Electrochemical mass transfer at a heated electrode in a vertical annular flow cell

Chemical Engineering and Processing 43 (2004) 921–928

Electrochemical mass transfer at a heated electrode in a vertical annular flow cell C.F. Oduoza∗ School of Engineering and Built Environment, University of Wolverhampton, Telford Campus TF2 9NT, UK Received 21 January 2003; received in revised form 1 August 2003; accepted 4 August 2003

Abstract Mass transfer data for the simulation of high-speed wire plating with simultaneous heat transfer was carried out in an axisymmetrical tubular flow cell with developing laminar flow mass transfer characteristics. Current–potential curves for ferricyanide ion reduction at different flow rates showed well-defined plateaux. At low flow rates and depending on
T, the interaction between forced and concentration driven natural convection was evident, with the latter effect dominating. At higher flow rates, forced convection became dominant. A plot of limiting current against cathode tube temperature at different Reynolds numbers showed an increase in mass transfer with flow rate with data tending to converge at higher temperatures for all flow rates. Data were compared with those of other workers for mass transfer, and also with the analytical Leveque solution for an annular duct and produced reasonable agreement. Data for heat transfer showed a pronounced effect of thermally driven natural convection at lower Reynolds number, but progressively merged with the Leveque solution at higher Reynolds number. © 2003 Elsevier B.V. All rights reserved. Keywords: Heat and mass transfer; Wire plating; Annular cell; Forced convection; Heated; Electrode

1. Introduction The concentric annular geometry has many engineering applications, such as in heat exchanger and nuclear reactor design. Similarly, mass transfer to fluids flowing through annuli is frequently encountered in industrial processes; electrochemical reactors, wire plating cells, condensers, transpiration and film cooling in ducts can be given as examples. Convective heat transfer to fluid flow in long straight ducts of constant cross section is of special interest to mechanical and chemical engineers, and considerable effort has been devoted to its understanding and prediction. With the advent of high-speed computational facilities and sophisticated mathematical techniques and experimental methods numerous problems in this area of study can now be solved. Modern electroplating plants carrying out high-speed metal deposition processes often involve complex geometric shapes and complex hydrodynamics, as well as multi-ion solutions with proprietary additives. In many cases resistive ∗

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heating effects, resulting from the high current densities applied in the process, accompany wire plating. A practical example is the case of automated high-speed zinc plating for steel wire, which involves considerable current flow along the wire and a consequent significant thermal effect. This means that the wire becomes hot relative to the plating bath and the ionic mass transfer involved in the electroplating process occurs in the presence of temperature gradients. In the case of relatively low relative velocities between wire and bath the occurrence of forced convection with simultaneous concentration and thermally driven free convection is a possibility. This complex situation has been simulated in the present work in a concentric annular cell with a central ‘core’ electrode acting as a model for electrodeposition on a wire. The electrodeposition process has been simulated using the cathodic reduction of ferricyanide ions from aqueous NaOH at nickel electrodes. The work is intended to assist in the validation of any modelling work, which takes account of the influence of temperature gradients in transport processes in axisymmetrical cells. High-speed wire plating cells are extremely complex systems where many interacting effects are found. Firstly,

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temperature gradients are known to affect local viscosity and therefore modify velocity profiles, especially in laminar flows. Secondly, the drawn wire produces a Couette flow, which interacts with the imposed pumped flow in the cell. Thirdly, the hydrodynamics in the cell are not well defined and the geometry is complex. Fourthly, the temperature gradients occurring also produce density gradients, which, in turn, produce natural convection. Fifthly, and especially at the low flow rates favoured by modellers, the occurrence of natural convection effects driven by concentration differences within the fluid are also known to be significant and have been studied previously [1]. The precise mode of these various interacting influences then further depends on the relative directions of the different flows; i.e. forced flow may be antidirectional to natural convection and the two types of natural convection may be opposed or co-directional. There is also the additional complication in mass transfer of the effect of temperature variation on diffusivity and other physical properties. Electrochemical mass transfer studies in similar annular ducts under isothermal conditions have been reported previously [2]. The variation of the mass transfer coefficient with electrode length was demonstrated and results for streamline flow were compared with the Leveque equation modified to account for the annular geometry. Wragg and Ross [3] in another study described superposed free and forced convective mass transfer in an electrochemical system. Their experimental data was successfully correlated by the equation;
  0.219 1.96Re Sc de Grm Sc de 0.75 Sh = + 0.04 L L for an annulus ratio of 0.5. Wragg [4] studied combined free and forced convective ionic mass transfer in the case of opposed flow in a vertical annular flow cell and proposed criteria for the dominance of both forms of pure flow, and also for the occurrence of the point of minimum transfer rate. Wedekind and Kobus [5] have developed a conversion scheme extending the predictive capability of a previously developed theoretical model for average heat transfer in the case of a flat plate to those of circular disks for both assisting and opposing flow situations. Celata et al. [6] have also proposed a new method for the calculation of heat transfer coefficient in upward mixed convection heated flow. They confirm the reduction in heat transfer rate for mixed convection in upward heat flow is mainly due to the laminarisation effect in the near wall region (buoyancy effect). According to these workers, the newly proposed method can also be used to predict data from different geometries and channels. In a separate work, Chouikhi et al. [7] have studied local mass transfer rates at the wall of a pipe downstream of constricting nozzles in turbulent pipe flow with varying Schmidt number. Sarac et al. also produced physical properties data

for the ferro–ferricyanide system in aqueous NaOH at temperatures (10–90 ◦ C), which is particularly pertinent to this work where properties are required at elevated temperature [8]. An electrochemical study of mass transfer at heated electrodes which also used the ferri–ferrocyanide as test reaction has been described previously by Sarac et al. [9]. Simultaneous heat and mass transfer in free convection was investigated using the electrochemical limiting diffusion current technique where the concentration and thermal buoyancy effects were both aiding and opposing at horizontal cylinder electrodes. They also measured natural convection electrochemical mass transfer rates under the influence of simultaneous aiding thermal convection at horizontal cylinder electrodes. The experiments were performed using the limiting diffusion current method with the ternary copper sulphate–sulphuric acid–water system. The convective flow structure was observed by Schlieren photography and results were correlated by the equations Sh = 0.53(Grm Sc)0.25 ,

7 × 107 < Grm Sc < 4 × 109

Sh = 0.018(Grm Sc)0.4 ,

4 × 107 < Grm Sc < 1011

In a separate work, Wragg et al. [10] have studied mass transfer distributions near cell entries and corners in electrochemical flow cells. In the current work, we chose to simulate the process using a vertical concentric annular cell with stationary electrodes, an upward forced flow spanning the laminar and low turbulent regime, a heated core producing an upward thermally driven natural convection and a concentrationally downward driven natural convection due to the nature of the electrochemical reaction at the cell inner tube (core) electrode. This test reaction was the cathodic reduction of ferricyanide ions from an equimolar mixture of potassium ferri and ferrocyanide in 1 M NaOH aqueous solution. The flow was partially developed (fully at low Reynolds numbers), the mass transfer boundary condition was that of constant surface ionic concentration, and the thermal boundary condition was that of constant heat flux over the heated section, heating being applied by resistive heaters located inside the inner core electrode.

2. Experimental Fig. 1 shows the experimental setup which consisted of a vertical annular cell, with a central nickel tube of 7.11 mm (d2 ) outside diameter (6.57 mm internal diameter) acting as the cathode, and a nickel pipe of o.d. 50 mm (d1 ) and 42.72 mm internal diameter acting as anode, arranged concentrically. The hydraulic diameter was calculated as d2 − d1 , where d2 and d1 are the internal diameters of the outer pipe and the outside diameter of the inner tube respectively. The outer nickel pipe was flanged to PVC inlet and outlet sections. The inner cathode tube was fitted with an

C.F. Oduoza / Chemical Engineering and Processing 43 (2004) 921–928

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Thermocouple lead ( T a)

Electrolyte Outlet

25 mm

20 cm

Thermocouple ( Tb)

Nickel Pipe (Anode) Thermocouple ( T c)

Active length of cathode (10 cm)

Heating Element

50 cm

90 cm

100cm

Nickel Tube ( Cathode ) Nickel Flange PVC Flange

Electrolyte inlet

PVC

25 mm

5 cm Heater Lead Fig. 1. Axisymmetrical tubular cell.

internal cartridge heater/thermocouple assembly (manufactured by Plastic Moulding Supplies), over a 100 mm length to simulate a hot wire. Each cartridge heater was alternating current (AC) compatible, had a diameter of 6.5 mm. Thermocouples (Farnell) were positioned in the bulk fluid near the lower end and downstream of the heated section. A thermocouple reading was also taken from the heated core and there was also a provision for adjusting the depth of the thermocouples in the electrolyte. The cell was particularly designed for the study of mass transfer at a cathode with simultaneous thermal effects affecting the physical properties and involving the possible significance of combined forced convection and thermally and concentration driven free convection mass transfer. The hydrodynamic entrance length (Le ) for an annulus is equivalent to 0.0288 Re de , where Re is the Reynolds num-

ber and de the duct equivalent diameter. The length of the cell was 90 cm and the flow development length before the test section was 40 cm. For the cell under study, the hydrodynamic entrance length for a fully developed flow cannot be achieved beyond a Reynolds number of 400. At a Reynolds number of 2000 the required hydrodynamic entrance length is 2.05 m. The inner nickel core was coated in lacquer apart from a 100 mm length, which acted as the electrochemically active zone and corresponded to the position of the internal cartridge heater. 2.1. Experimental procedure The test cell was connected to a flow rig consisting of a 100 dm reservoir equipped with a cooling coil, a Beresford PV 121 chemical pump and a rotameter covering a flow

C.F. Oduoza / Chemical Engineering and Processing 43 (2004) 921–928

rate range of 0.01–220 l/min. The flow rig was equipped with several drains and an emergency dump tank. The plating process was simulated using the cathodic reduction of ferricyanide ions from an equimolar solution of 0.0038 M K4 Fe(CN)6 and K3 Fe(CN)6 in aqueous 0.5 M sodium hydroxide. The electrolyte entered the bottom of the cell and exited at the top. Oxygen-free nitrogen was bubbled through the electrolyte for several hours prior to experiments to remove dissolved oxygen; a nitrogen blanket was also maintained above the electrolyte during runs. It was possible to regulate the temperature in the reservoir by means of heat exchangers supplied with a mixture of cold and hot tap water. Electrical power to the cell was partly supplied by a Thandar TS 3023S dual channel precision dc power supply and current output monitored through a Thurlby Thandar 1906 computing multi-meter with auto ranging facilities. Before each experiment, voltage was applied beyond the limiting current region of 1.2 V for about 5 min to liberate hydrogen. This had the effect of removing surface oxides. By applying a potential of 800 mV between anode and cathode, i.e. mid-plateau region of current–potential curves, and simultaneously turning on the cartridge heater to heat the test region of the cathode to the specific temperature, limiting current values for a range of Reynolds numbers were determined. Ts (surface temperature) was approximated via the inner core thermocouple reading (Ta ) in Fig. 1; i.e. the temperature drop through the thin wall of the core electrode was neglected. Mass transfer coefficients for the defined length of cathodic test section were determined from current–voltage curves, which were found to display well, defined limiting currents. Measurements were also made both for natural convection (stationary electrolyte condition) and forced conditions at different flow rates. Flow varied between Re 300 and 15,000 and electrode temperature adjusted to 23, 40, 60, 80 ◦ C etc. was set and flow rate was increased in four steps from 0 to 38.5 l/min at each fixed electrode temperature.The equation, KL =

IL z F A c∞

was then applied to calculate KL . The heat transfer coefficient was calculated from the equation of heat transfer rate Q = h A(Ts − Tb ).

I /mA

924

7 6 5 4 3 2 1 0 0

0.3

0.6

0.9

1.2

1.5

Voltage (V) Fig. 2. Current–potential curves from the axisymmetrical cell (isothermal) [(䊊) Re = 0, (䊏) Re = 1370].

The electrolyte is easily decomposed in the presence of light to hydrogen cyanide, which both poisons the electrode and alters the concentration. The ferricyanide solution was therefore analysed frequently especially at the end of experiments by UV spectrophotometry at a wavelength of 419 nm to validate the concentration of the electrolyte. The reactions in this system are as follows: 4− Fe(CN)3− 6 + e → Fe(CN)6

at the cathode

3− Fe(CN)4− 6 → Fe(CN)6 + e

at the anode

Table 1 shows the physical properties of the electrolyte at different temperatures as calculated from the data of Sarac et al. [8].

3. Results and discussions Typical current–potential curves from the axisymmetrical cell for both zero flow (natural convection only) and forced flow with a Reynolds number of 1370 for isothermal conditions are shown in Fig. 2. Well-defined plateaux were obtained in both cases. A plot of mass transfer coefficient against flow rate is shown in Fig. 3. The strong influence of electrode temperature is immediately apparent with the data for the four different temperatures lying on distinct separate curves (Table 2). Mass transfer is seen to increase with temperature for a fixed flow rate, this being largely due to decreased viscosity and increased ionic diffusion coefficient in the near electrode layer. The points group closer together due to the incorporation of temperature sensitive physical property terms in Re. The data are also seen to be tending to converge at higher Re. The curves show particularly interesting

Table 1 Physical properties of electrolyte at different temperatures

Density(kg/m3 ) Dynamic Viscosity (kg/m s) Diffusivity of ferri ion (m2 /s) Schmidt number Kinematic viscosity

23 ◦ C

32 ◦ C

42 ◦ C

52 ◦ C

1024 0.955 × 10−3 7.4 × 10−10 1257 0.93 × 10−7

1018 0.73 × 10−3 9.4 × 10−10 766 0.72 × 10−7

1015 0.64 × 10−3 12.0 × 10−10 538 0.63 × 10−7

1011 0.55 × 10−3 14.0 × 10−10 378 0.54 × 10−7

Mass transfer coefficient (m/s)

C.F. Oduoza / Chemical Engineering and Processing 43 (2004) 921–928

925

3.0E-05

2.0E-05

1.0E-05

0.0E+00

5000

0

10000

15000

20000

25000

300 00

35000

Reynolds Number Fig. 3. Plot of mass transfer coefficient against Reynolds number at different temperatures [(䉱) 23 ◦ C (Tmean = 23 ◦ C), (䊉) 40 ◦ C (Tmean = 32 ◦ C), (䉫) 60 ◦ C (Tmean = 42 ◦ C), (+) 80 ◦ C (Tmean = 52 ◦ C)].

trends in the lower flow rate region. At 23 ◦ C electrode temperature (isothermal conditions) there is a steady increase in temperature with flow rate, whereas at the two higher temperatures increase in flow rate causes an initial decrease, followed by an increase, so that a minimum is observed. This may reflect the interaction between the (concentrationally) downward driven free convection and the upward forced flow which has produced minima in similar curves in other work with the copper–copper sulphate system [4]. It is somewhat puzzling that similar effects are not seen for the lower temperature curves. The lower flow rate range of these data is shown as Fig. 4 which shows the individual points in the very low Re range more clearly. The distinct decrease in mass transfer with flow rate for the two higher electrode temperatures is clearer in this form. It may be that at and near zero forced flow the mass transfer rate dependence on thermally driven natural convection is well established and becomes destabilised as the forced flow increases. Fig. 5 shows a comparison of the mass transfer results in the form of a plot of Sh against Re Sc de /L for four different electrode temperatures with those of Carbin and Gabe for isothermal conditions [11], and with the analytical Leveque solution for mass transfer to the inner wall of an annulus of the present dimensions in fully developed laminar flow [3]:   Re Sc de 0.33 Sh = 2.216 L

Carbin and Gabe proposed the equation:  0.35 de Shav = 3.93Re0.32 Sc0.33 L to correlate data for mass transfer in an annular cell using the copper–copper sulphate system with longitudinal flow. It should be noted however, that the hydrodynamic conditions in the Carbin and Gabe cell [11] were not well characterised. The four sets of data show two distinct regions when plotted in this form. At relatively lower values of Re Sc de /L the curves are relatively flat indicating little influence of forced convection and the dominance of natural convection. At higher values of Re Sc de /L the plots have a positive linear slope and tend to merge together towards one correlating line; this region is indicative of dominant forced convection. Ross and Wragg [2] have pointed out that when evaluating 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. Mass transfer, therefore, may vary as the average mean velocity 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. Our present data were acquired partially in a developing flow regime. A plot of heat transfer coefficient (HTC) against
T at different flow rates is shown in Fig. 6. Heat transfer is seen

Table 2 Heat transfer data Grh (Temperature, ◦ C)

Grh

Grh Grh Grh Grh

1.0993E 5.6443E 1.4700E 3.3100E

(23) (32) (42) (52)

Gr(Comb)9

Grm + 07 + 07 + 08 + 08

2.61E 2.61E 2.61E 2.61E

+ 06 + 06 + 06 + 06

7.1426E 3.1350E 7.4445E 1.5576E

+ 07 + 08 + 08 + 09

Pr (Temperature, K)

k

de

cp

6.5 (10) 5.0 (20) 4.39 (33) 3.75 (61)

6.00E − 04

0.0356

4.12

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Mass transfer coefficient (m/s)

3.0E -05

2.0E -05

1.0E -05

0.0E + 00 0

100

200

300

400

500

Reynolds Number Fig. 4. Plot of mass transfer coefficient against low Reynolds number at different temperatures [(䉱) 23 ◦ C (Tmean = 23 ◦ C), (䊉) 40 ◦ C (Tmean = 32 ◦ C), (䉫) 60 ◦ C (Tmean = 42 ◦ C), (+) 80 ◦ C (Tmean = 52 ◦ C)].

Sherwood .Number (Kd e/D)

1000

100

(Sh = 2.216(ReScde/L)^0.33[Leveque]) (Carbin and Gabe[11]) 10 1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

ReScde/L Fig. 5. Plot of Sherwood number against Re Sc de /L [(䉱) 23 ◦ C, (䊉) 40 ◦ C, (䉫) 60 ◦ C, (+) 80 ◦ C].

HTC (W/m2 K)

900

600

300 0

20

40

60

80

Temperature difference (K) Fig. 6. Plot of heat transfer coefficient against temperature difference at different flow rates [(䉫) 0.1 l/min, (䊏) 0.8 l/min, (䉱) 3 l/min, (䊏) 29 l/min].

to increase with increase in temperature difference at all flow rates. It appears very high flow rates at low
T only make a small difference in heat transfer at the sort of temperatures used in this work. Fig. 7 portrays a plot of Nu against Re at different temperature difference and shows that at higher flow rates the effect of temperature difference becomes less significant as data gradually merge towards a single line correlation. These data show that differences due to thermal convection are prominent at low flow rates but become less significant at high flow rates. The plot of Nusselt number against Graetz and Grashof number is shown in Fig. 8. The effect of combined natural and forced convection on heat transfer shows data comparatively collapsed into a

C.F. Oduoza / Chemical Engineering and Processing 43 (2004) 921–928

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Nusselt Number

100

10 1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

Reynolds Number

Fig. 7. Plot of Nusselt number against Reynolds number at different mean temperatures [(䉫) 10 K, (䊐) 20 K, (䉱) 33 K, (䊏) 61 K].

Nusselt Number

100

10 1.0E+04

1.0E+05 G z+(Gr h Prd e / L)

1.0E+06 0.75

Fig. 8. Plot of Nu against Gz + Gr(Grh Pr de /L)0.75 [(䉫) 10 K, (䊐) 20 K, (䉱) 33 K, (䊏) 61 K].

a flowing electrolyte to the core of an annulus in fully developed streamline and turbulent flow, and varying the geometry of the annular flow cell, length of the working electrode, solution flow rate and electrolyte concentration at a single temperature. This research on the other hand has produced mass transfer data with simultaneous heat transfer carried out in an axisymmetrical tubular flow cell with developing laminar flow mass transfer characteristics. Data for heat transfer showed a pronounced effect of thermally driven natural convection at lower Reynolds number and progressively merging with the Leveque solution at higher Reynolds number.

4. Conclusions single profile. Fig. 9 is the equivalent plot to Fig. 8 showing the effect of combined natural and forced convection on mass transfer where the Nusselt number is a dimensionless mass-transfer rate [12]. Mass transfer is seen to increase with increase in temperature and also with flow rate. At higher flow rates temperature difference is no longer significant as data start to converge into a single file. Axial flow in the annular space between two concentric cylinders provides a convenient situation for experimental studies of mass transfer. Ross and Wragg in an earlier experiment [2] have measured the rates of mass transfer from

Nusselt Num ber

100

10 1.00E +06

1.00E +07 G z+(G r m Scd e /L) 0.75

Fig. 9. Plot of Nu against Gz + Gr(Grh Sc de /L)0.75 [(䉱) 23 ◦ C, (䊉) 40 ◦ C, (䉫) 60 ◦ C, (+) 80 ◦ C.

The objective of this piece of work was to provide heat and mass transfer data for the simulation of high-speed wire plating under varying temperature conditions. Such data would enable the validation of any modelling work, which takes account of the influence of temperature gradients in transport processes in axisymmetrical cells. Based on this study we can draw the following conclusions: • It is certain that the rate of deposition (mass transfer) in a wire plating process is affected by temperature increase resulting from resistive heating which as a consequence affects the physical properties of the electrolyte. • Data also show that the prevailing complex hydrodynamics regime is determined by a combination of forced convection, concentrationally driven natural convection (co- or antidirectional to forced convection), thermally driven natural convection as well as the cell orientation whether vertically or horizontally aligned. • Mass transfer is affected by temperature difference between wire and bulk electrolyte increasing with increase in temperature difference after an initial minimum is observed at low flow rates due to interaction between the (concentrationally) downward driven free convection and upward forced convection.

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• Mass transfer data under different electrode temperatures were compared with those of other workers and with the analytical Leveque solution for mass transfer (to the inner wall of an annulus in fully developed laminar flow) and also for heat transfer. • Heat transfer increases with increase in temperature difference at low flow rates with marginal change only at higher flow rates. Also the effects of thermal convection are prominent at low flow rates. The outcome from this study should contribute to information necessary to model high-speed wire plating in industrial manufacturing set up as well as provide helpful hints to researchers working on heat and mass transfer in axisymmetrical cell geometries.

Re Sc Shde Tb Ts v W z

Reynolds number (de v ρ/µ) Schmidt number (ν/D) Sherwood number based on duct equivalent diameter (KL de /D) bulk liquid temperature (K) surface temperature (K) mean fluid velocity in cell or cell channel (m/s) mass flow (kg/s) number of electrons exchanged in electrode reaction

Greek letters ρ fluid density (kg/m3 ) ν kinematic viscosity of electrolyte (m2 /s) µ dynamic viscosity (kg/s m)

Acknowledgements This work was performed under Brite-Euram III Contract Number: BRPR-CT95-0008. The author also wishes to thank Professor A.A. Wragg for his support and helpful hints during the preparation of this paper.

Appendix A. Nomenclature A C cp d1 d2 de D F Grh Grm Gr (Comb) Gz h IL k KL L Nu Pe Pr Q

electrode surface area (m2 ) bulk species concentration (mol/m3 ) specific heat capacity (kJ/kg K) internal diameter of pipe (m) outside diameter of inner tube (m) equivalent diameter of annular duct, d1 − d2 (m) diffusion coefficient (m2 /s) Faraday constant (96,487 C/mol) Grashof number for heat transfer Grashof number for mass transfer combined Grashof number for mass transfer as defined in [9] Graetz number (Wcp /kL) heat transfer coefficient (W/m2 K) limiting electrolysis current (A) thermal conductivity of electrolyte (W/m K) mass transfer coefficient (m/s) electrode length (m) Nusselt number (hde /k) Peclet number (Re Sc) Prandtl number (cp µ/k) heat transfer rate (W)

References [1] C.F. Oduoza, M.A. Patrick, A.A. Wragg, Mixed convection mass transfer studies in cases of opposing and aiding flow in a parallel plate electrochemical cell, J. Appl. Electrochem. 28 (7) (1998) 697– 702. [2] T.K. Ross, A.A. Wragg, Electrochemical mass transfer studies in annuli, Electrochim. Acta 10 (1965) 1093–1106. [3] A.A. Wragg, T.K. Ross, Superposed free and forced convective mass transfer in an electrochemical flow system, Electrochim. Acta 12 (1967) 1421–1428. [4] A.A. Wragg, Combined free and forced convective ionic mass transfer in the case of opposed flow, Electrochim. Acta 16 (1971) 373– 381. [5] G.L. Wedekind, C.J. Kobus, Int. J. Heat Mass Transfer 39 (1996) 2843–2845. [6] G.P. Celata, F.D. Annibale, A. Chiaradia, M. Cumo, Upflow turbulent mixed convection heat transfer in vertical pipes, Int. J. Heat Mass Transfer 39 (1998) 4037–4054. [7] S.M. Chouikhi, M.A. Patrick, A.A. Wragg, Mass transfer downstream of nozzles in turbulent pipe flow with varying Schmidt number, J. Appl. Electrochem. 17 (1987) 1118–1128. [8] H. Sarac, A.A. Wragg, M.A. Patrick, Physical properties of the ternary electrolyte potassium ferri–ferrocyanide in aqueous sodium hydroxide solution in the range 10–90 ◦ C, J. Appl. Electrochem. 23 (1993) 51–55. [9] H. Sarac, A.A. Wragg, M.A. Patrick, Natural convection mass transfer at a horizontal cylinder electrode with an opposed thermal buoyancy effect, Electrochim. Acta 38 (1993) 2589–2598. [10] A.A. Wragg, D.J. Tagg, M.A. Patrick, Diffusion-controlled current distributions near cell entries and corners, J. Appl. Electrochem. 10 (1980) 43–47. [11] D.C. Carbin, D.R. Gabe, Electrochemical mass transfer in an annulus, Electrochim. Acta 19 (1974) 653–654. [12] J.S. Newman, Electrochemical Systems, in: International Series in the Physical and Chemical Engineering Sciences, second ed., Prentice-Hall, Englewood Cliffs, 1991, pp. 253.