Mass transfer in a rotating cylinder cell—I. Laminar flow

Eloctrochimica

MASS

Acta,

1972,

Vol.

17,

TRANSFER

pp.

1121

to

1127.

Pergamon

Press.

Printed

in Northern

Iceland

IN A ROTATING CYLINDER LAMINAR FLOW*

CELL-I.

D. FL GABE and D. J. ROBINSON~ Department of Metallurgy, University of Sheffield, Sheflield, England Abstract-The relationship between mass transfer and rotation rate for a concentric cylinder cell, in which the inner cylinder rotates and the outer cylinder remains stationary, has been derived for a simple model assuming linear velocity gradients near the rotating cylinder surface. A previous treatment by Kimla and Str&felda has been shown to be in error, but when corrected it takes essentially the same form as that derived in this paper. relation entre le transfert de masse et la vitesse de rotation dans une celluIe cylindrique dont le cylindre interieur tourne et le cylindre exterieur coaxial reste lixe, a 6th d&uite d’un modele simple supposant des gradients de vitesse lineaires au voisinage de la surface cylindrique en rotation. Une 6tude anterieure de Kimla et Strafelda apparait err-on& mais, apres correction, eile prend essentiellement la mcme forme que celle ici Btablie. R&zmm~~

Zusammenfassang-Es wird der Zusammenhang zwischen Stofftransport und Drehzahl fiir eine konzentrische Zyllnderzelle, bei welcher der innere Zylinder rotiert und der Bussere stillsteht, abgeleitet. Die Geschwindigkeitsgradienten in der NIhe der rotierenden ZylinderobertBche werden Es wird gezeigt. dass eine friihere Behandlung durch Kimla und dabei als linear angenommen. Str&felda nicht richtig war. Werden deren Resultate korrigiert, so erhllt man jedoch im wesentlichen dasselbe wie in diesem Artikel. INTRODUCTION

of mass transfer under conditions of forced convection have been carried out in many types of cell making use of rotating and vibrating electrodes, fluid flow through annuli, parallel plate electrode configurations, conical electrode annuli etc.l The choice of system and its geometry depend a great deal on the particular application envisaged; for example, for voltammetry a smooth reproducible flow pattern is desirable, implying laminar flow and use of a rotating disk electrode or flow in an annulus. The use of rotating concentric cylinders to study both heat and mass transfer has received support from time to time, but its widespread use in electrochemistry has been limited owing to the overwhelming tendency for the flow regime to be essentiahy turbulent in character at all but the slowest rotation speeds. However, an application of metallurgical interest is in the maximization of mass transfer for industrial processing in which limitations are set by appearance and microstructure of the product. In this context turbulent flow is advisable and relatively large demountable cathodes desirabIe so that deposits may be examined in a microscope. For this application either parallel plate electrodes set in a long channel or a rotating cylinder around which a cathode sheet can be wrapped have been used-the latter is considered to be more suitable because turbulence is attained at lower flow rates. Theories of mass transfer in various cell geometries under a range of flow patterns have been reviewed by Newman 24 but the best understood systems are those involving the rotating disk electrode. Empirical relations between mass transfer and rotation rate have been derived for both anodic (dissolution, corrosion and redox) reaction+s and cathodic electrodepositiong-ll reactions; in most cases the central rotor acted as “working” electrode although it is probable that rotation of the outside cylinder STUDIES

l Manuscript received 16 November 1970; as amended, 3 May 1971. t Present address: Cominco Ltd., Trail, Canada.

1121

1122

D.

R. GABB and

D.

J. ROBINSON

and making measurements at the static inner electrode offers a more stable configuration.**la In practical terms rotating the inner cylinder offers greatest convenience and this arrangement has found further application in examining coatings produced under conditions of flow and agitation.l%l6 Bazan and Arvi3’ have pointed out that there are three possible definitions of the dimensionless Reynolds number Re for a concentric cylinder cell,

where U is the peripheral velocity of the cylinder, d the characteristic length and Y the kinematic viscosity. The characteristic dimension may be taken as (i) the electrode diameter, (ii) the electrode height, as in an annulus system, or (iii) the inter-electrode spacing. Eisenberg et aP showed by means of mass-transfer correlations that (i) was the correct definition and this has been used in subsequent investigations.ll On this basis the laminar-turbulent flow transition occurs at a value of Re in the range 5&200 depending upon surface roughness etc; for a 4-cm dia. cylinder in aqueous solutions at room temperature (Y a l-2 x 1O-2 cm2/s) Re = 200 corresponds to u = O-6 cm/s or a rotation speed of 3-O rev/mm. Therefore, while laminar flow in a rotating cylinder cell may be fairly unrealistic in a practical sense it could be of importance in the use of such electrodes for voltametric analyses1 and theoretically lays a basis for a treatment of the more complex case of turbulent flow which is of interest when high rates of mass transfer are required. The case of laminar flow is considered in this paper and turbulent flow will be considered in a subsequent study. THE

THEORETICAL

MODEL

Ions may be transported through solution by molecular diffusion, convection, or by migration under the inlluence of an electric field. When sufficient supporting electrolyte is present to make the resistance insignificant the equation of convective diffusion= describes the steady-state transport of ions,

0. grad ci = D,. V2cj,

(I)

where D is the velocity distribution, ci the concentration of species i and Di its diffusion coefficient. This equation applies strictly to laminar flow, which can be described as linear near the solid wall, an approximation first shown to be realistic by Leveque.ls If #I is the velocity gradient then the velocity U = /?y (see Fig. 1). The case of concentric cylinders can be treated simply by considering a segment of the inner cylinder, radius R, of length M, such that M < 2arR. The annulus can thus be considered to be a straight channel (Fig. 2) and rectilinear co-ordinates may be used to describe fluid flow in the x-direction and mass transfer in the y-direction. Whereupon the convective diffusion equation becomes

Mass transfer in a rotating cylinder cell-I.

Distance,

FIG. 1. Velocity

Laminar flow

1123

y

distribution between inner rotating cylinder and outer stationary cylinder.

Fro. 2. Theoretical model based on

a

segment (width M)

of the rotating electrode.

00 and c = c,, at y = 0. In order The boundary conditions are that c = cb as y to reduce the number of independent variables to one, substitution of a dimensionless parameter, X may be made. The form of X may be found by trial and error to be

II 1

On substitution

1/s

B

x=y-

9Dx

(2) becomes

_3pdc=& dX

.

dXB’

(3)

(4)

1124

R. GABE and D. J. ROBINSON

D.

the solution being (when the boundary

conditions

1

c-cc,==-

x

0.893

The diffusion cd at the electrode

so

exp (-Xx”)

equivalent

dc [ dY

i=---

zFD(Cb -

u=o

1 zF D(cb -

can be derived from

law.

(5).

We may

l/S

$

5,)

0.893

by integrating

[ 9Dx

1

(7)

*

(7) over the whole surface from 0 to M,

c,)

@

[--]

0.893

M

1

dc dX dX’dY’

where dX/d Y is given by (3) in effect and dc/dX then obtain

The average cd is obtained

(5)

and J the mass flux as defined by Fick’s

dc -=dY

i=

dX.

surface is

i=zFJ=zFD

where ZF is the Faraday But since

are suitably modified)

1’31M,-1/a

dx,

whence i = For this particular given by1sm20

04307~FD*‘~(c,

-

case the velocity

c~)M-~‘~/~~“.

distribution

(8)

across the inter-cylinder

gap is

(9) where r is the distance variable and w is the angular velocity; y = 0 (ie r = RI) is B =

--w,[l

thus the gradient #I at

+

where the minus sign indicates that the velocity decreases as y increases. of the value for fl in (8) gives the diffusion cd, i =

O.~O~ZFD”~(C,

or in terms of dimensionless Sh =

-

Substitution

c,)M_~/~~~/~

groups RI

0.64 f

Re . SC . M

.

(1 +

R~~lRza)

(1 -

R,=/R,‘)

“’

1

’

(11)

Mass transfer in a rotating cylinder cell-I.

Laminar flow

1125

where the Sherwood number Sh = (k&D), the Reynolds number Re = (W/V), the Schmidt number SC = (Y/D), Y is the kinematic viscosity and kL the limiting mass transfer coefficient, is given by (iL/zI;cJ. EXPERIMENTAL

SIGNIFICANCE

As has already been mentioned, laminar flow is not a normal feature of the concentric cylinder system, but if the rotor (inner) diameter is large and the gap width small reasonable rotation speeds may be used within the laminar regime. At the same time the approximation inherent in the theoretical treatment is improved, in that the curvature of the rotor is neglected and that by setting M = 2ar4 the full circumference may be used as the mass-transfer surface. A similar analytical study has been reported by Kimla and &rafeldalO, who used a more complex treatment to derive a limiting current for a segment, I = O~~~ZFC,,LM~‘~D~‘~&~ which in dimensionless

. (1 -

R12/R22)-1’3,

(12)

1

(13)

terms becomes Sh = O-38

Re. (1 -

SC

R1 ‘I3

R12/Rz2) iii

’

where M was the segment width in the direction of flow and L its height. Agreement with experimental data was claimed up to 30 rev/min despite the fact that the cell geometry was such that R1 = O-7 cm and R, = 2.5 cm (a gap of 1.8 cm) and M = O-2 cm and L = O-2 cm. It may be noted that the Taylor number criteriorP predicts that instability occurs when the Taylor number Ta = 29-3 ; this indicates a critical angular velocity Oarit = 3 rev/min. Further studies using rotating wire cathodes, which represent an extreme case of the rotating cylinder, have yielded similar relationships. Ibl et aZ22 showed that the limiting current was directly proportional to concentration. Kambara et aZ23 showed that at high rotation rates the limiting current was proportional to c&s but that below the critical speed it was proportional to 01i4 in a relationship of the form

In this particular case, however, the concept of the cylinder where R 3 L is not obeyed and so a strict comparison may not be fair. Clearly, (12) and (13) have a similar form to those derived in this paper [(lo) and (11) respectively]. The constant differs by a factor of 1.68 but the more important error is the omission by Kimla and %rafeldalO of the additional term (1 + RX2/h2), which arises through their assuming an incorrect equation for the velocity distribution U [cf (9)] which was quoted as U =

co1R12(r/R12 -

1/r)(l

-

R12/Rz2)-1.

The error introduced by this omission will depend upon the values for R, and R,. This may account for the apparent excellent agreement over a range of flow velocities which normally would be turbulent and therefore outside the scope of the present treatment.

D. R. GABIX and D. 3. ROBINSON

1126

In practice because the transition from laminar to turbulent flow takes place at such low rotation speeds the rotating cylinder cell is used only in the turbulent regime. authors gratefully acknowledge fiuanoial support from the Athlone FeHowships Committee (for D. J. R.) and the Science R esearoh Council and the British Steel Corporation (Strip Mills Division), and facilities made available by Professors A. G. Quarreli and G. W. Green-

AcknowZea&ements-l%e

woo& NOMENCLATURE

ZF M L Re SC Sh

inner and outer cylinder radii characteristic length radial lengths (co-ordinate) angular velocity peripheral velocity velocity distribution 241~7, velocity gradient concentration of ion species i bulk concentration concentration at electrode surface kinematic viscosity diffusion coefficient mass-transfer flux limiting mass-transfer coefficient current density current Faraday equivalent per mol segment length height of segment Reynolds number = Ud/v Schmidt number = D/v Sherwood number = k&D

TU

Taylor number = Re

Rl. Ra d

x,

Y,

r

w

u

8” Ci

cb

c&l v

D J

h ;

6:

5

::ll’=

REFERENCES 1. R. N. ADAMS, EZectrochemistry ut Solid EZectrodes. Dekker, New York (1969). 2. J. N BWMAN, Ind. engng. Chem. Fund. 5,525 (1966). 3. J. NEWMAN, in Achtances in Electrochemistry and EZectrochemicaZ Engineering, ed. P. D~LA~IAY and C. W. Toasts, Vol. 5, p. 87. WiIey, New York (1967). 4. J. NEWMAN, Xnd. e”gng. Gem. 60,12 (1968). 5. M. E IFiENBERG, C. W. TOBIAS and C. R. WILKI~, J. elecfroclrem. Sot. 100, 513 (1953); 101. 306 (1954); 103, 413 (1956). 6. A. C. MAKRIDES, 3. elecrrochem. Sot. 107, 869 (1960). 7. Z. A. FOROULIS and H. H. UHLIG, J. electrochem. Sot. 111, 13 (1964). 8. E. HEITZ, WerkstoflKorr. 15, 63 (1964). 9. A. J. ARvfA and J. S. W. CARROZZA, Electrochim. Acta 7, 65 (1962). 10. A. KIMLA and F. STRapeLDA, CoZZn. Czech. them. Commun. 32,56 (1967). 11. D. J. ROBINSON and D. R. GABE, nuns. 1-t. Met. Finishing 48, 35 (1970); 49, 17 (1970). 12. J. JORDAN, AnaZyt. Chem. 27,1708 (1955). 13. D. A. SWALHEIM, mans. eZectrochem. Sbc. 86, 395 (1944). 14. J. EDWARDS

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Mass transfer in a rotating cylinder cell-T.

Laminar flow

1127

15. L. R. BEARD, D. R. GABE and S. H. MELBOURNE, Trans. Inst. Met. Finishing 44, 1 (1966). 16. D. R. GAEZE,Met. Fin. J. 16, 340. 370 (1970). 17. J. C. BAZAN and A. J. ARV~A, Ecectrochim. Actu 9, 667 (1964). 18. V. G. L~~I~Kz, Physiochemical Hydroa’ynumics, Prentice-Hall, Englewood Cliffs, New Jersey (1962). 19. J. LEVEQUE, Ann. Min. C’arbur. Paris (ser. 12) 13, 201, 305, 381 (1928). 20. R. B. BIRD, W. E. S~WART and E. N. LIGFI-~FOOT, Transport Phenomena. WiIey, New York (1962). 21. G. I. TAYLOR, Phil. Tkans. R. Sac. A223,289 (1923). 22. N. IBL, K. BUOB and G. TRUMPLER, Heh. chh. acta 37, 2251 (1954). 23. T. KAMBARA, T. TSUKAM OTO and I. TACHI, J. elecrrochem. Sot., Japan 19, 199, 297 (1951).