, 667 (1964). 3. J. L&&QUIZ,Ann. Mines Carbur., Paris (12) 13,201, 305, 381 (1928). 4. C. S. LIN, E. B. DENTON,H. S. GASKILLand G. L. PUTMAN,Ztistr. Engng. Chem. 43,2136 (1951). 5. P. VAN SHAW, L. P. REISSand T. J. HANRATIY,A. I. Ch. E. Journal 9,362 (1963). 6. W. H. INTON and T. K. SHERWOOD,Chem. Engng. Prog. 46,258 (1950). 7. E. S. C. MEYERINKand S. K. FRIEDLANDER, Chem. Engng. Sci. 17, 121 (1962). 8. W. L. FRIENDand A. B. METZNER,Amer. Inst. Chem. Eqprs. J. 4,393 (1958). 9. J. G. KNUDSEN,Amer. Inst. Chem. Engnrs. J. 8, 565 (1962).
1106
T. K. Ross and A. A. WIUOO
10. R. R. ROTHFUS,C. C. MONRAD,K. G. SIKCHIand W. J. HEIDFBER, Industr. Enpg. Chem. 47, 913 (1955). 11. H. S. HEATON, W. C. REYNOLDS and W. M. KAYS, Znt. J. Heat Mass Trumfer 7, 763 (1964). 12. A. A. WRAOO,Ph.D. Thesis, University of Manchester (1964). 13. M. E~NEZERQ,C. W. TOBUSand C. R. WILKE,J. Electrochem. Sot. 103,413 (1956). 14. A. R. GOFW~Nand A. COLE, J. Phys. Chem. 40,733 (1936). 15. L. GRAETZ,Ann. Phys. Chem. 25,337 (1885). 16. G. WRANGLBN and 0. NILSS~N,Electrochim. Acta 7, 121 (1962).
Psrgaunn Press Ltd.
Printed in Northern Ireland
ELECTROCHEMICAL MASS TRANSFER IN ANNULI* T. IL Department
Ross and A. A.
STUDIES
WRAGG
of Chemical Engineering, Faculty of Technology, University of Manchester, England
Abstract-Rates of mass transfer in conditions of transport control and in the mass transfer entry region have been determined by measuring limiting currents for the deposition of copper from acidified solutions of copper sulphate on to copper cathodes of different lengths. The cathodes formed part of the inner wall of an annular flow cell and conditions were such that in both streamline and turbulent flow the hydrodynamic conditions were fully developed at the mass-transfer section. The variation of the mass transfer coefficient with electrode length has been clearly demonstrated, and results for streamline flow have been compared with a form of the L&&que equation suitably modified to account for the annular geometry. The data have been successfully correlated by the equation? Sh = 1.76(Re. SC . dcIL))“” for an annulus radius ratio of 05. For amnrli of radius ratios 0.25 and O-125 the constant in the correlatingequation was found to increase in the manner expected from velocity-profile considerations. In turbulent flow with an annulus diameter ratio of 05, experimental data have been correlated by the equation St = 0.276 Sc-a12Re-o’m(d,L)1’s. R&urn&-Des mesures de courants-hmites d%lectrodeposition de Cu en solution acide CuSO, sur cathodes Cu de differentes longueurs ont permis de determiner des vitesses de transfert massique dans les conditions oh elles se manifestent r@latrices. Le coefficient de transfer? massique apparait dependre nettement de la longueur de l’electrode et l’on compare les r&ultats avec les previsions issues de l’equation de LRveque, convenablement modifiee pour tenir compte dune geometric ici annulaire. L’equation Sh = 1,76(Re . SC . dJL)1’8 en rend convenablement compte pour un armeau de rapport radial 0,5. La constante numerique s’accroit de facon previsible pour des rapports 0,25 et 0,125. En regime turbulent, avec un rapport 0,5, l’equation devient St = 0,276 Sc-*/sRe-o:4B(d,L)11s. ZusanunenfassnnR-Stofftransportgeschwindigkeiten unter Bedingungen der Transportkontrolle im Anlaufbereich des Stofftiberganges werden durch Grenzstrommessungen bei der Kupferabscheidung aus sauren Kupfersulfatliisungen auf Kupferkathoden verschiedener Abmessungen ermittelt. Die Kathoden waren Teil der inneren Wand einer ringfbrmigen Striimungszelle. Die Redingungen waren derart gewahlt worden, dass sowohl voll ausgebildete laminare wie such turbulente hydrodynamische Striimungen am 01% des Stofftibergang-Messgebietes enielt werden konnten. Die Aenderung der Stoffiibergangskoefbzienten mit der AnstrLimlinge konnte deutlich gezeigt werden. Die Ergebnisse im laminaren Gebiet konnten fiir ein Radienverhaltnis von 0,5 mit einer ftir die Geometrie des Ringspaltes modifizierten Form der Ltveque-Gleichung Sh = 1,76(Re . SC . dJL)‘ls erfolgreich korreliert werden. Ftir Radienverhlltnisse von 0,25 und 0,125 vergriissert sich die Konstante der Gleichung in einer aus Striimungsprotil-Retrachtungen zu erwartenden Weise. * Manuscript received 1 October 1964 t See Notation list at end of paper 4
1093
T. K. Ross and A. A. WRAGG
1094
Im turbulenten Gleichung
Gebiet mit einem Radienverhlltnis
von 0,s lassen sich die Ergebnisse durch die
St = 0,276 Sc-*18Re-o,4~(d%Y L)l/* darstellen. INTRODUCTION
RECENT paperGs in which ionic mass-transfer experiments are described give an incomplete account of the principles of mass transfer from flowing solutions, especially with regard to hydrodynamic factors. For instance, the experimental arrangements are such that there is uncertainty as to the degree of hydrodynamic development at the mass-transfer section in the flow cell,l and this important factor is not adequately considered in the analysis and correlation of the results. Further, despite the fact that in many of the experiments hydrodynamic conditions are fully, or almost fully, developed, the theoretical relationship with which the experimental results are compared1 applies only in conditions of developing velocity distributions. The theoretical expression with which the results are found to agree is in fact that for which the hydrodynamic and mass-transfer boundary layers develop from the same point, a condition tihich did not apply in the experiments described. Moreover the results from outside the laminar region were included in the overall correlation and the authors2 appear to misunderstand the conditions of the Ledque solution,3 which in fact predicts that I& varies as Uz” in fully developed hydrodynamic conditions. The claim that the experimental dependence of KL on Vi” indicates fully developed conditions is thus mistaken. The only other work on solid-liquid mass transfer in an annulus is that of Lin et a1.,4 who worked with four different electrochemical reactions. Hydrodynamic conditions were fully developed and these workers reported their mass transfer measurements to be in agreement with the Ltv&que solution in streamline flow, and with the Chilton-Colburn equation in turbulent flow. The experimental work was however restricted to one annulus geometry and one electrode length. The case of the circular tube has been treated by Van Shaw, Reiss and Hanratty,6 who also used an electrochemical technique and concentrated on the turbulent flow region. Further useful work on mass transfer in circular ducts involving the dissolution of tubes of varying lengths, cast from substances such as benzoic acid, has been contributed by Linton and Sherwood,6 and more reliably by Meyerink and Friedlander,’ who investigated fully developed streamline and turbulent conditions. In view of the limitations of earlier investigations the present work was undertaken to provide a more comprehensive treatment. Rates of mass transfer from a flowing electrolyte to the core of an annulus in fully developed streamline and turbulent flow were measured, the goemetry of the annular flow cell, the length of the working electrode, the solution flow rate, and the electrolyte concentration being varied at a single temperature. THEORETICAL
DISCUSSION
Streamline Jlow
The simple solution laminar flow past a flat and applied to the case profile terms. For mass
developed by L&&que3 for the heat-transfer coefficient in plate can be conveniently expressed in mass-transfer terms of tubes or annuli by suitable modification of the velocitytransfer in laminar, unidirectional, fully developed flow we
Electrochemical
mass transfer studies in annuli
1095
can write, in the notation given at the end of the paper,
(1) where the x and y directions are parallel and perpendicular to the direction of flow respectively. In the region near the wall the variation of U, with y is approximately linear, and the equation yields the following solution for the mean mass-transfer coefficient between surface and fluid KL = 0.807 D & “’ ( DL ) '
(2)
where @is the velocity gradient at the wall. For a tube, the velocity at any point is given by
u,= 2U,(l-
(f)“),
(3)
so that by differentiation (4) and the LevCque solution for a tube can be written as Sh=y=
1*614(Re.Sc.g)liq
where Re = dump and SC = -k P PD.
(5)
In an annulus the velocity distribution is given by the equation
u, = 2u,
(6)
where a = rl/rz = annulus radius ratio, and for the inner wall it can be shown that
where
(‘3)
T. K. Ross and A.
1096
A. WRAGG
and d, = annulus equivalent diameter
=
4 x cross sectional area wetted perimeter
= d2 - dl. Thus for mass transfer to the inner wall of an annulus the L&Cque solution gives Sh = K;de - 1.614 [&)]lj3( Re . SC . :)““, where
Re
=
(9)
dJd -. P
Lin et aZ.4 compared their experimental laminar flow data with a form of (5) in which the equivalent diameter of the annulus was substituted for the tube-diameter term, and excellent agreement was obtained. However, this procedure has been criticized by Friend and Metzner,s who point out that the correct form of the Levv&que solution with which this experimental data should be compared is (9), which, for rl/rz = O-5 (the radius ratio used by Lin et al.) becomes Sh=
1.939(Re.Sc.%r3.
(10)
It is clear therefore that the results in question are nearly 17 per cent lower than the more correct theoretical solution, and cannot be considered to agree with the theory as well as the authors make it appear. Turbulent$ow
The basic differential equation of diffusion and convection in the system may be written
The superscript bar indicates time-averaged values, and the subscript f fluctuating velocity and concentration components. Following the procedure of Linton and Sherwood6 and Van Shaw, Reiss and Hanratty,5 three of the terms on the right hand side of the equation are assumed to be negligible and the equation simplifies to
(12) similar to (1). Assuming an approximately linear velocity distribution in the region close to the wall, the solution for the mass transfer coefficient is ,
(13)
Electrochemical
mass transfer studies in ammli
1097
where @’is the velocity gradient at the wall in turbulent flow. The velocity near the wall can be conveniently expressed in terms of the friction factor, f, (14) whence (15) Using the Blasius formula for the friction factor and substituting for r in (13) gives for the tube St = K$ = 0.276
~~-213
Re-0.42
(@)1/3.
m
In applying this procedure to the annulus it is necessary to use the appropriate equation for the friction factor, and this involves making the distinction between the friction factors at the inner and outer walls of the annulus, fiand f2respectively. This question has recently been discussed by Knudsen,g who presents a method by which the friction factor at both walls of the annulus may be determined for different values of Re and r,/r,. This treatment, based on the work of Rothfus et aLlo who fist defined two distinct annulus friction factors, yields the following expressions, both of which approximate the exact values of the core friction factor within 5 per cent in the Reynolds number range 4 x 18 and 106 with the present system. (17) and
fl=
O-079 Re- 025(~j@“”
($
:;)),
where il. = rmsX/r2 and
Insertion of the appropriate values of a and il for a radius ratio of two, and substituting in (15) and then (13) yields St = z
= 0.25 Re-w40
~~-213
(d,/L)l/3
(1%
m
and St = z
= O-2995 Re-0.42 Sc-2/3(d8/L)1’3. m EXPERIMENTAL
Apparatus Rates of mass transfer were determined by measuring limiting electrolysis current for the cathodic deposition of copper from solutions of cupric sulphate containing excess sulphuric acid. The vertical annular flow cell consisted of an outer tube of
T. K. Ross and A. A. WRAOO
1098
TABLE 1. SUMMARY
OF ANNULUS
Annulus number Diameter of tube, da, cm Length of tube, cm Diameter of core, d,, cm Equivalent diameter, de, cm Entrance length, cm Diameter ratio, da/d1 Value of [$(a)1118in equation (9) Electrode lengths, cm
AND ELECTRODE OIZQME~
1 2.54 152 1.21 1.27 125 2
1.201 o-05, 0.10 0.05, 1.0 3.81, 7.62 15.24
2 5.08 214 2.54 2.54 165 2
1.201 1-o 3-81 15.24
3
4
2.54 152 0.635 1905 115
5.08 214 0.635 4.445 125
4
1.289 1.0 3-81 15.24
8
1.418 1.0 3.81 15.24
copper,
flanged at both ends to Perspex inlet and outlet sections which served to hold the Perspex core concentrically within the copper tube. The copper cathode sections on which the mass-transfer process took place formed part of the core of the annulus, and were situated at such a distance from the cell entrance as to ensure fully developed flow conditions. (The annulus hydrodynamic entry length problem has received attention from Heaton et al.,ll and the results of this treatment were used in ascertaining the required entry length in the present work.) Twenty equivalent diameters were allowed as an exit length. The outer tube acted as anode in the electrolysis cell, and the electrolyte flowed vertically upwards. Four annulus assemblies were used and the geometries are summarized in Table 1, which also gives details of the electrode lengths used with each annulus. The cell assembly and electrical circuit is illustrated in Fig. 1. The source of variable voltage was a transistorized power supply capable of a current throughput of 5 A.
Detail
of wow-tight --
fitting
F-10. 1 Diagram of apparatus, including details of amndus assembly.
Electrochemicalmass transfer studiesin annuli
1099
Any one of a series of 8 ammeters covering a range of O-10 ,uA to O-10 A could be selected to measure the electrolysis current. The potential of the cathode was measured relative to that of the anode by means of an electronic millivoltmeter. (Since the area of the anode was large compared with that of the cathode, it constituted a convenient reference electrode for the purpose of limiting current measurements.) A decade resistance box was also incorporated in the circuit. The rate of solution flow was measured by the use of one of four specially calibrated Rotameters equipped with stainless steel floats. A thermometer was situated in the electrolyte reservoir and cooling of the electrolyte was effected by a film of cold water on the outside of the reservoir surface. During the course of the work a variety of centrifugal pumps, fabricated in both plastic and stainless steel, were used. Procedure
The most extensive series of experiments was performed with Annulus 1, which permitted seven different electrode lengths to be studied. Prior to each experimental run the electrode surface was treated with increasingly fine grades of emery cloth, degreased with trichlorethylene and washed with alcohol before being secured in the core assembly and introduced into the annulus. After being made up to 1.5 M in H,SOI and 0.01 or 0.05 M in CuSO,, the electrolyte was treated for about 90 min with oxygen-free nitrogen and its temperature adjusted to 18°C. A low current density was passed through the cell for a few minutes to achieve a uniform fresh deposit of copper, and then current/potential curves were obtained for different flow rates by increasing the electrical potential across the cell in steps of about 40 mV until hydrogen evolution commenced. The steady state current corresponding to each value of potential was noted. Readings were taken in laminar flow with all seven electrodes and in turbulent flow with the 15.24-cm, 3*81-cm and l-cm electrodes, and were repeated three times for each electrode at each flow rate. Samples of solution were withdrawn at frequent intervals for analysis. Confirmatory experiments in streamline and turbulent flow were performed with Annulus 2 in which the diameter ratio was also 0.5. The effects of varying diameter ratio were studied by employing Annuli 3 and 4 with an electrolyte concentration of 0.01 M in streamline flow. In the case of Annulus 4, for which the ratio of entry length to equivalent diameter is low, Reynolds numbers were restricted to a maximum of 1,300. Secondary experiments involving low flow rates and widely varying electrolyte concentrations were performed in order to examine natural convection effects. By this means results obtained in conditions where natural convection effects were known to be significant could be excluded from the final streamline flow correlation. Tables of results, a more detailed account of the apparatus and experimental procedure, and a fuller exposition of the theory, are given elsewhere.12 RESULTS Typical current/potential curves from which values of the limiting current may conveniently be estimated are shown in Fig. 2. Since electrical migration effects are rendered negligible by the presence of a large excess of indifferent electrolyte, and since at the limiting rate of deposition the concentration of copper ions is reduced virtually
1100
T. K. Ross and A. A. WRAGG
mV
AE,
FIG. 2. Current/potential curves obtained using Anmdus 1 (Table 1).
to zero at the electrode surface, the mass-transfer coefficient due to diffusion and convection, KL, is related to the limiting current by the equation KL=-.
k .zFAC,
Figure 3 shows the manner of variation of the observed mass-transfer coefficient with Reynolds number; the change in gradient between the regions of streamline and turbulent flow is clear. A plot of the average value of the mass-transfer coefficient obtained for varying electrode lengths for three different values of Re is given in Fig. 4. This affords an excellent illustration of the variation of the rate of mass transfer with electrode length in the mass-transfer entrance region. The mass-transfer coefficient
/
100 / 50 /'$
2:r,AF
Annulus I : S, q2450 L =3*8lcm:&,=OOlM 50-
20 KS
I
I
IO'
104
Re Fw. 3. Variation of average mass-trans~u”,~lient
with Reynolds Number, employing
Electrochemical
1101
mass transfer studies in annuli
can be seen to tend toward an infinite value at the leading edge of the electrode, (corresponding to zero mass-transfer boundary-layer thickness), and to decrease very rapidIy over the first few cm. These results were obtaitied with Annulus 1.
Variation af average mass
transfer coefficient with electrodelength Sc=2450 0: =l27cm
0
I
I I
I
2
I
3
II 4
5
I 6
IIll
7
8
9
Electrodelength,L.
FIG. 4. Variation
KJ
I II
I
I2
I
I3
I I I
I4
L
I5
cm
of the average mass-transfer coeficient (d, = 1.27 cm, SC = 2450).
with electrode
length
Laminar flow Further calculations based on the experimental data were performed in the manner suggested by the LMque solution as expressed by (9). Using the viscosity and density data of &e&erg, Tobias and Wilke13 and values of diffusivity based on the data of Gordon and Cole,%he relevant dimensionless groupswere evaluated. Figure 5 shows a plot of all the laminar flow results obtained with Annuli 1 and 2. The appropriate annulus L&&que solution corresponding to these conditions (lo), is represented by the upper broken line, while the Iower broken line represents the circular tube equation (5). All the results fall between these two lines and are well represented by the equation Sh = 1*76(Re. SC. c&/L)~~.
(22)
Experimental values of Re . SC . d,/L have varied between 1oQand IOB,so that a thorough examination of the mass-transfer entrance region has been made. Comparison of the present results with those of Lin et aL4shows that they lie closer to the annulus L%que equation, falling an average of about 9 per cent below the line compared with the 17 per cent discrepancy of the data of Lin et al. It is suggested that this may be due to the fact that the desired conditions of transport control are realized
T. K. Ross and A. A. WRAGG
1102
more closely with the electrolytic system employed in the present work. In experiments using circular pipes, Linton and Sherwood6 reported results in agreement with the LBv&que solution, and it would be expected that for an annulus the results would be above, and not (as Lin et al. report) on the same line. The close agreement with the prediction of the LBv&que solution is not unexpected, since the assumptions involved are not unrealistic in the system examined.
106 Re.
IO'
SC d. /L
FIG. 5. Results obtained in streamline flow when r = 05
The data obtained with the annuli of different geometries were treated in a similar way, and the results are best represented by Sh = 1*80(Re . SC . d&)1/3
(23)
for rl/ra = 0.25, and Sh = 2*03(Re . SC . de/Q113 (24) for rl/ra = 0.125. Comparison of (22), (23) and (24) indicates that as the value of rl/ra decreases, the correlating line on the plot of Sh against Re . SC . d,/L lies higher, but in each case is some 9 to 13 per cent lower than the appropriate form of (9). This fact demonstrates the inadequacy of correlations for annular flow mass transfer, such as that of Lin et ~1.~ which merely substitute the annulus equivalent diameter term into the equation applicable to a tube. Turbulentpow Calculations based on the experimental data for turbulent flow were performed in the manner suggested by (19) and (20). The results obtained with Annulus 2 are presented in Fig. 6 in the form of a plot of (KJU,> Sc213against Re, the solid line representing the solution for a tube, (16). Coincidence with this equation is obtained except for the case of the 15.24cm electrode, for which there is a marked deviation as the Reynolds number increases due to the rapid onset of fully developed mass transfer conditions. The empirical Chilton-Colbum equation jo = St . Sc213= O-023 Re-“‘2,
(25)
Electrochemical mass transfer studies in annuli
1103
with which the turbulent flow data of Lin et al4 agreed, is shown as a broken line in the diagram. The results obtained with both Annuli 1 and 2 are given in Fig. 7, where (KJ iJ,J . Sc2J3 (L/d,)“3 is plotted against Re. The results are compared with (16) for the
Reynolds FIG.
6.
number
Results obtained in turbulent flow with Annulus 2.
case of a tube with de substituted ford, and also with the approximations to the exact annulus solution represented by (19) and (20). It can be seen there that is good agreement with the tube equation, whereas the results lie some 9 per cent below the theoretical annulus line. This finding is similar to that of Van Shaw, Reiss and Hanratty,6
Ix
2r
FIG.
7.
Results obtained in turbulent flow with Annuli 1 and 2.
whose results for the circular tube experiments lay, on average, 7 per cent below the theoretical line. The turbulent flow results obtained with the two Annuli of radius ratio 1 to 2, are therefore well represented by the equation St = !$ = 0.276 Sc-2/sRe-o42(d,/L)l/s m
T. K. Ross and A. A. WRAOO
1104
Mass transfer entry length in laminar flow Although only relatively short electrodes have been used in the present study, it is possible to estimate the magnitude of the mass-transfer entrance length from the fact that close agreement with a theoretical analysis is obtained. The solution of Graetz,15 which is not restricted by the assumption of a linear velocity gradient, provides a more extensive solution than that of LCv&que, and while the two solutions coincide when
IO 7-
Gmetzsolution
3-
100
I _ 1000
Re. Scd/x
Fm. 8. Representation of solutions of Lkdque and Graetz for a tube.
Re . SC d/x > 200, the Graetz solution predicts that the total Sherwood number (Sh,) approaches a constant value at low values of Re . SC d/x. This corresponds to the fully developed mass-transfer condition. The situation is somewhat complicated in the present system by the anode mass-transfer boundary conditions, but if the simpler case of the round tube is considered (Fig. 8), the mass-transfer conditions are fully developed when Re . SC d/x is less than about 8. For a typical electrolytic process with SC = 2,000 and Re = 500, x > 125,000d (27) for fully developed conditions. The magnitude of the entry length for mass transfer is thus far greater than the corresponding length in most heat-transfer situations, because of the high value of the Schmidt number, which represents the ratio between momentum and mass diffusivities. CONCLUSIONS
When inspecting the results of experiments designed to measure mass transfer in circular or annular ducts, it is important to be clear about the different relationships which apply to various flow regimes. As the expressions presented in the theoretical portions of this and other paperP make clear, the mass transfer coefficient (KL) may vary as the average liquid velocity (Cl,,,) to powers of 0*33,0.50, 0.58 or 0.60 as the flow is developed streamline, developing streamline, developed turbulent or developing turbulent respectively. If, therefore, a range of results obtained under all these conditions is correlated it is not surprising that an approximate dependence on Uk” is reported.1,2 Such treatments do not however contribute to an understanding of the relationships between mass transfer and various hydrodynamic conditions. The present work has confirmed the dependence of KL on lJL33 and U,o’58for fully developed streamline and turbulent flow conditions respectively.
Electrochemicalmass transfer studies in annuli
AckrwwZe&ments-We us (A. A. W.), and to
1105
wish to acknowledge the receipt of a D.S.I.R. studentshipawarded to one of thank Dr. B. Gay for helpful discussionsduring the research.
NOTATION Electrode surface area, cm2 A Annulus radius ratio, r1/r2 a Concentration of transferred species, mole/ems (Subscripts b and s indicate C bulk and surface values respectively) Tube diameter, cm d 4 4 Annulus inner and outer diameters, cm Annulus equivalent diameter, d, - d,, cm 4 Diffusion coefficient, cm2/s Friction factor Faraday i i Electrolysis current, A Limiting current, A iL j-factor for mass transfer jD Average mass transfer coefficient, cm/s KL Electrode length, cm L Hydrodynamic entrance length, cm LS Radial distance from conduit axis, cm r Inner and outer annulus radii, cm h/r2 Radial distance of point of maximum velocity from annulus axis, cm rmax Reynolds number, dU,p/p Re Schmidt number, ,u/p D SC Sherwood number, K,d/D Sh Stanton number, KL/Um St Average liquid velocity in x direction, cm/s u?n Point liquid velocity in x direction, cm/s G Velocity in y direction, cm/s V Distance in direction of flow, cm X Distance perpendicular to direction of flow, cm Y Z Number of faradays in electrodeposition of one mole of ionic species. Velocity gradient at conduit wall, s-l Ratio of radial distance of point of maximum velocity to outer radius (rmax/r2) Viscosity, g/s.cm Kinematic viscosity, cm2/s Density, g/cm3 REFERENCES 1. J. C. B.u.~N and A. J. ARVfA, Electrochim. Acta 9, 17 (1964). 2. J. C. BAZAN and A. J. ARVfA, Electrochim. Acta 9, 667 (1964). 3. J. L&&QUIZ,Ann. Mines Carbur., Paris (12) 13,201, 305, 381 (1928). 4. C. S. LIN, E. B. DENTON,H. S. GASKILLand G. L. PUTMAN,Ztistr. Engng. Chem. 43,2136 (1951). 5. P. VAN SHAW, L. P. REISSand T. J. HANRATIY,A. I. Ch. E. Journal 9,362 (1963). 6. W. H. INTON and T. K. SHERWOOD,Chem. Engng. Prog. 46,258 (1950). 7. E. S. C. MEYERINKand S. K. FRIEDLANDER, Chem. Engng. Sci. 17, 121 (1962). 8. W. L. FRIENDand A. B. METZNER,Amer. Inst. Chem. Eqprs. J. 4,393 (1958). 9. J. G. KNUDSEN,Amer. Inst. Chem. Engnrs. J. 8, 565 (1962).
1106
T. K. Ross and A. A. WIUOO
10. R. R. ROTHFUS,C. C. MONRAD,K. G. SIKCHIand W. J. HEIDFBER, Industr. Enpg. Chem. 47, 913 (1955). 11. H. S. HEATON, W. C. REYNOLDS and W. M. KAYS, Znt. J. Heat Mass Trumfer 7, 763 (1964). 12. A. A. WRAOO,Ph.D. Thesis, University of Manchester (1964). 13. M. E~NEZERQ,C. W. TOBUSand C. R. WILKE,J. Electrochem. Sot. 103,413 (1956). 14. A. R. GOFW~Nand A. COLE, J. Phys. Chem. 40,733 (1936). 15. L. GRAETZ,Ann. Phys. Chem. 25,337 (1885). 16. G. WRANGLBN and 0. NILSS~N,Electrochim. Acta 7, 121 (1962).