Ind. Eng. Chem. Res. 1991, 30, 264-267
264
Deming, S. N.; Morgan, S. L. Data Handling in Science and Technology; Elsevier Science Publishing Company, Inc.: New York, 1987;Vol. 3, pp. 197-210. TVA. New Developments in Fertilizer Technology. Tennessee Valley Authority ( U S . ) Bul. Y-107,1976.
TVA. New Developments in Fertilizer Technology. Tennessee Valley Authority (U.S.) Bul. Y-136,1978.
Received for review March 26, 1990 Accepted July 19, 1990
Mass-Transfer Correlation for Flow over Cylindrical Microelectrodes David J. Earl, Harlan J. Kragt, Christopher W. Macosko, and H e n r y S . White* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
An annular electrochemical flow cell was designed in which electrolyte flow occurred longitudinally with respect to a cylindrical platinum microelectrode. Convective mass-transfer properties of the annular cell were determined by measurement of faradaic currents corresponding to the masstransfer-limited oxidation of ferrocyanide in water (Sc = 1390) and ferrocene in acetonitrile (Sc = 180). Measurements were made by using electrodes of 25-pm, 127-pm, and 1.0-mm diameter, by using a flow cell of 4%" diameter, and a t flow velocities corresponding to Reynolds numbers between 1and 900. The generalized mass-transfer correlation, based on the electrode diameter, was determined to be Shd = (0.342 f 0.008)Red1/2S~1/3. Introduction Electrochemical measurements based on forced convection have numerous applications in chemical analysis (Bard and Faulkner, 1980) and in hydrodynamic studies (Newman, 1973; Hannatty and Campbell, 1983). The magnitude of a faradaic current i, measured as a function of an applied potential, reflects the average molar flux of electrons to the surface
- 1- - (Ne) nFA which results in oxidation or reduction of an electroactive species. In eq 1, F is the Faraday constant, n is the equivalent number of electrons transferred per mole, and A is the electrode area. The faradaic current is also a measure of the mass-transfer flux of reactant to the surface. For example, the current corresponding to the oxidation of a soluble and chemically stable species, e.g., R + 0 + ne-, can be written as
where (NR) = k,(CRbUlk- C *q, k , is the average masstransfer coefficient, and CRbjkand CRSd are the bulk and surface concentrations of R, respectively. If the electrochemical system is reversible, i.e., if the electron-transfer kinetics at the surface are very rapid, the surface concentrations of 0 and R are described by the Nernst equation, E = Eo'+ (RT/nF) In (COsUrf/CRBU~ ( E O ' is the formal reaction potential), leaving k , as the only unknown parameter in eq 2. A t applied potentials sufficiently positive of E O ' , the surface concentration of the reduced species is driven essentially to zero. Thus, measurement of the faradaic current provides a direct method of obtaining k , for a particular electrode geometry and flow condition. This simple relationship provides an exquisitely simple method for establishing generalized mass-transfer correlations in complex geometrical systems. In this article, a set of electrochemical measurements based on eq 2 is used to establish the generalized masstransfer correlation for an annular flow cell, Figure 1. The key element of the cell is a cylindrical platinum microelectrode (diameter = 25 km, 127 pm, or 1 mm) oriented 0888-5885/91/2630-0264$02.50/0
along the central axis of a long glass tube that serves as the main body of the electrochemical cell. An electrolyte solution containing a soluble electroactive species is pumped through the cell at a specified flow rate, and the resulting current at the Pt wire is measured. Because the electrolyte flow is parallel to the working electrode, the resulting mass-transfer correlation obtained from measurement of current in the laminar flow regime is expected to have a similar form as that for laminar flow over a flat plate Shd = KRed1/zSC'/3 (3) Here, Shd is the Sherwood number (=k,d/Dab), K is an experimentally determined constant, Red is the Reynolds number (=u,,d/u), uav, is the average fluid velocity, u is the kinematic viscosity, Dab is the mass diffusivity of the electroactive species, Sc is the Schmidt number ( = v / o a b ) , and d is the characteristiclength. For flow over a flat plate, K = 0.664 and d is the length of the plate. For the concentric annular geometry shown in Figure 1, K is anticipated to be significantly different than that for a flat plate due to (i) the enhanced radial flux of reactive species to the microcylindrical electrode and (ii) the development of the annular flow profile inside the annular cell. Mass-transfer correlations have been established by using cylindrical electrodes (Muller, 1947; Ranz, 1958) where electrolyte flow is orthogonal to the electrode axis. In addition, a theoretical investigation has been made (Sioda, 1989) of tubular flow perpendicular to a cylindrical electrode. However, this report is the first to characterize the configuration where flow through a tube is parallel to a cylindrical microelectrode. Heat-transfer correlations exist for longitudinal flow over a bed of cylinders (Axford, 1965) but, because they are obtained for fully developed mass-transport profiles over bundles of large diameter rods, do not apply to the cell arrangement of Figure 1. Experimental Section Materials. The body of the cell was constructed from a l-m-long, 0.48-cm-i.d. Pyrex glass tube, Figure 1. The length and diameter of the tube were chosen in order to eliminate entrance and exit flow effects on the flow behavior of the test area at the middle of the cell. The glass cell was modified with ports to allow attachment of a 0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 1,1991 265 Reference Electrode Working Electrode
Connections to Pump
Flow in
Silver Epoxy 2.5"
'Iatinum
Wire
500 r
Counter Electrode
Re,= 1320
Flow out
Counter Electrode
2.0"
4 Copper Wire
EPOXY
Figure 1. Top: Electrochemical cell. Bottom: An exploded view of the cell near the Pt electrode.
reference electrode and to connect the cell to the flow source. C-Flex tubing (TM Concept, Inc.) was used to connect the pump and the reference electrode to the cell. The ends of the flow cell were sealed with vinyl rubber septa. The C-Flex tubing and the septa were tested for solvent compatibility by soaking in acetonitrile for 72 h. A Masterflex peristaltic pump (6-600 rpm; Cole-Parmer, Chicago, IL) was used throughout the experimentation. A laboratory-built saturated calomel reference electrode (SCE) was connected to the cell at -35 cm downstream from the working electrode. A lOemesh Pt gauze (Aldrich Chemical Co.), fashioned in the shape of a hollow cylinder, was used as the counter electrode. The Pt gauze electrode lined the inside wall of the glass cell surrounding the working electrode, providing a nearly radial concentric current distribution between the working and counter electrode. The working electrode (Figure 1)consisted of a platinum wire of diameter d = 25 pm, 127 pm, or 1mm and length 1 = 0.5-2.0 cm (Aldrich Chemical Co.). The electrode was connected to a copper wire lead with EPOTEK H20E silver epoxy (Epoxy Technology Inc., Billerica, MA). Both ends of the cylindrical electrode were sealed in a 2-mm-0.d. Pyrex glass tubing with Epoxi-Patch 56C (Dexter Corporation, Olean, NY). In some measurements, the downstream end of the Pt wire was left unattached. The test solutions used in these studies were (i) an aqueous potassium ferrocyanide (K,Fe(CN),) solution with KC1 as supporting electrolyte and (ii) ferrocene ((C5H5)2Fe) in acetonitrile with tetrabutylammonium perchlorate (TBAP) as supporting electrolyte. The water used in experimentation (18 MQ) was purified by a Labconco Model 90004 Water Prodigy. K,Fe(CN), (ACS. grade, Spectrum Chem. Manufacturing Corp., Renaldo Beach, CA), KC1 (Analytical grade, Mallinckrodt Inc., Paris, KY), (C5H5)2Fe (Strem Chemicals Inc., Newburyport, MA), and acetonitrile (HPLC grade, Fisher Scientific, Fair Lawn, NJ) were used as received. Tetrabutylammonium perchlorate (Electrometric Grade, SouthwesternAnalytical Chemicals, Inc., Austin, TX)was recrystallized in spectra grade ethyl acetate (EM Science, Cherry Hill, NJ) and oven dried at 90 OC. Procedure. Before each trial, the cell was cleaned by pumping copious amounts of clean solvent (either water or acetonitrile) through the cell. The cell was drained and the electrolyte solution was pumped through the cell. The average flow velocity for each pump speed and electrolyte solution was determined from volumetric flow rate measurements.
-100
I
t
L
0.6
0.8
1.0
Voltage vs SCE (VI
Figure 2. Cyclic voltammograms for the oxidation of 10 m M K4Fe(CN)6at a 127-pm-diameter Pt electrode. Red is the Reynolds number for each flow velocity based on the electrode diameter. The solution contained 1.0 M KCl as supporting electrolyte. Potential scan rate: 0.2 V/s.
Cyclic voltammetric curves were obtained over a range of flow velocities. The potential of the microcylinder Pt electrode was cycled at 0.2 V/s between 0.0 and 1.0 V vs SCE for the K,Fe(CN), system and between 0.0 and 1.8 V vs SCE for the (C5H5)2Fesystem. The flow rate was randomly selected for each successive measurement so that time-dependent trends that might result from electrode fouling or changes in electrolyte composition would be apparent. No such trends were apparent. Measurements of the mass-transfer-limiting current were made at the positive limit of the voltammetric curve. Cyclic voltammograms were obtained by using a Princeton Applied Research (Princeton, NJ) Model 175 universal programmer and a Model 173 Potentiostat and recorded with a Kipp & Zonen X-Y recorder.
Results and Discussion The electrochemical reactions
+
F c ~ ( C N )* ~ ~Fc?(CN)~~e-
E"' = 0.185 v VS SCE
and (C5H5)2Fe== (C5H5)2Fe++ e-
E"' = 0.31 V vs SCE
were used to establish a generalized mass-transfer correlation for the annular flow cell. Both redox couples are soluble in their respective solvent (Fe(CN),Q-/Fe(CN)lin H 2 0 and (C5H5)2Fe/(C5H5)2Fe+in acetonitrile) and chemically stable on experimental timescales reported here. In all measurements, an inert supporting electrolyte was added to the solutions (in 100-fold excess of the redox species concentrations) to reduce the migrational component of the flux. Figure 2 shows the voltammetric behavior observed by using a 1.1-cm-long, 127-pm-diameter Pt wire with a 10 mM Fe(CN)e4- (1.0 M KC1) solution flowing through the annular cell. The cyclic voltammogram in the absence of flow (Red = 0) is shown for comparison and displays the characteristic anodic and cathodic current peaks expected for a diffusional controlled response. In the absence of flow, the diffusion current at potentials positive of E"' approaches a pseudesteady-state value of -40 pA, in good agreement with the value (37 p A ) extrapolated from numerical values of Jaegar and Clarke (1942) for a purely diffusional flux to a cylindrical body and from a recent digital simulation in our laboratory (Norton and White, 1990).
266 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 150
100 A I27 pm x 2.0 cm Femrene Oxidation + I27 pm x 1.85 cm
c WY
50
0
0
50
o
25pmxIOcm
150 200 ReiRSc'l-)
100
250
300
350
o 127pmxZOcm A 127pmx2Ocm
+ 127 pm x
c WY
0
20
1 85 cm
40
60
100
80
Re"Scl' Figure 3. (a) Mass-transfer correlation for the oxidation of Fe(CN)*'- in water and (C5H5)2Fein acetonitrile. (b) Enlargement of lower left region of (a). The line drawn through the symbols represents the best fit to data. Error bars represent fl standard deviation.
At finite flow velocities, a steady-state, sigmoidal-shaped current-potential curve is obtained. Well-defined limiting current plateaus are observed for both Fe(CN)64-and (C5H5)2Feoxidation and for all combinations of wire lengths and diameters. In each case, the limiting current was found to be proportional to the square root of the average flow velocity. Values of the average mass-transfer coefficient, k,, were calculated as a function of the flow velocity from the limiting currents (ia,) positive of E O ' . In this potentid region, the concentration of Fe(CN)64- (or (C5H5)2 Fe) at the electrode surface is reduced to essentially zero, allowing eq 2 to be rewritten as (4)
We find that, for the same range of flow velocities, k, increases significantlyas the electrode diameter decreases, demonstrating an enhanced rate of radial mass transfer to the smaller electrodes. Based on this observation, representative data from 110 measurements of the limiting current have been plotted in Figure 3 as Shd vs Red1f2Sc'f3, where Shd and Red are calculated by using the electrode diameter as the characteristic length. Literature values of the diffusivities (K,Fe(CN),, 6.5 X lo4 cm2 s (von Stackelberg et al., 1953); (C5H5)2Fe,2.4 X 10- cm2/s (Kuwana et al., 1960)) and solvent kinematic viscosities (H20,9.02 X cm2/s; acetonitrile, 4.32 X cm2/s at 24.5 OC (Perry and Green, 1984; Weast, 1982))were used in the calculations. Fi ure 3 demonstrates a good correlation of Shd vs Redl 2sc1/3for electrode lengths between 0.5 and 2 cm and diameters between 25 pm and 1 mm. In addition, Sc values for the two redox systems are widely different (1390
6
7
for Fe(CN)4- in H20; 180 for (C5H5)2Fein acetonitrile), providing a test of the assumed dependence of Shd on the 1/3 power of the Sc number. A linear regression of the form y = mX to the entire data set, part of which is shown in Figure 3, yields Shd = 0.342(*o.008)Red1f2Sc'/9.The error value is derived from the standard deviations in the slope and represents a 95% confidence level. It is interesting to note that the observed correlation has no dependence on the diameter of the flow cell. We believe that this is a consequence of the electrode diameters being relatively small (40 "C) and nitric acid concentrations (>25 wt %), the reaction is not easily controlled due to foam formation and nitrogen oxide generation. In these experiments, cold nitric acid was added slowly to the reaction mixture in order to keep the temperature constant. After a certain time, the reaction mixture was cooled and filtered through a medium porosity fritted porcelain funnel 0 1991 American Chemical Society
