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Theory of collection efficiencies in the double tubular hydrodynamic electrode
Journal of
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 583 (2005) 318–326 www.elsevier.com/locate/jelechem
Theory of collection efficiencies in the double tubular hydrodynamic electrode Mary Thompson, Richard G. Compton
*
Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, Oxfordshire OX1 3QZ, United Kingdom Received 6 April 2005; received in revised form 29 May 2005; accepted 13 June 2005 Available online 8 August 2005
Abstract The theory of collection efficiencies at a double tubular electrode is explored through numerical simulation, including an analysis of the effects of axial diffusion. The theory is compared with analytical results in the high flow rate limit, using the Levich equation and the Matsuda theory for collection efficiency. For lower flow rates the dependence of the collection efficiency on absolute and relative lengths and radii of the tubular electrodes is investigated. 2005 Elsevier B.V. All rights reserved. Keywords: Double tubular electrode; Collection efficiency; Computational electrochemistry
1. Introduction This work deals with the theory of collection efficiencies obtained from a double tubular electrode, where two electrodes are embedded, in series, flush with the wall of an insulating tube through which a solution of electroactive species can flow (Fig. 1). The tubular electrode lends itself to numerical analysis due to the welldefined flow pattern within it. Much theoretical and experimental work at single tube electrodes has been reported previously, building on work which gave analytical results for the Le´veˆque (thin diffusion layer) limit [1–11]. Double electrodes have been studied extensively, due to their usefulness as probes of the rate of some homogeneous reaction which may occur ÔbetweenÕ the two electrodes [12–14]. The double channel electrode is a close analogue of the double tubular electrode and much *
Corresponding author. Tel.: +44 0 1865 275413; fax: +44 0 1865 275410. E-mail address: [email protected] (R.G. Compton). 0022-0728/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2005.06.018
experimental and theoretical work has been reported for this arrangement [12,14–17]. Similarly, the rotating ring disc electrode (RRDE) has also been extensively researched [18–21]. Work has been undertaken on the double tube electrode; however the theory reported is confined to analytical solutions in the high flow rate Le´veˆque limit, and experimental data is limited, most studies to date having used the channel electrode setup. An important practical advantage of the double tube electrode over the channel is the relative facility of its fabrication, even in microdimensions, where the use of microtubes relative to microdiscs remains relatively unexplored. This arrangement can be achieved by simply layering metal sheets or foil between layers of insulating material, and punching or drilling a hole through all the layers, as recently reported by Davis et al. [22] for a non-flowing system. 1.1. Tubular electrode theory The mass transport equation for convection and diffusion in cylindrical coordinates is [23,24]
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
2. Modelling the double tubular electrode
Flow direction
r
y
xe1
2.1. Model with axial diffusion neglected
xe2
gap
x Fig. 1. A schematic of the double tubular electrode, showing coordinate system and dimension labels.
2 oc o c 1 oc o2 c oc ¼D þ þ vx ; 2 2 ot or r or ox ox
ð1Þ
where c is the concentration of electroactive species, D is its diffusion coefficient, r and x are the coordinates in the radial and axial directions, respectively, and vx is the convective flow velocity, defined by the equation r2 vx ¼ v0 1 2 ð2Þ q for laminar flow in a tube where v0 is the axial velocity at the tube centre and q is the tube radius. Conditions for achieving laminar flow have been discussed previously [4]. Levich made some simplifying assumptions to the mass transport equation, leading to the equation for the limiting current in a tubular electrode [4–6,23,24] 2
2
1
ilim ¼ 5.24 105 nF ½Abulk x3e D3A V 3f ;
ð3Þ
where F is the Faraday constant, [A]bulk and DA are the bulk concentration and diffusion coefficient of the electroactive species, respectively, and Vf is the volume flow rate of solution through the tube. First it is assumed that the axial diffusion term may be neglected; a good approximation when the ratio q/xe is large enough and the flow rate is sufficiently fast that this term is negligible compared with the much greater effect of convection in the axial direction. The radial diffusion term may also be ignored providing the tube radius is large enough that the cylindricity of the system may be neglected and therefore the tube may be approximated to a channel setup. Finally, the parabolic flow is approximated to a linear profile according to the Le´veˆque approximation [23,25] vx
2v0 y ; q
319
ð4Þ
where y = q r, and this holds when the diffusion layer is very thin compared with the tube radius. The effects of the radial and axial diffusion in the tubular electrode have been extensively studied and reported in a previous publication [4]. It is found that for all but the fastest flow rates it is important to consider the radial ð1r oro Þ term, and that for much slower flow rates and smaller q/xe ratios the axial term becomes increasingly important.
The mass transport equation given above (Eq. (1)) has oc/ot set to 0, as only the steady-state solution is required (at both electrodes). Two species are involved; at the upstream electrode A is oxidised to B, and downstream B is reduced back to A. The normalised equations [4] describing these species are oa o2 a 1 oa PsY oa ¼ ð2p Y Þ ¼ 0; oT oY 2 ðp Y Þ oY 2p oX 2 ob DB o b 1 ob ¼ 2 oT DA oY ðp Y Þ oY PsY ob ð2p Y Þ ¼0 2p oX
ð5Þ
ð6Þ
2
o c with the axial diffusion ðox 2 Þ term omitted. The normalised variables are defined by
p ¼ q=xe1 ; 4V f x2e1 ; pDA q3 X ¼ x=xe1 ; Y ¼ y=xe1 ; Ps ¼
a ¼ ½A=½Abulk ; b ¼ ½B=½Abulk ; tDA T ¼ 2 . xe In order to solve the set of equations boundary conditions must be applied for the upstream end of the cell, the tube centre and the electrode surfaces, and these are given in Table 1. It is assumed that the generator electrode is held at a potential such that all A reaching its surface is oxidised and hence a = 0; likewise at the collector electrode all B reaching the surface is reduced and b = 0. Upstream of the generator [A] = [A]bulk and [B] = 0, and there is a no flux condition at the tube centre and at insulating surfaces. The equations above are discretised using the backwards implicit (BI) method, to allow their simultaneous numerical solution ai;j1 ðky 1 þ ky 2 Þ þ ai;j ð2ky 1 kx Þ þ ai;jþ1 ðky 1 ky 2 Þ ¼ kx ai1;j ; DB DB ðky þ ky 2 Þ þ bi;j 2 ky kx bi;j1 DA 1 DA 1 DB þ bi;jþ1 ðky ky 2 Þ ¼ kx bi1;j ; DA 1 where
ð7Þ
ð8Þ
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Table 1 Boundary conditions Species
Tube centre oa/oY = 0 ob/oY = 0
A B
ky 1 ¼
Electrode 1, y = 0 a=0 oa ob DA oY ¼ DB oY
1 ; DY 2
ð9Þ
1 ; 2ðp Y ÞDY PsY ð2p Y Þ kx ¼ . 2pDX
ð10Þ
ky 2 ¼
ð11Þ
A uniform finite difference grid is constructed over the tube length (Fig. 2) and an extended form of the Thomas algorithm [26] is used to solve the matrix system obtained, to give steady-state concentration profiles for species A and B. The limiting currents are evaluated using the concentration profiles of a and b according to the equations ilim 1 ¼ 2pqFnDA ½Abulk
Nx1 X ai;1 ai;0 dx; dy i¼1
ð12Þ
ilim 2 ¼ 2pqFnDA ½Abulk
Nx X bi;1 bi;0 dx; dy i¼Nx2
ð13Þ
Gap, y = 0
Electrode 2, y = 0
Inlet, x = 0
oa/oY = 0 ob/oY = 0
oa DA oY
a=1 b=0
¼
ob DB oY
b=0
2.2. Model with axial diffusion included The program which simulates the limiting currents with axial diffusion taken into account uses a straightforward variation of the method described in a previous publication [4]. The mass transport equation now includes the axial diffusion term (Eq. (1)), and is discretised and solved via the alternating direction implicit (ADI) method using a pseudo-time variable. As in the previous publication the transformation nðX Þ ¼ arctanðaX Þ is used to allow the same grid to be used for any electrode size, and as before, a is set to a value of 0.8. Fig. 3 depicts the grid used for the program including axial diffusion, with Nx1, Nx2, Nx3 and Nx4 given by hp i arctanðaX a Þ =dn; Nx1 ¼ ð14Þ 2 ð15Þ Nx2 ¼ Nx1 þ ½arctanðaX b Þ arctanðaX a Þ=dn; Nx3 ¼ Nx2 þ ½arctanðaX c Þ arctanðaX b Þ=dn; Nx4 ¼ Nx3 þ ½arctan ðaX d Þ arctanðaX c Þ=dn;
where Nx1 and Nx2 are defined in Fig. 2.
X X1
X2
X3
j=Ny
R
centre of tube
p
Flow direction
Y
j=0 i=0
wall of tube i=Nx1
i=Nx2
i=Nx
1 Fig. 2. The uniform grid used to simulate the double tube electrode without axial diffusion.
Fig. 3. The uniform grid used to simulate the double tube electrode including the axial diffusion term.
ð16Þ ð17Þ
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
where
xe1 þ gap þ xe2 Xa ¼ ; ð18Þ 2xe1 xe þ gap þ xe2 þ 1; ð19Þ Xb ¼ 1 2xe1 xe þ gap þ xe2 gap þ1þ ; ð20Þ Xc ¼ 1 xe 2xe1 1 xe þ gap þ xe2 gap þ xe2 Xd ¼ 1 þ1þ . ð21Þ 2xe1 xe1 The transformed mass transport equation is discretised, resulting in the following finite difference equations for solution: kþ1
kþ1
kþ1
2 2 ai1;j ðkn1 þ kn2 þ kn3 Þ þ ai;j 2 ð1 þ 2kn1 kn2 kn3 Þ þ aiþ1;j ðkn1 Þ
¼ aki;j1 ðky 1 þ ky 2 Þ þ aki;j ð1 2ky 1 Þ þ aki;jþ1 ðky 1 ky 2 Þ; ð22Þ akþ1 i;j1 ðky 1 ¼
kþ1 ky 2 Þ þ akþ1 i;j ð1 þ 2ky 1 Þ þ ai;jþ1 ðky 1
kþ12 ai1;j ðkn1
kþ1 kn2 kn3 Þ þ ai;j 2 ð1 2kn1
kþ12 ðkn1 Þ þ aiþ1;j
þ ky 2 Þ
DT =2 ; DY 2 DT =2 ; ky 2 ¼ 2ðp Y ÞDY DT =2 2 4 kn1 ¼ ða cos nÞ; Dn2 DT =2 ða2 sin 2ncos2 nÞ; kn2 ¼ Dn
ð30Þ
with the boundary conditions equivalent to those used in the non-axial program. Good convergence is achieved for grid sizes of Nx = Ny = 1000, which are used throughout. For the case including axial diffusion the program is terminated when the steady-state is reached, or when the difference between consecutive current values at both electrodes is small enough ikþ1 ik < 1 107 . ik
ð31Þ
Programs written in Delphi Pascale and run on a PC with 1 GB of RAM typically took less than 1 min per simulation for the non-axial program; the time taken for simulations including axial diffusion increased as the flow rate used was decreased, as a longer time is needed to reach steady-state.
þ kn2 þ kn3 Þ ð23Þ
for species A, implicit in n and y, respectively, and DB kþ12 bi1;j ðkn1 þ kn2 Þ þ kn3 DA DB DB kþ12 kþ12 þ bi;j 1 þ ð2kn1 kn2 Þ kn3 þ biþ1;j kn1 DA DA DB DB ¼ bki;j1 ðky þ ky 2 Þ þ bki;j 1 2 ky DA 1 DA 1 DB þ bki;jþ1 ðky ky 2 Þ ; ð24Þ DA 1 DB DB ðky 1 ky 2 Þ þ bkþ1 1þ2 ky bkþ1 i;j1 i;j DA DA 1 DB þ bkþ1 ðky 1 þ ky 2 Þ i;jþ1 DA DB kþ12 ¼ bi1;j ðkn1 kn2 Þ kn3 DA 1 DB kþ þ bi;j 2 1 þ ð2kn1 þ kn2 Þ þ kn3 DA 1 DB kþ2 þ biþ1;j kn1 ð25Þ DA for species B, again implicit in n and y, respectively. The variables in these equations are given by ky 1 ¼
DT =2 PsY ð2p Y Þacos2 n kn3 ¼ Dn 2p
321
ð26Þ ð27Þ ð28Þ ð29Þ
3. Results of simulations without axial diffusion effects 3.1. Dimensionless current For convenience the dimensionless forms of the two limiting currents are used, denoted by Nu1 and Nu2 respectively: ilim1 ; 2pqFDA ½Abulk ilim2 Nu2 ¼ . 2pqFDA ½Abulk
Nu1 ¼
ð32Þ ð33Þ
The dependence of Nu1 is as described previously for the single tubular electrode [4] where the only important factors are p and Ps. At the downstream electrode the current passed is due solely to the species produced upstream, so it follows that Nu2 will also depend on the length of the gap between the two electrodes. For a given flow rate, this length will determine the amount of diffusion towards the tube centre where convection is greater and will carry the species away from the vicinity of the electrode faster; therefore the longer the gap, the lower the current at the downstream electrode. Fig. 4 shows how Nu1 and Nu2 vary with the dimensionless parameters p and Ps, for fixed electrode lengths and tube radius. It is seen that the value of p has a much larger effect when p is small, but for larger q : xe1 ratios little change is seen in the Nu vs. Ps curve. The curve shape observed at low values of p is more marked at the downstream electrode. This is the region corresponding to the transition between the thin diffusion layer and thin-layer limits, as discussed previously for a single electrode [17]. The surface showing how the collection efficiency, N, varies with p and Ps will be discussed in Section 3.3.
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and with reference to Fig. 1, the variables xn correspond as
1
x0 ¼ 0; x1 ¼ xe1 ;
0
x2 ¼ xe1 þ gap; x3 ¼ xe1 þ gap þ xe2 .
log Nu1
2
Cook [28] assumed a uniform flux condition at the upstream electrode, and that the downstream electrode was potentiostated such that [B] = 0 at the electrode surface; these conditions result in a different expression for the collection efficiency 1 þ k2 k2 N ¼ 1 ð1 þ k2 þ k3 ÞF þ ðk2 þ k3 ÞF k3 k3 3 2 i 1 1 32 k33 h þ ð1 þ k2 Þ3 k32 ; ð36Þ 2p
-1
-2
5
0.4
6
4
0.8
p
3
1.2
2
1.6
a
1
Ps
log
2.0
2
where x2 x1 ; k2 ¼ x1 x0 x3 x2 k3 ¼ x1 x0
log Nu2
1
0
5
0.4
6
4
0.8
p
3
1.2
b
2
1.6 2.0
log
1
Ps
Fig. 4. The dependence of Nu1 and Nu2 on the parameters p and Ps; X2 = 2 and X3 = 3.
ð39Þ
These theories then depend only on the ratios of gap : xe1 and xe2 : xe1 , and at high flow rates are in very good agreement with the numerical results obtained using the program which neglects axial diffusion, as shown in Fig. 5. 3.3. Collection efficiency: below the Le´veˆque limit The full, normalised, steady-state equation for the mass transport in a tube is given by
3.2. Analytical results Various analytical theories have been developed for the Le´veˆque thin diffusion layer regime. The equation found by Matsuda [27] and Braun [18] for the collection efficiency, N, at both channel and tubular double electrodes, results from the constant concentration boundary condition at both upstream and downstream electrodes: " 2 n2 x3 x2 3 x1 x0 x3 x2 N¼ G þG n1 x1 x0 x2 x1 x2 x1 23 # x3 x0 x1 x0 x3 x2 G ; ð34Þ x1 x0 x3 x0 x2 x1 where 1
ð38Þ
and the function F(h) is given by Albery [29] ! 1 3 1 1 3 32 ð1 þ h3 Þ 3 1 1 2h 1 ln þ tan F ðhÞ ¼ þ . 1 2p 4 4p 1þh 2 3
-1
-2
ð37Þ
1
32 1þh 3 2h3 1 tan1 ln GðhÞ ¼ 1 3 þ 1 2p 4p ð1 þ h3 Þ 32
! þ
1 4
ð35Þ
0¼
o2 c 1 oc o2 c PsY ð2p Y Þ oc þ ; 2 ðp Y Þ oY oX 2 2p oX oY
ð40Þ
where the variables are defined in Section 2.1. From this equation it can be seen that the concentration is dependent on four parameters, c = c(X, Y, p, Ps). With reference to Fig. 2, the expression for the collection efficiency can be written R X 3 oc X
oY y¼0
dX
0
oY y¼0
dX
N ¼ R X21 oc
x
ð41Þ
; x þgap
x þgapþx
where X 1 ¼ xee1 ; X 2 ¼ e1xe and X 3 ¼ e1 xe e2 . Since 1 1 1 X1 is always equal to 1, it is now clear that the collection efficiency is dependent on the four dimensionless parameters, X2, X3, p and Ps. While the upstream electrode current depends solely on p and Ps, the downstream
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
323
1.0
0.40
program neglecting axial diffusion Matsuda Cook
0.35
N 0.5
N 0.30
0.25 0.0 0.4 0.8
0.20
1.2
p
1.0
1.5
2.0
X
2.5
3.0
1.6
a
2
1
2.0
4
3
5
6
s
log P
2
Fig. 5. Comparison of the Matsuda and Cook theories with results obtained using the program neglecting axial diffusion, in the high flow rate limit.
one, and hence the collection efficiency, is also a function of X2 and X3. Two working surfaces may now be obtained for the collection efficiency to illustrate the effect of each of these parameters; one for constant length dimensions but varying p and Ps, and one with p and Ps constant but changing the values of X2 and X3. These surfaces are shown in Fig. 6(a) and (b), respectively. From Fig. 6(a) it can be seen that at the lowest flow rates, and for small values of p, a thin-layer regime is reached at both electrodes. All of species A in the vicinity of electrode 1 is oxidised, and subsequently all of B is reduced at electrode 2; hence in this regime the collection efficiency achieves its maximum value of 1. As the flow rate, and hence Ps, is increased, both electrode responses undergo a transition between the thin layer and thin diffusion layer limits. It is seen that the Nu1 curve in Fig. 4(a) is always increasing with Ps, as increased convection compensates for the lack of time for all of A in the electrode vicinity to diffuse to the surface and be oxidised. However, the Nu2 curve, shown in Fig. 4(b), exhibits a region where the gradient is shallower than that seen in the Le´veˆque limit. Here, much of species B produced upstream diffuses to the centre of the tube where convection is fastest, and is carried away before reaching the downstream electrode surface. Thus the collection efficiency decreases from the thinlayer limit until both electrodes are in the Le´veˆque regime and the collection efficiency levels off to a value that agrees with the Matsuda analysis. Fig. 7 shows concentration profiles obtained from the program neglecting axial diffusion, for three different flow rates. Fig. 7(a) shows the thin diffusion layer limit found at the highest flow rates, (b) illustrates the intermediate case where the diffusion layer thickness is
0.6
N
0.4
0.2 2.0 2.5 3.0 X 3.5 2
b
4.0
7 4.5
6 5
8
9
10
X3
Fig. 6. Working surfaces for the dependence of N on (a) the parameters p and Ps; X2 = 2 and X3 = 3 and (b) the parameters X2 and X3; p = 1 and log Ps=3.
becoming comparable to the tube radius, and (c) shows the thin layer limit, where first all of species A is oxidised at the upstream electrode and then all of B is reduced so that at the second electrode a = 1 over the whole of the tube radius. Fig. 6(b) shows the working surface for N as a function of X2 and X3, when p and Ps are held constant (p = 1, log Ps = 3) with the results obtained using the program that neglects the axial diffusion term. For a constant value of X2, the collection efficiency increases as X3 increases; a larger downstream electrode means a higher current and hence higher value of N. As X2 is increased, with X3 held constant, the collection efficiency is decreased, due to the increased length and therefore time available for B to diffuse away from the tube walls before reaching the vicinity of the downstream electrode, and also to the fact that in order to keep X3 constant, the size of the downstream electrode must be reduced as the gap size increases.
324
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326 1.5 1.0
log Nu
0.5 0.0 -0.5
log Nu1 without axial diffusion term log Nu2 without axial diffusion term log Nu1 with axial diffusion term log Nu2 with axial diffusion term
-1.0 -1.5 -1
0
1
2
3
4
log Ps Fig. 8. The effect of axial diffusion on the limiting current at each electrode; X2 = 2, X3 = 3, p = 1.
Fig. 7. Concentration profiles for species A when axial diffusion is neglected for flow rates of: (a) 1 · 101, (b) 1 · 104, and (c) 1 · 108 cm3 s1; X2 = 2, X3 = 3, p = 0.5.
4. Axial diffusion effects It is clear that axial diffusion becomes more important as the flow rate decreases, as its effect becomes greater compared with convection in the axial direction. The
analysis in Section 3.3 shows that when axial diffusion is considered, the collection efficiency is still a function of the four dimensionless parameters X2, X3, p and Ps. In order to rationalise the effect of axial diffusion it is informative to examine the individual current responses, shown in Fig. 8. As discussed above, the generator current depends only on p and Ps, while the collector current also depends on X2 and X3. Therefore with p, X2 and X3 held constant, single curves may be obtained for Nu1 and Nu2 vs. Ps. At high Ps the currents are enhanced by the axial diffusion term, due to the increase in mass transport to the electrode. However the greater effect is observed at the downstream electrode where the current depends solely on the species produced upstream, as axial diffusion means that more of species B will reach this electrode and be reduced before diffusing too far towards the tube centre where it is quickly removed by convection. Thus, over a wide range of Ps values, the collection efficiency is higher than predicted when axial diffusion is neglected. Fig. 9 shows curves for the collection efficiency as a function of Ps, where p, X2 and X3 are all kept constant, resulting from the programs both with and without consideration of the axial diffusion term. The shape of this curve without axial diffusion has been discussed in Section 3.3. At the higher values of Ps the collection efficiency is enhanced by axial diffusion, due to the greater relative effect of the increase in mass transport found at the downstream electrode compared with that upstream. At lower Ps where the effect of axial diffusion starts to become more dominant compared with the convection term, the curves cross and the collection efficiency is lower than predicted in the absence of axial diffusion. This is due to some of the electroactive species being carried away from the electrode ends by diffusion, and N therefore does not reach the maximum value of 1 predicted for the thin-layer limit until much slower flow
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
rates are used, and the time to reach steady-state is greatly increased. This is illustrated in Fig. 10 which shows the concentration profile of species A resulting from simulations including the effect of axial diffusion (p = 1, X2 = 2, X3 = 3, Ps = 1.1). The normalised concentration at the downstream electrode does not reach 1 over the whole radius as was seen in the profile in Fig. 7(c); if axial diffusion dominates, much of species B produced upstream does not travel downstream to the collector electrode, and the collection efficiency is greatly diminished. Inspection of Fig. 8 shows that the axial diffusion term is relatively unimportant in terms of the limiting current for high Ps at the upstream generator electrode. However, from Fig. 9 it can be seen that the axial diffusion term, because of its greater effect on the current at the second electrode, has an important influence on the collection efficiency over a much wider range of p and Ps than that observed on the current at a single electrode.
1.1 1.0 0.9 0.8 0.7
N 0.6 0.5
325
5. Conclusions In the high flow rate Le´veˆque regime the results obtained through numerical simulation agree well with the analytical theories of Matsuda/Braun [27,18] and Cook/Albery [28,29], and if xe1 ¼ xe2 it is solely the ratio of gap/xe that influences the collection efficiency. For a broader range of conditions the collection efficiency has been shown to be dependent on four dimensionless parameters, X2, X3, p and Ps. In the absence of axial diffusion N ranges from the thin-layer limit where N is equal to 1, through a period where much of B produced at the generator electrode has diffused away from the tube wall before it reaches the collector electrode and the collection efficiency decreases, to the Matsuda/Cook limit in the Le´veˆque regime. However, at lower values of Ps axial diffusion must be taken into account. When this term is dominant compared to convection, the collection efficiency cannot reach the limit of 1 predicted when this term is not considered until the flow rate is much slower. Much of species B formed at the upstream electrode will no longer reach the vicinity of the downstream electrode and will not be converted back to A. Because of its greater effect at the second, downstream electrode, the axial diffusion term has an important effect on the collection efficiency over a wide range of Ps values, so that when working with collection efficiencies this term must be included at much higher flow rates than needed when observing only the limiting currents at a single tubular electrode.
0.4 0.3
N without axial diffusion term N with axial diffusion term
0.2 -4
-2
0
Acknowledgement
2
4
6
M.T thanks the EPSRC for financial support.
log Ps Fig. 9. The effect of axial diffusion on the collection efficiency; X2 = 2, X3 = 3, p = 1.
[A]/[A] bulk
1.0
0.5
0.0 -1.5 -1.0
1.0 0.8 -0.5
0.6 0.0
0.4 0.5
0.2
1.0 1.5
Y
0.0
Fig. 10. Concentration profile for species A when axial diffusion is included; p = 1, X2 = 2, X3 = 3, log Ps = 1.1.
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[15] B. Fosset, C.A. Amatore, J.E. Bartelt, A.C. Michael, R.M. Wightman, Anal. Chem. 63 (1991) 306. [16] R.G. Compton, B.A. Coles, A.C. Fisher, J. Phys. Chem. 98 (1994) 2441. [17] M. Thompson, O.V. Klymenko, R.G. Compton, J. Electroanal. Chem. 576 (2005) 333. [18] R. Braun, J. Electroanal. Chem. 19 (1968) 23. [19] D.R. Rosseinsky, P.M.S. Monk, J. Chem. Soc., Faraday Trans. 86 (1990) 3597. [20] K. Endo, Y. Katayama, T. Miura, Electrochim. Acta 50 (2005) 2181. [21] P.J. Daly, D.J. Page, R.G. Compton, Anal. Chem. 55 (1983) 1191.
[22] N.G. Karousos, R. Carrington, S.J. Wilkins, C. Livingstone, J. Davis, Electrochem. Commun. 7 (2005) 500 (in press). [23] V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. [24] A.J. Bard, L.R. Faulkner, Electrochemical methods, fundamentals and applications, John Wiley and Sons, 2001. [25] M.A. Le´veˆque, Ann. Mines, Mem., Ser. 12 13 (1928) 201. [26] R.G. Compton, M.B.G. Pilkington, G.M. Stearn, J. Chem. Soc. Faraday Trans. 1 84 (1988) 2155. [27] H. Matsuda, J. Electroanal. Chem. 16 (1968) 153. [28] R.M. Cook, DPhil Thesis, Oxford, 1973. [29] W.J. Albery, S. Bruckenstein, Trans. Faraday Soc. 62 (1966) 1920.
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 583 (2005) 318–326 www.elsevier.com/locate/jelechem
Theory of collection efficiencies in the double tubular hydrodynamic electrode Mary Thompson, Richard G. Compton
*
Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, Oxfordshire OX1 3QZ, United Kingdom Received 6 April 2005; received in revised form 29 May 2005; accepted 13 June 2005 Available online 8 August 2005
Abstract The theory of collection efficiencies at a double tubular electrode is explored through numerical simulation, including an analysis of the effects of axial diffusion. The theory is compared with analytical results in the high flow rate limit, using the Levich equation and the Matsuda theory for collection efficiency. For lower flow rates the dependence of the collection efficiency on absolute and relative lengths and radii of the tubular electrodes is investigated. 2005 Elsevier B.V. All rights reserved. Keywords: Double tubular electrode; Collection efficiency; Computational electrochemistry
1. Introduction This work deals with the theory of collection efficiencies obtained from a double tubular electrode, where two electrodes are embedded, in series, flush with the wall of an insulating tube through which a solution of electroactive species can flow (Fig. 1). The tubular electrode lends itself to numerical analysis due to the welldefined flow pattern within it. Much theoretical and experimental work at single tube electrodes has been reported previously, building on work which gave analytical results for the Le´veˆque (thin diffusion layer) limit [1–11]. Double electrodes have been studied extensively, due to their usefulness as probes of the rate of some homogeneous reaction which may occur ÔbetweenÕ the two electrodes [12–14]. The double channel electrode is a close analogue of the double tubular electrode and much *
Corresponding author. Tel.: +44 0 1865 275413; fax: +44 0 1865 275410. E-mail address: [email protected] (R.G. Compton). 0022-0728/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2005.06.018
experimental and theoretical work has been reported for this arrangement [12,14–17]. Similarly, the rotating ring disc electrode (RRDE) has also been extensively researched [18–21]. Work has been undertaken on the double tube electrode; however the theory reported is confined to analytical solutions in the high flow rate Le´veˆque limit, and experimental data is limited, most studies to date having used the channel electrode setup. An important practical advantage of the double tube electrode over the channel is the relative facility of its fabrication, even in microdimensions, where the use of microtubes relative to microdiscs remains relatively unexplored. This arrangement can be achieved by simply layering metal sheets or foil between layers of insulating material, and punching or drilling a hole through all the layers, as recently reported by Davis et al. [22] for a non-flowing system. 1.1. Tubular electrode theory The mass transport equation for convection and diffusion in cylindrical coordinates is [23,24]
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
2. Modelling the double tubular electrode
Flow direction
r
y
xe1
2.1. Model with axial diffusion neglected
xe2
gap
x Fig. 1. A schematic of the double tubular electrode, showing coordinate system and dimension labels.
2 oc o c 1 oc o2 c oc ¼D þ þ vx ; 2 2 ot or r or ox ox
ð1Þ
where c is the concentration of electroactive species, D is its diffusion coefficient, r and x are the coordinates in the radial and axial directions, respectively, and vx is the convective flow velocity, defined by the equation r2 vx ¼ v0 1 2 ð2Þ q for laminar flow in a tube where v0 is the axial velocity at the tube centre and q is the tube radius. Conditions for achieving laminar flow have been discussed previously [4]. Levich made some simplifying assumptions to the mass transport equation, leading to the equation for the limiting current in a tubular electrode [4–6,23,24] 2
2
1
ilim ¼ 5.24 105 nF ½Abulk x3e D3A V 3f ;
ð3Þ
where F is the Faraday constant, [A]bulk and DA are the bulk concentration and diffusion coefficient of the electroactive species, respectively, and Vf is the volume flow rate of solution through the tube. First it is assumed that the axial diffusion term may be neglected; a good approximation when the ratio q/xe is large enough and the flow rate is sufficiently fast that this term is negligible compared with the much greater effect of convection in the axial direction. The radial diffusion term may also be ignored providing the tube radius is large enough that the cylindricity of the system may be neglected and therefore the tube may be approximated to a channel setup. Finally, the parabolic flow is approximated to a linear profile according to the Le´veˆque approximation [23,25] vx
2v0 y ; q
319
ð4Þ
where y = q r, and this holds when the diffusion layer is very thin compared with the tube radius. The effects of the radial and axial diffusion in the tubular electrode have been extensively studied and reported in a previous publication [4]. It is found that for all but the fastest flow rates it is important to consider the radial ð1r oro Þ term, and that for much slower flow rates and smaller q/xe ratios the axial term becomes increasingly important.
The mass transport equation given above (Eq. (1)) has oc/ot set to 0, as only the steady-state solution is required (at both electrodes). Two species are involved; at the upstream electrode A is oxidised to B, and downstream B is reduced back to A. The normalised equations [4] describing these species are oa o2 a 1 oa PsY oa ¼ ð2p Y Þ ¼ 0; oT oY 2 ðp Y Þ oY 2p oX 2 ob DB o b 1 ob ¼ 2 oT DA oY ðp Y Þ oY PsY ob ð2p Y Þ ¼0 2p oX
ð5Þ
ð6Þ
2
o c with the axial diffusion ðox 2 Þ term omitted. The normalised variables are defined by
p ¼ q=xe1 ; 4V f x2e1 ; pDA q3 X ¼ x=xe1 ; Y ¼ y=xe1 ; Ps ¼
a ¼ ½A=½Abulk ; b ¼ ½B=½Abulk ; tDA T ¼ 2 . xe In order to solve the set of equations boundary conditions must be applied for the upstream end of the cell, the tube centre and the electrode surfaces, and these are given in Table 1. It is assumed that the generator electrode is held at a potential such that all A reaching its surface is oxidised and hence a = 0; likewise at the collector electrode all B reaching the surface is reduced and b = 0. Upstream of the generator [A] = [A]bulk and [B] = 0, and there is a no flux condition at the tube centre and at insulating surfaces. The equations above are discretised using the backwards implicit (BI) method, to allow their simultaneous numerical solution ai;j1 ðky 1 þ ky 2 Þ þ ai;j ð2ky 1 kx Þ þ ai;jþ1 ðky 1 ky 2 Þ ¼ kx ai1;j ; DB DB ðky þ ky 2 Þ þ bi;j 2 ky kx bi;j1 DA 1 DA 1 DB þ bi;jþ1 ðky ky 2 Þ ¼ kx bi1;j ; DA 1 where
ð7Þ
ð8Þ
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Table 1 Boundary conditions Species
Tube centre oa/oY = 0 ob/oY = 0
A B
ky 1 ¼
Electrode 1, y = 0 a=0 oa ob DA oY ¼ DB oY
1 ; DY 2
ð9Þ
1 ; 2ðp Y ÞDY PsY ð2p Y Þ kx ¼ . 2pDX
ð10Þ
ky 2 ¼
ð11Þ
A uniform finite difference grid is constructed over the tube length (Fig. 2) and an extended form of the Thomas algorithm [26] is used to solve the matrix system obtained, to give steady-state concentration profiles for species A and B. The limiting currents are evaluated using the concentration profiles of a and b according to the equations ilim 1 ¼ 2pqFnDA ½Abulk
Nx1 X ai;1 ai;0 dx; dy i¼1
ð12Þ
ilim 2 ¼ 2pqFnDA ½Abulk
Nx X bi;1 bi;0 dx; dy i¼Nx2
ð13Þ
Gap, y = 0
Electrode 2, y = 0
Inlet, x = 0
oa/oY = 0 ob/oY = 0
oa DA oY
a=1 b=0
¼
ob DB oY
b=0
2.2. Model with axial diffusion included The program which simulates the limiting currents with axial diffusion taken into account uses a straightforward variation of the method described in a previous publication [4]. The mass transport equation now includes the axial diffusion term (Eq. (1)), and is discretised and solved via the alternating direction implicit (ADI) method using a pseudo-time variable. As in the previous publication the transformation nðX Þ ¼ arctanðaX Þ is used to allow the same grid to be used for any electrode size, and as before, a is set to a value of 0.8. Fig. 3 depicts the grid used for the program including axial diffusion, with Nx1, Nx2, Nx3 and Nx4 given by hp i arctanðaX a Þ =dn; Nx1 ¼ ð14Þ 2 ð15Þ Nx2 ¼ Nx1 þ ½arctanðaX b Þ arctanðaX a Þ=dn; Nx3 ¼ Nx2 þ ½arctanðaX c Þ arctanðaX b Þ=dn; Nx4 ¼ Nx3 þ ½arctan ðaX d Þ arctanðaX c Þ=dn;
where Nx1 and Nx2 are defined in Fig. 2.
X X1
X2
X3
j=Ny
R
centre of tube
p
Flow direction
Y
j=0 i=0
wall of tube i=Nx1
i=Nx2
i=Nx
1 Fig. 2. The uniform grid used to simulate the double tube electrode without axial diffusion.
Fig. 3. The uniform grid used to simulate the double tube electrode including the axial diffusion term.
ð16Þ ð17Þ
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
where
xe1 þ gap þ xe2 Xa ¼ ; ð18Þ 2xe1 xe þ gap þ xe2 þ 1; ð19Þ Xb ¼ 1 2xe1 xe þ gap þ xe2 gap þ1þ ; ð20Þ Xc ¼ 1 xe 2xe1 1 xe þ gap þ xe2 gap þ xe2 Xd ¼ 1 þ1þ . ð21Þ 2xe1 xe1 The transformed mass transport equation is discretised, resulting in the following finite difference equations for solution: kþ1
kþ1
kþ1
2 2 ai1;j ðkn1 þ kn2 þ kn3 Þ þ ai;j 2 ð1 þ 2kn1 kn2 kn3 Þ þ aiþ1;j ðkn1 Þ
¼ aki;j1 ðky 1 þ ky 2 Þ þ aki;j ð1 2ky 1 Þ þ aki;jþ1 ðky 1 ky 2 Þ; ð22Þ akþ1 i;j1 ðky 1 ¼
kþ1 ky 2 Þ þ akþ1 i;j ð1 þ 2ky 1 Þ þ ai;jþ1 ðky 1
kþ12 ai1;j ðkn1
kþ1 kn2 kn3 Þ þ ai;j 2 ð1 2kn1
kþ12 ðkn1 Þ þ aiþ1;j
þ ky 2 Þ
DT =2 ; DY 2 DT =2 ; ky 2 ¼ 2ðp Y ÞDY DT =2 2 4 kn1 ¼ ða cos nÞ; Dn2 DT =2 ða2 sin 2ncos2 nÞ; kn2 ¼ Dn
ð30Þ
with the boundary conditions equivalent to those used in the non-axial program. Good convergence is achieved for grid sizes of Nx = Ny = 1000, which are used throughout. For the case including axial diffusion the program is terminated when the steady-state is reached, or when the difference between consecutive current values at both electrodes is small enough ikþ1 ik < 1 107 . ik
ð31Þ
Programs written in Delphi Pascale and run on a PC with 1 GB of RAM typically took less than 1 min per simulation for the non-axial program; the time taken for simulations including axial diffusion increased as the flow rate used was decreased, as a longer time is needed to reach steady-state.
þ kn2 þ kn3 Þ ð23Þ
for species A, implicit in n and y, respectively, and DB kþ12 bi1;j ðkn1 þ kn2 Þ þ kn3 DA DB DB kþ12 kþ12 þ bi;j 1 þ ð2kn1 kn2 Þ kn3 þ biþ1;j kn1 DA DA DB DB ¼ bki;j1 ðky þ ky 2 Þ þ bki;j 1 2 ky DA 1 DA 1 DB þ bki;jþ1 ðky ky 2 Þ ; ð24Þ DA 1 DB DB ðky 1 ky 2 Þ þ bkþ1 1þ2 ky bkþ1 i;j1 i;j DA DA 1 DB þ bkþ1 ðky 1 þ ky 2 Þ i;jþ1 DA DB kþ12 ¼ bi1;j ðkn1 kn2 Þ kn3 DA 1 DB kþ þ bi;j 2 1 þ ð2kn1 þ kn2 Þ þ kn3 DA 1 DB kþ2 þ biþ1;j kn1 ð25Þ DA for species B, again implicit in n and y, respectively. The variables in these equations are given by ky 1 ¼
DT =2 PsY ð2p Y Þacos2 n kn3 ¼ Dn 2p
321
ð26Þ ð27Þ ð28Þ ð29Þ
3. Results of simulations without axial diffusion effects 3.1. Dimensionless current For convenience the dimensionless forms of the two limiting currents are used, denoted by Nu1 and Nu2 respectively: ilim1 ; 2pqFDA ½Abulk ilim2 Nu2 ¼ . 2pqFDA ½Abulk
Nu1 ¼
ð32Þ ð33Þ
The dependence of Nu1 is as described previously for the single tubular electrode [4] where the only important factors are p and Ps. At the downstream electrode the current passed is due solely to the species produced upstream, so it follows that Nu2 will also depend on the length of the gap between the two electrodes. For a given flow rate, this length will determine the amount of diffusion towards the tube centre where convection is greater and will carry the species away from the vicinity of the electrode faster; therefore the longer the gap, the lower the current at the downstream electrode. Fig. 4 shows how Nu1 and Nu2 vary with the dimensionless parameters p and Ps, for fixed electrode lengths and tube radius. It is seen that the value of p has a much larger effect when p is small, but for larger q : xe1 ratios little change is seen in the Nu vs. Ps curve. The curve shape observed at low values of p is more marked at the downstream electrode. This is the region corresponding to the transition between the thin diffusion layer and thin-layer limits, as discussed previously for a single electrode [17]. The surface showing how the collection efficiency, N, varies with p and Ps will be discussed in Section 3.3.
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and with reference to Fig. 1, the variables xn correspond as
1
x0 ¼ 0; x1 ¼ xe1 ;
0
x2 ¼ xe1 þ gap; x3 ¼ xe1 þ gap þ xe2 .
log Nu1
2
Cook [28] assumed a uniform flux condition at the upstream electrode, and that the downstream electrode was potentiostated such that [B] = 0 at the electrode surface; these conditions result in a different expression for the collection efficiency 1 þ k2 k2 N ¼ 1 ð1 þ k2 þ k3 ÞF þ ðk2 þ k3 ÞF k3 k3 3 2 i 1 1 32 k33 h þ ð1 þ k2 Þ3 k32 ; ð36Þ 2p
-1
-2
5
0.4
6
4
0.8
p
3
1.2
2
1.6
a
1
Ps
log
2.0
2
where x2 x1 ; k2 ¼ x1 x0 x3 x2 k3 ¼ x1 x0
log Nu2
1
0
5
0.4
6
4
0.8
p
3
1.2
b
2
1.6 2.0
log
1
Ps
Fig. 4. The dependence of Nu1 and Nu2 on the parameters p and Ps; X2 = 2 and X3 = 3.
ð39Þ
These theories then depend only on the ratios of gap : xe1 and xe2 : xe1 , and at high flow rates are in very good agreement with the numerical results obtained using the program which neglects axial diffusion, as shown in Fig. 5. 3.3. Collection efficiency: below the Le´veˆque limit The full, normalised, steady-state equation for the mass transport in a tube is given by
3.2. Analytical results Various analytical theories have been developed for the Le´veˆque thin diffusion layer regime. The equation found by Matsuda [27] and Braun [18] for the collection efficiency, N, at both channel and tubular double electrodes, results from the constant concentration boundary condition at both upstream and downstream electrodes: " 2 n2 x3 x2 3 x1 x0 x3 x2 N¼ G þG n1 x1 x0 x2 x1 x2 x1 23 # x3 x0 x1 x0 x3 x2 G ; ð34Þ x1 x0 x3 x0 x2 x1 where 1
ð38Þ
and the function F(h) is given by Albery [29] ! 1 3 1 1 3 32 ð1 þ h3 Þ 3 1 1 2h 1 ln þ tan F ðhÞ ¼ þ . 1 2p 4 4p 1þh 2 3
-1
-2
ð37Þ
1
32 1þh 3 2h3 1 tan1 ln GðhÞ ¼ 1 3 þ 1 2p 4p ð1 þ h3 Þ 32
! þ
1 4
ð35Þ
0¼
o2 c 1 oc o2 c PsY ð2p Y Þ oc þ ; 2 ðp Y Þ oY oX 2 2p oX oY
ð40Þ
where the variables are defined in Section 2.1. From this equation it can be seen that the concentration is dependent on four parameters, c = c(X, Y, p, Ps). With reference to Fig. 2, the expression for the collection efficiency can be written R X 3 oc X
oY y¼0
dX
0
oY y¼0
dX
N ¼ R X21 oc
x
ð41Þ
; x þgap
x þgapþx
where X 1 ¼ xee1 ; X 2 ¼ e1xe and X 3 ¼ e1 xe e2 . Since 1 1 1 X1 is always equal to 1, it is now clear that the collection efficiency is dependent on the four dimensionless parameters, X2, X3, p and Ps. While the upstream electrode current depends solely on p and Ps, the downstream
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
323
1.0
0.40
program neglecting axial diffusion Matsuda Cook
0.35
N 0.5
N 0.30
0.25 0.0 0.4 0.8
0.20
1.2
p
1.0
1.5
2.0
X
2.5
3.0
1.6
a
2
1
2.0
4
3
5
6
s
log P
2
Fig. 5. Comparison of the Matsuda and Cook theories with results obtained using the program neglecting axial diffusion, in the high flow rate limit.
one, and hence the collection efficiency, is also a function of X2 and X3. Two working surfaces may now be obtained for the collection efficiency to illustrate the effect of each of these parameters; one for constant length dimensions but varying p and Ps, and one with p and Ps constant but changing the values of X2 and X3. These surfaces are shown in Fig. 6(a) and (b), respectively. From Fig. 6(a) it can be seen that at the lowest flow rates, and for small values of p, a thin-layer regime is reached at both electrodes. All of species A in the vicinity of electrode 1 is oxidised, and subsequently all of B is reduced at electrode 2; hence in this regime the collection efficiency achieves its maximum value of 1. As the flow rate, and hence Ps, is increased, both electrode responses undergo a transition between the thin layer and thin diffusion layer limits. It is seen that the Nu1 curve in Fig. 4(a) is always increasing with Ps, as increased convection compensates for the lack of time for all of A in the electrode vicinity to diffuse to the surface and be oxidised. However, the Nu2 curve, shown in Fig. 4(b), exhibits a region where the gradient is shallower than that seen in the Le´veˆque limit. Here, much of species B produced upstream diffuses to the centre of the tube where convection is fastest, and is carried away before reaching the downstream electrode surface. Thus the collection efficiency decreases from the thinlayer limit until both electrodes are in the Le´veˆque regime and the collection efficiency levels off to a value that agrees with the Matsuda analysis. Fig. 7 shows concentration profiles obtained from the program neglecting axial diffusion, for three different flow rates. Fig. 7(a) shows the thin diffusion layer limit found at the highest flow rates, (b) illustrates the intermediate case where the diffusion layer thickness is
0.6
N
0.4
0.2 2.0 2.5 3.0 X 3.5 2
b
4.0
7 4.5
6 5
8
9
10
X3
Fig. 6. Working surfaces for the dependence of N on (a) the parameters p and Ps; X2 = 2 and X3 = 3 and (b) the parameters X2 and X3; p = 1 and log Ps=3.
becoming comparable to the tube radius, and (c) shows the thin layer limit, where first all of species A is oxidised at the upstream electrode and then all of B is reduced so that at the second electrode a = 1 over the whole of the tube radius. Fig. 6(b) shows the working surface for N as a function of X2 and X3, when p and Ps are held constant (p = 1, log Ps = 3) with the results obtained using the program that neglects the axial diffusion term. For a constant value of X2, the collection efficiency increases as X3 increases; a larger downstream electrode means a higher current and hence higher value of N. As X2 is increased, with X3 held constant, the collection efficiency is decreased, due to the increased length and therefore time available for B to diffuse away from the tube walls before reaching the vicinity of the downstream electrode, and also to the fact that in order to keep X3 constant, the size of the downstream electrode must be reduced as the gap size increases.
324
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326 1.5 1.0
log Nu
0.5 0.0 -0.5
log Nu1 without axial diffusion term log Nu2 without axial diffusion term log Nu1 with axial diffusion term log Nu2 with axial diffusion term
-1.0 -1.5 -1
0
1
2
3
4
log Ps Fig. 8. The effect of axial diffusion on the limiting current at each electrode; X2 = 2, X3 = 3, p = 1.
Fig. 7. Concentration profiles for species A when axial diffusion is neglected for flow rates of: (a) 1 · 101, (b) 1 · 104, and (c) 1 · 108 cm3 s1; X2 = 2, X3 = 3, p = 0.5.
4. Axial diffusion effects It is clear that axial diffusion becomes more important as the flow rate decreases, as its effect becomes greater compared with convection in the axial direction. The
analysis in Section 3.3 shows that when axial diffusion is considered, the collection efficiency is still a function of the four dimensionless parameters X2, X3, p and Ps. In order to rationalise the effect of axial diffusion it is informative to examine the individual current responses, shown in Fig. 8. As discussed above, the generator current depends only on p and Ps, while the collector current also depends on X2 and X3. Therefore with p, X2 and X3 held constant, single curves may be obtained for Nu1 and Nu2 vs. Ps. At high Ps the currents are enhanced by the axial diffusion term, due to the increase in mass transport to the electrode. However the greater effect is observed at the downstream electrode where the current depends solely on the species produced upstream, as axial diffusion means that more of species B will reach this electrode and be reduced before diffusing too far towards the tube centre where it is quickly removed by convection. Thus, over a wide range of Ps values, the collection efficiency is higher than predicted when axial diffusion is neglected. Fig. 9 shows curves for the collection efficiency as a function of Ps, where p, X2 and X3 are all kept constant, resulting from the programs both with and without consideration of the axial diffusion term. The shape of this curve without axial diffusion has been discussed in Section 3.3. At the higher values of Ps the collection efficiency is enhanced by axial diffusion, due to the greater relative effect of the increase in mass transport found at the downstream electrode compared with that upstream. At lower Ps where the effect of axial diffusion starts to become more dominant compared with the convection term, the curves cross and the collection efficiency is lower than predicted in the absence of axial diffusion. This is due to some of the electroactive species being carried away from the electrode ends by diffusion, and N therefore does not reach the maximum value of 1 predicted for the thin-layer limit until much slower flow
M. Thompson, R.G. Compton / Journal of Electroanalytical Chemistry 583 (2005) 318–326
rates are used, and the time to reach steady-state is greatly increased. This is illustrated in Fig. 10 which shows the concentration profile of species A resulting from simulations including the effect of axial diffusion (p = 1, X2 = 2, X3 = 3, Ps = 1.1). The normalised concentration at the downstream electrode does not reach 1 over the whole radius as was seen in the profile in Fig. 7(c); if axial diffusion dominates, much of species B produced upstream does not travel downstream to the collector electrode, and the collection efficiency is greatly diminished. Inspection of Fig. 8 shows that the axial diffusion term is relatively unimportant in terms of the limiting current for high Ps at the upstream generator electrode. However, from Fig. 9 it can be seen that the axial diffusion term, because of its greater effect on the current at the second electrode, has an important influence on the collection efficiency over a much wider range of p and Ps than that observed on the current at a single electrode.
1.1 1.0 0.9 0.8 0.7
N 0.6 0.5
325
5. Conclusions In the high flow rate Le´veˆque regime the results obtained through numerical simulation agree well with the analytical theories of Matsuda/Braun [27,18] and Cook/Albery [28,29], and if xe1 ¼ xe2 it is solely the ratio of gap/xe that influences the collection efficiency. For a broader range of conditions the collection efficiency has been shown to be dependent on four dimensionless parameters, X2, X3, p and Ps. In the absence of axial diffusion N ranges from the thin-layer limit where N is equal to 1, through a period where much of B produced at the generator electrode has diffused away from the tube wall before it reaches the collector electrode and the collection efficiency decreases, to the Matsuda/Cook limit in the Le´veˆque regime. However, at lower values of Ps axial diffusion must be taken into account. When this term is dominant compared to convection, the collection efficiency cannot reach the limit of 1 predicted when this term is not considered until the flow rate is much slower. Much of species B formed at the upstream electrode will no longer reach the vicinity of the downstream electrode and will not be converted back to A. Because of its greater effect at the second, downstream electrode, the axial diffusion term has an important effect on the collection efficiency over a wide range of Ps values, so that when working with collection efficiencies this term must be included at much higher flow rates than needed when observing only the limiting currents at a single tubular electrode.
0.4 0.3
N without axial diffusion term N with axial diffusion term
0.2 -4
-2
0
Acknowledgement
2
4
6
M.T thanks the EPSRC for financial support.
log Ps Fig. 9. The effect of axial diffusion on the collection efficiency; X2 = 2, X3 = 3, p = 1.
[A]/[A] bulk
1.0
0.5
0.0 -1.5 -1.0
1.0 0.8 -0.5
0.6 0.0
0.4 0.5
0.2
1.0 1.5
Y
0.0
Fig. 10. Concentration profile for species A when axial diffusion is included; p = 1, X2 = 2, X3 = 3, log Ps = 1.1.
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