Mass transfer at the electrodes of concentric cylindrical reactors combining axial flow and rotation of the inner cylinder

Mass transfer at the electrodes of concentric cylindrical reactors combining axial flow and rotation of the inner cylinder


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MASS TRANSFER AT THE ELECTRODES OF CONCENTRIC CYLINDRICAL REACTORS COMBINING AXIAL FLOW AND ROTATION OF THE INNER CYLINDER F. COEURET and J. LEGRAND Laboratoire d’Etudes A&odynamiquea et Thcrmiqucs (LA CNRS N” 191). 40, avenue du Recteur Pincau, 86022 Poitiers Ctdex, France (Receiued

11 June 1980, in revisedfarm August 1980)

Abstract-The paFor reports results obtained during mperimantal studies of the overall mass transfer between a liquid and the walls of a type of electrochemical reactor which consists of coaxial cylinders and theaxialflow of electrolyte with the rotation of the inner cylinder. Three intercylinder gap which associates widths, 2.5 mm; 5.0 mm and 7.5 mm respectively, were studied. The experimental variations of the Sherwood number Sh as a function of the axial Reynolds number Re, and of the Taylor number Ta are aualyscd, principnRy the laminar vortex regittm is covered in the experiments and empirical correlations are deduced. Two domains appear asclearlyseparated: one, at small Re, values is such that Sh is only dependent on Ta, the other for Re, > 300 is characteri& by a positive influence of both parameters Re, and Ta.


D p L N Nu :

Re. RI &



= [l + Na+(l - N)l/log N]/[2+(1 - Ne)/log N] diffusion coeRicient = Ra - RI, intercylinder gap width friction factor overall mass transfer coefficient length of the transfer surface = &I& Nuaselt number with 2e as characteristic length electrolyte volumetric flow rate

overall fractional conversion = u. 2/v, axial Reynolds number radius of the inner rotating cylinder internal radius of the outer stationary cylinder electrode surface area = v/D, Schmidt number = k. 2elD. Sherwood number


= y-(e,R,)

Ta. ” Y

critical Taylor number


Ii* , Taylor number

super5cial slectrolyte velocity in the annulus kinematic viscosity angular velocity of the inner cylinder = (N - 1)/N{ [NZ.ln N + (1 - N’)/2]/ [(1+N2).lnN+(1-N2)]},geometricfactor

possibility to act on two indepan&nt parameters (axial flow rate, rotation) (ii) existence of particular vortex regimes which may favour local mixing of the products generated at the electrodes or the transport of a product from one electrode to the other (case of certain electro-organic syntheses) Fz ~lsaiaiu~f using intercylinder gap widths of a It seems that this type of device has not been studied for its application in electrochemistry although ita two particular casts which are the electrolyte flow alone in the annulus[3,4] and the rotation alone of the inner cylinder (rotating cylinder electrode)[5-lo] have been extensively studied. However, similar problems have been examined, principally in the case of heat transfer with air, with a view to determine the heat transfer rate between rotor and stator of electrical motors or generators and therefore the cooling of these systems[ll-131. Other works which concern heat transfer in water or in oil[12,14] or mass transfer by evaporation[lS] can be quoted. Only some of them propose empirical correlations, generally For turbulent flow[lZ, 141. Kataoka

INTRODUCIION The concept of an electrochemical reactor with coaxial cylindrical electrodes and combining the rotation of the inner cylinder with the forced axial flow of the electrolyte has been reported in the literature but the Jack of data about it has also been pointed out[ l]. In spite of the disadvantages due to the cylindrical configuration and the rotating of an electrode, the advantages it presents deserve intereat[2]:

(i) high mass transfer coefficients even at low axial Bow rates, and thus high conversion rates, owing to the

er a/.[161

have extended

to the case of

combined axial flow and rotation, the work of Mizushina et a/.[171 which was concerned with rotation only; it seems that both papers have been the first to take advantage of the electrochemical method for local mass transfer studies at the wall of the stationary outer cylinder. They examined the range of low axial Reynolds number and researched the relation between the axial motion of the vortices and the periodic variations of the instantaneous local mass transfer coefficients. The timeaveraged local mass transfer coefficients were obtained by integration; their variations with both parameters characterizing the rotation and the axial flow respectively were presented Paper[lb] was published just after the beginning of 865




our research, the motivations of which were different and used an intercylinder space of 18 mm and a rotating cylinder of a 29 mm radius; the axial Reynolds numbers were lower than 50 and the hydrodynamical regime was the laminar vortex flo_w regime. The most recent work seems to be[133 which concerns heat transfer between the air and the inner surface of the stationary outer cylinder; the study was for turbulent flow and considered small intercylinder gap widths (N = l&/R, equal to 0.8 and 0.955 respectively). The experimental uncertainties are too high for quantitative conclusions to be drawn; the work presents an interesting development similar to Reynolds analogy but numerical calculations were erroneous. A simple equation could have been deduced before the calculations (see equation A-4 in the appendix). It must be remembered that the flow resulting from the combination of axial flow and rotation of the inner cylinder is characterized by two dimensionless criteria[l8]: The axial Reynolds The Taylor


Re, = u .2e/v


As can be seen from Fig. la several types of Bow can be obtained according to the value of the couple Re,-Ta although this view is very schematic as it is confirmed by the divergences between authors concerning the demarcation lines between regimes[l 1). The true laminur regime is not of interest from the point of view of transfer. Our aim has been to explore the laminar vortex regime in the case of very small intercylinder spaces and Reynolds numbers Re, compatible with the concept of an open electrochemical reactor. Figure 1b schematizes the Taylor vortices which appear for Re, = Oat the critical Taylor number Ta, (Ta, = 40 when Re, = 0) and which result from

J. LEGRAND instability of the flow; the perfectly toroidal vortices of Figure lb are rather ideal and in fact several states can be observed (aacending spiral, tangential waves, etc.)[ 191 according to the values of Ta and of the ratio e/R,. The superposition of an axial fiow causes the displacement of the vortices and their deformation. The present work deals with the exparimental study of the overall mass transfer at the rotating cylinder in three cells characterized by different values of e and also at the internal surface of the outer cylinder in one of the cells; a brief article[20] had presented the first results obtained in one of the cells at the beginning of the study and also reported the intluence of the Schmidt number SC. The purpose here is to draw up the balance of all the studies and to analyse the results thereof. A further atticleE will report the results of a local study the aim of which is to interpret the overall results. EXPERIMENTAL The experimental technique has already been described in[20]; the mass transfer coeffieicnts are obtained by reduction of potassium ferricyanide in polarographic conditions and at 30°C; NaUH 0.5 N is the conducting support and the solution contains also potassium ferzocyanide (at a high concentration 0.25 M in order to guarantee the diffusional limitation at the cathode only)_ The transfer surface is made of nickel or copper but always plated with gold and then with platinum; in every case it has a kngth of L: 10 cm and it always covers the median part of the 29.6 cm height of the cell. The radius of the inner rotating cylinder was R, = 1.75 cm and three gap widths were used (0.25 cm; 0.5 cm; 0.75 cm); only the space e = 0.5 cm was considered for the d&rmination of the mass transfer coefficient at each wall of the annulus. by 5, when it The transfer surface is designated

Si ,rotating s2,


Turbulent flow

Turbulent flow WIT v~trces




Fig. I. Schematicviews: (a) of the domains of existence of the different hydroclyrtamk regimes; (b) of the cell configuration with representation of Taylor vortices (Re. = 0).

Mass transfer at the electrodes of concentric cylindrical reactors concerns the inner rotating cylinder ati by SZ when it concerns the outer cylinder. The variation range of the exuerimental variables is tabulated below: rotation frequency (rev./mm.)


sc Ta Re.

0.143;0.286; Ct.429 138%up to 6450 in[20] 0 and 135-3700 0 and 25820

Qlivcr,t 0,


RESULTS Ta = 0 and Re, = 0 These two particular cases have been studied by numerous authors; therefore they were considered here only in order to test the quality of the experimental technique by comparison of the results with well known correlations. 1. Liquidflow alone in the annulus (Ta = 0). The experimental data obtained for axial flow alone has been compared to two equations previously used by Ross and Wragg[4]. Both are deduced from the Leveque solution: the one (Fig. h) is the Lcveque solution applied directly to the annulus, the other is the Leveque solution applied, as in[3], to the tube with the substitution of the tube diameter by the equivalent annulus diameter (Fig. 2b). Figure 2 gives, for Sc = 1380 (the only value used in these tests), this comparison which seems as satisfactory as in[4], This also confirms that the hydrodynamic entry length is sufficiently long. 2. Rotation alone of the inner cylinder (Re, = 0). A partial comparison of mass transfer results corresponding to the rotating cylinder has already been made in[lO] with the correlations proposed for the turbulent flow regime; it was not convincing except in an order of magnitude sense. It appears that the set of data is

globally described with a 20 per cent margin by the Eisenberg correlation into which is introduced a corrective term[ IO]. However if such a weak precision might be sufficient as a description, considering the approximative character of the correlation itself, a quantitative comparison with the works concerning heat or mass transfer in the laminar Couette flow with vortices must be made. Such a comparison was the object of a separate paper[22] which emphasized divergences between authors but made apparent the fact that, at least for the smallest spaces (e = 2.5 and 5.0 mm) Sh varies as Ta “’ The divergences mentioned, which are certainly due to geometrical factors, especially the distance e, draw attention to the need of supplementary data even extrinsically to the spirit of the present article.

Combined rotation and axial flow 1. Presentation ofthe results. Besides Ta and Re, the thickness e has been one of the parameters and, in the case of e = 5.0 mm the transfer surface has also been changed. The most representative results are given below. Figure 3 gives for e = 2.5 mm the variations of Sh = k.2e/D as a function of Ta for several values of Re, covering the interval from 0 to 820. It can be seen that for a given Ta, when Re, is increased from zero upwards Sh decreases sharply but its value changes little there after, as long as Re, remains below 300. For Re. greater than approximately 300, an increase of Re, when Ta = constant improves the transfer; for Re, of around 500 the mass transfer becomes equivalent to that obtained for Re, = 0. A further increase of Re, improves the transfer further but it seems that the effect is attenuated when the values of Ta are the highest. For the case (e = 5.0mm; mass transfer to the stationary cylinder), Fig. 4 separates the two domains






‘“” ‘”

(Re.Sc.=.)J L


Fig 2. Case of axial flow alone. Comparison of tbc mass transfer results with correlations from the literature: (a) Levique’s solution for an annulus (b) LevEque’ssolution for a tube.





log, To

Fig. 3. Results for muss transfer at the inner rotating

cylinder (e = 2Smm).

oRe,=320 0



-467 -540 = 614 z.666

A I 0








Lop, To

Fig. 4. Results for mass transfer

at the outer stationary cylinder (e = 5.0 mm) (a) for Re, c 300; (b) for Re, x+300.

Mass transfer

at the Ckctrodea of cancentric cylindrical reactors

Re, < 300 and Re, > 300. The same unnments as above can be mode, however the maximum Re, value experilncntiy employed barely achieves a transfer comparable to the one obtained for Re, = 0. This figure can be compared to Fig. 4 oQ20] which reports similar results for the same cell but for mass transfer to the rotating cylinder. Figure 5 presents likewise the results obtained for e = 7.5 mm. More or less the same developments can be observed although more attenuated, particularly for Rc, < 300; here agaia this value of Re, = 300 may be adopted as a transition value. One can say that from




70 W.

Re, - 28 up to Re, = 2Q0, Sk dazreases too slightly with Rc, for any kind of i&unce of Rc, to be apparent. 2. Empirical correlations. Among other results,[20] reported that theinfluence of Sc was as [email protected];

also ttn empirical correlation had been proposed for each of the two domains separated by Re, = 300. The complementary results given in the present paper also show the importance of this value of Rc, and therefore correlations have also been sought for each domain; the exponents of Ta and of Re, have been determined



Fig. 5. Results for mass transfer at the inner rotating cylinder (e = 7.5 mm) (a) for Re, c 300, (b) Re, z-300.

Table 1. Summary of the empiriml correlations deduced

Conditions of TmnBfn: surfaa

outer mtationary cyliad= (S,)



for Re, < 300

ror Rc” > 300


31 < Re, < 812 130 Ta c690

Sh = 0% Ta”2 SC+

Sh = 0.12 Re,“’ To’,“’ SC’/’

25 < Rc, c 800 2oO
Sh = 0.38 Taljf SC”

Sh = 0.12 Re,“’ Tao,‘* SC’@


28 < Re, -C 72U 7OOtTa ~3700

Sh = 0.59 Ta”l Sc”’

Sh = 0.08 Rc,“~ lkrra SC”


29 c Rc, c 760 ZW
Sh - 0.42 To”’ SC”’

Sh = 0.05 Rc,“’ Ta1f2 SC”’

Inner rotating cylinder (S,)


e (mm)






by regressive analysis. The correlations obtained have been assembled in Table 1; it is remarkable to observe that for Re, < 300 the exponent of Ta is equal to 0.5 (with correlation coefficients varying between 0.95 and

1). For the Ra, > 300 range it is dedbced that Re, acts as Re,“’ and Ta with an exponent equal to 0.4 or 0.5. Figure. 6 compares tht results o$rained for Re, -C 300; the number of experimental points and the

Fig. 6. Correlation of all the results obtained for Re, -c 300.

Fig. 7. Correlation of all the results obtained for Re, > 300


Mass transfer at the electrodes of concentric cylindrical reactors partial covering of the explored fields make it possible to give only the contours enclosing all the experimental points (the number of these points is specified for each contour). In[20] the contents of one of these contour6 can be found. It can be seen in Figure 6 that the empirical correlation: Sh = 0.4 Ta1’2. !3c”3


can be accepted in order to describe mass transfer when Re,, < 300 for e = 2Smm or 5.0mm at both surfaces S, or S,. As for Re, = 0, the case e = 7.5 mm leads to mass transfer better than for the other cells. The comparison of the correlations deduced for Re, z 300 is not as simple, given the two values 0.4 and 0.5 deduced for the Ta exponent. However the accuracy of the experiments and the dispersion of the experimental points around the correlation (see Fig. 6 oflI20]) allow, in order to facilitate a comparison with semi-theoretical solutions (see appendix) adoption of a Ta0,47 dependency thus leading to the comparison of Fig. 7. It seems that the form of expressions (A-5) and (A-6) cannot be retained for the results corresponding to e = 7.5 mm. For e = 5.0 mm it appears that the transfer is more or less the same to the rotating cylinder and to the outer stationary cylinder. The straight line drawn in Fig. 7 is the one which seems to best describe the data corresponding to the smallest spaces (e = 2.5 and 5.0 mm); the numerical coefficient 0.07 which figures in it means that (A-S) and (A-6) overestimate the performances from the point of view of mass transfer.

DISCUSSION 1. Comparison with datafrom the literature. Firstly it must be emphasized that it is experimentally proved that at a fixed Ta, Sh decreases when Re, increases from zero to about Re, = 300. Kataoka et a[.[161 have reported the same effect illustrating it even better since they observe a more progressive decrease than in our case; they were able to impose values for Re, varying only from 5 to 50 but sufficiently to show an attenuation of the influence of Re,. The lowest values obtained for ShjSc ‘I3 by Kataoka are in agreement with the correlation Sh = 0.4 Ta”‘. !SC”~which describes the results for small P in Fig. 6; such an agreement is surprising because it does not exist when the data obtained for Re, = 0 are considered[22]. From the local instantaneous information collected at the wall of the outer stationary cylinder, Kataoka shows that the progressive increase of Re, has the consequence of deforming the distribution of the local instantaneous coefficients (regular sinusoidal variation when Re, = 0) and also reducing notably the mean values and the amplitudes. Low axial flow rates (small Re,) seem to have a stabilizing effect on the flow, which would be in agreement with[23]. Up to now the critical value Re, = 300 has not received explanation but in[21] it will be shown that local studies allow one to see it as a transition value between two flow structures. The weak negative influence of Re, on Sh or on Nu has also been pointed out by other authors[ 11,151. In the first domain corresponding to low Re+ values (30 < Re, < 300), the practically indiscermble in-

fluence of Re, proves the preponderant importance of the rotation over the axial flow. The OEher domain (Re, > 300) reveals an influence of both parameters Re, and Ta, which is not in disagreement qualitatively with the one which may be deduced from analogies but which has not received a quantitative interpretation. The experiments have always outlined differences between the results obtained in the cell with e = 7.5 mm and those deduced in the other cells; the reason for this is probably due to a difference in the hydrodynamic flow or in the vortex structure. 2. Consequences concerning electrochemical applications. Applications in electrolysis of the device studied in this paper would probably use small intercylinders gaps. Although the data corresponding to these conditions are few, whether for Re, = 0 or for the general case, it seems that: -it is possible to establish satisfactory empirical correlations -at least for low e values, the mass transfer coeficient is the same at the outer stationary wall and at the rotating inner cylinder. Once more it is extremely important to emphasize the fact that at low Re, values the mass transfer coefficient is not only almost independent of Re, but also even lower (about ,/2 times) t,hati when Re, ,= 0. Indeed the application to design of the correlations established for Re, = 0 would result in electrode surface areas being highly underestimated. In this laminar vortex flow regime at low Re,, the fractional conversion of the electrolyte should he able to reach high values. Whatever the hydrodynamic regime an advantage is the possibility of controlling the mass transfer coefficient through two parameters, but especially by the rotation of the inner cylinder. The overall fractional conversion R through a piston-type electrochemical reactor is:



-k.S Q 1

where S represents the electrode surface where the transformation takes place and Q the electrolyte flow rate. The correlations deduced in the present work, whether for Re, -Z 300 or for Re, > 300 are such that the gain obtained for the ratio k/Q by bringing into play the rotation is directly proportional to Ta. In order to illustrate the problem, let us consider the cell with e = 5.0mm: for Re, = 1000 and Ta = 0 (forced axial flow alone in the annulus) one would find (using the expressions given in Fig. 2) that Sh = 20, an incompacably smaller value than the one which would be possible to obtain for Re, Q loo0 but with the rotation (Figs 6 and 7).

CONCLUSIONS The electrochemical reactor with coaxial cylinders, combining the rotation of the inner cylinder and the axial flow of the electrolyte is an open reactor which allows a high fractional conversion. However a variety of hydrodynamic states may exist which renders its study complicated. The results obtained are to be



cbntribution to the knowledge of such devices but they are sufficiently encouraging to justify further research. considered as partial

REFERENCES I. D. J. Pickett,

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.


19. 20.

Electrochemical Reactor Design, Elsevim (1977). J. Legrand, The&, University of Poitiers, France (1980). C. S., Lii E. B. Denton, H. S. Gaskill and G. L. Putnam, Ind. Engag Chem. 43,2136 (1951). T. K. Ross and A. A. Wragg, Electrochim. Acta 10, 1093 (1965). M. Eisenberg, C. W. Tobias and C. R. Wilke, J. electr&lefn sot. lot, 306 (1954). J. Comet and R. Kappaser, Trans. Inst. them.Engrs 47, T 194 (1969). D. J. Robinson and D. R. &be, Trans. Inst. mef. Fin. 48, 35 (1970). D. R. Gabe and D. J. Robinson, Electrochimica Acro 17, 1121 (1972). D. R. Gabc and D. J. Robinson, Electrochimica Acta 17, 1129 (1972). D. R. Gabe, J. appl. Electrochcm. 4, 91 (1974). K. M. Becker and J. Kaye. J. Hear Transfer 84,97 (1962). F. Tacbibana and S. Fukui, Bull, J. Sot. mech. Engrs. 7 (26), 385 (1964). D. A. Simmers and 1. E. R. Coney, Int. J. Heat Mass Traa.$r 22, 679 (1979). A. A. Mosyak, B. G. Rykova, P. D. Kostov and G. I. Gruxintsev, Hear Transfe Soviet Res. 9, 110 (1977). J. R. Flower, N. Mackod and A. P. Sbahbenderian, Chem. Engng Sci. 24, 637 (1969). K. Kataoka, H. Doi ahd T. Komai, Inr. J. Hear Muss Transfer 20, 57 (1977). T. Mizuabina, R. Ito, K. Kataoka, S. Yokoyama, Y. Nakajimaand A. FukukaKagaku Kogaku32,795 (1968). Work summarised in Adacs in Heat Traqfer, Vol. 7, chapter by T. Mizusbina, Academic Press (1971). F. Kreith Convection heat transfer in rotating systems, in Advances in Heat Transfer Vol. 5. Academic Press (1968). D. Coles, J. fluid Mech. 21, 385 (1965). J. Legrand, P. Dumargue and F. Coeuret, Elecrrochim. Acta 25, 669 (1980).

21. J. Legrand,

M. Billon and F. Coeuret, to be published. 22. F. Coewet and J. Lcgrand, 1. qpl. Elecctrochem. 10, 785 (1980). 23. T. H. Hugues and W. Reid, Phil. Trans. R. Sac. Load. A 263, 57 (1968).

which follows from the corresponding expressions for the Poiaeuilk flow alone and for the Couette tlow alone respectively. In expression (A-l) the term [Ta/Ta,l’.‘7 replacea the ratio of shear stresses.(studied flow/Poiseuilk Pow) and is deduced experimentally. It is suppolcd in[13] that all therceie.tmm to heat transfer at the outer stationary cylinder is situated ia the lamittar a&layer (transfer by conduction) on this wall and in the b&r layer (transfer by forced convection in this twwt *) whichfollowsit. The remainderofthegapiaaupposed tobettt a constant temperature. By means of some supplementary simplifying hypothasea, the authors determine for Pr = 1 and N = RI/RI near one (e -+ 0) the total temperature drop near the transfer surfaa and deduce an expression [email protected] the Nusselt number Nu. This expression, designated by (35) and (36) is reducible for Pr = 1 to: (1 - N)3’*





2(1 -IV)’ =r.-




Ta’ [ Ta,




Ta: = 171.74 Re,,“*9’



from which (A-2) can be written as:

Nu =




This expression shows, contrary to the theoretical results of 1131, that Re, has a positive influence. Also (A-4) is quantitatively in disagreement with the experimental results of[lf]. By transposition to mass transfer (high Schmidt numbers) the following solution is obtained: Sh = 0.137s.Re.L’3





Ta0.L’ Sc’Jf.


Reynolds analogy

From Simmers’s experimental work mentional in[l3] and relative to studies of wall stresses and velocity pmfiks, it seems that the Taylor vortices, once present in the annular space, behave as if there was no axial tlow. Thus, if the Reynolds analogy is applied by choosing for the mean flow velocity the peripheric velocity R, .w at the surface of the rotating cylinder, it follows easily that: Nu = 0.092~.

For small gap widths e, Simmers and Coney[l3] have dcduad the following expression for the friction factor in the cast of Taylor vortices and axial flow:




Simmers and Coney used this expression in order to examine the variations of Nu with Re. (2 300) and Ta; they noticed that at a tixed Ta, Nu decrea& wbeu Re, iacrrssed However it can he shown rapidly[2] that the tabulated vah& of Re, and Ta, employed in this numerical calculation could have b&n empirically related very satisfactorily by:







The same kind of deduction, but in the spirit of the ChiltonColburn analogy for high SC, would give:


Sh = 0.092~.Re.“3.Tao-‘7.Sc’i’.

Apart from its numericalcoeiIicimt,


(A-7) is identical to (A-S).