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A study of roughness and mass transfer enhancement for the rotating cylinder electrode
Surface Technology, 24 (1985) 29 - 44
29
A STUDY OF ROUGHNESS AND MASS T R A N S F E R ENHANCEMENT FOR THE R O T A T I N G CYLINDER E L E C T R O D E * P. A. MAKANJUOLA and D. R. GABE Department of Materials Engineering and Design, University of Technology, Loughborough, Leics. LE11 3TU (Gt. Britain) (Received March 29, 1984; in revised form June 21, 1984)
Summary It is well established that enhanced mass transfer occurs at a rotating. cylinder electrode operated in the t ur bu l ent regime when cathodically deposited metal grows in a rough p o w d e r y or dendritic manner. In the absence o f quantitative data for the degree of roughness and t he consequent increase in the friction factor, such e nha nc em ent has been expressed empirically as an increase in the power e x p o n e n t of a simple mass transfer relationship K L = constant × U n where n is normally a b o u t 0.7 in t ur bul ent flow for a s m o o t h electrode. By using cylinder electrodes o f known induced roughness, which can be characterized by a true surface area, t o p o g r a p h y etc., it is possible to distinguish between the effects of increased surface area and microturbulence, whence it becomes possible to recognize and devise optimal m et hods of increasing the surface roughness in order to maximize mass transfer. For t he V-grooved cylinders investigated, the saturated or m a x i m u m enhanced mass transfer was correlated according to the pow er law relationship where n varied between 0.51 and 0.58. For six such cylinders the average increase in mass transfer at a high velocity, or degree o f enhancement, was 80% relative to a s m o o t h cylinder.
1. I n t r o d u c t i o n The work r epor t e d here is a cont i nuat i on of earlier studies concerned with the d e v e l o p m e n t of dendritic and powder electrodeposition o f metals on rotating cylinder electrodes [1, 2], which are known to allow enhanced mass transfer to develop during operation. Wide-ranging m et hods have been used in attemp ts to p r o m o t e higher mass or heat transfer rates or to reduce friction coefficients in flow systems. It is apparent t hat t he greatest strides have been achieved in t he field of heat transfer as t he t w o major reviews by Bergles [ 3 , 4 ] show. Some o f the m a ny augmentation techniques discussed *Paper presented at a meeting on Engineering Aspects of Electrochemistry, Loughborough, Gt. Britain, April 17, 1984. 0376-4583/85/$3.30
© Elsevier Sequoia/Printed in The Netherlands
30 for improving forced convection heat transfer include surface vibration, fluid vibration, mechanical aids, swirl flow devices, rough surfaces and surface tension devices. It is worth mentioning that only a handful of these methods are in any serious contention for intensification purposes, and two of these are considered in this paper. They come under the headings of mechanical aids (mechanical agitation of the cylinder electrode) and rough surfaces. The first known report on a rotating cylinder was by Brunner [5] whose observations and conclusions were generally qualitative in nature. Following Brunner, the established configuration was two concentric cylinders with the inner one rotating and the outer stationary, and Taylor [6] was the first to study fluid stability within the annular gap formed by such coaxial cylinders. Thereafter, Theodorsen and Regier [7 ] published friction data based on both smooth and rough inner rotating cylinders. In 1954, Eisenberg e t al. [8, 9] conducted the most thorough study to date on mass transfer to an inner rotating cylinder electrode with the outer cylinder serving as the counterelectrode. Their data for turbulent flow regimes were correlated by Sh = 0.079Re°'7°Sc °'a56
(1)
where Sh, Re and Sc are the Sherwood, Reynolds and Schmidt numbers respectively. A comparison of the above correlation with the friction results obtained by Theodorsen and Regier [7] for a smooth cylinder confirmed quite adequately the analogy between mass and m o m e n t u m transfer for this system and also confirmed that the critical dimension is the diameter of the inner cylinder. The technique for mass transfer measurement employed by Eisenberg et al. [8] was the electrochemical limiting current m e t h o d utilizing the ferriferrocyanide redox couple in alkaline solutions. Certain electrodeposition reaction systems can also be employed. The limiting current technique and its m a n y advantages over conventional mass and heat transfer measurement methods have been discussed by Landau [ 10] amongst others. A concise review of the rotating cylinder electrode was produced by Gabe [11]. Among many other topics, he discussed the emerging interest in the use of such an agitation cell in metal-finishing processes and for electrowinning metals from solutions of increasing dilution. The rotating cylinder causes fluid instability at quite low Reynolds numbers {below 200) and hence achieves high turbulence at moderate operating speeds. This characteristic is a useful attribute for maximizing electrodeposition rates while recovering metals from effluent solutions. Laminar flow conditions will not be considered for this reason. It is well established that a rough granular deposit results during the approach to and within the limiting current range when copper deposition at the cathode is the chosen system. Ibl and Schadegg [12] and Conway and Bockris [13] have studied this effect. Such a dendritic deposit is known to cause an enhancement in mass transfer coefficients as reported by Holland [14] and Gabe and Walsh [15]. Makrides and Hackerman [16] and King and Howard [17] observed a similar phenomenon during metal dissolution experiments. Such reactions have been proved to be
31 diffusion controlled [18]. The increase in dissolution rates was accompanied by a gradual surface roughening which resulted in a form of discrete roughness. Holland [14] proposed a simple velocity power law K (or I) = constant × U n
(2)
where n = 0.7 for a smooth rotating cylinder and n -+ 1.0 for a roughened cylinder. As we shall attempt to show, this power law has certain limitations. Makrides and Hackerman [16] and other workers have attributed the increase in mass transfer to an increase in the reaction surface area. However, as Levich [19] pointed out, if the average height o f the protuberances exceeded that of the hydrodynamic viscous sublayer, microturbulence will be expected at the solid-liquid interface. These irregular roughnesses are understandably difficult to characterize. Bergles [3, 4] suggested the use of artificially roughened surfaces produced by machining techniques, an example of which is the work of Kappesser et al. [20] who resorted to knurled pyramid-type roughness elements. As pointed out by Bergles [3], because of the variety of geometrical configurations it has not been possible to adopt a unified theory or a broad correlation for the effects of surface roughness on single-phase heat transfer; neither has it been achieved for mass transfer. In the present study we attempt to separate the effects due to interface microturbulence from any enlargement of the surface area. Various roughness types will eventually be tested during the duration of this research in order to understand what effects the critical roughness dimension, i.e. the size or height, and the orientation of the element have on an enhancement factor which is a function of these variables. Of these, only one is reported here, i.e. V-grooves normal to the direction of fluid flow. All the data will be obtained within a controlled environment to ensure that all the results are directly comparable.
2. Experimental details 2.1. The ro rating cylinder electrode rig The present rig has been described in detail elsewhere [15]. The general structure remains unaltered although a few adjustments have been made to accommodate the experiments reported here. In its original state the squaresectioned catholyte compartment was separated from the anolyte section by four cation exchange membranes, one on each side. The removable 1 1 catholyte chamber could be positioned in a 20 1 bath containing the supporting electrolyte. In this divided cell, lead alloy anodes {anodized to lead dioxide during use) were placed along the walls of the bath. In this form the cell is excellent for batch concentration decay experiments or for fast constant-potential mass transfer runs. However, for constant-concentration test runs a cylindrical copper anode of 8.0 cm diameter was fabricated. This was positioned inside the catholyte compartment such t h a t it could easily be
32 lifted o u t b e t w e e n successive runs. This could be d e s c r i b e d as an u n d i v i d e d cell w h e r e b y t h e electrical c o n t a c t s to t h e lead d i o x i d e a n o d e s w e r e disconn e c t e d while t h e adjoining b a t h served o n l y as a t e m p e r a t u r e c o n t r o l . This w h o l e a s s e m b l y c o u l d be raised vertically to enclose a test cylinder, w h i c h in t u r n c o u l d be u n s c r e w e d f r o m a rigid r o t o r s t e m b e t w e e n runs. T h e r o t o r was driven b y a 1/8 hp S e r v o m e x C o n t r o l s m o t o r . An auxiliary t a c h o m e t e r was s i t u a t e d e x t e r n a l l y to m a i n t a i n a c h e c k o n t h e r o t a t i o n a l speed. Electrical s u p p l y was c o n t r o l l e d using a Chemical E l e c t r o n i c s 20 V, 50 A m o d e l p o t e n t i o s t a t and fed t h r o u g h t w o silver-filled g r a p h i t e b r u s h e s to t h e c a t h o d e . An a d d i t i o n a l f e a t u r e was a p o t e n t i a l p i c k - u p b r u s h in c o n t a c t with a silver-plated c o p p e r ring. T h e ring was situated b e t w e e n t h e b r u s h a s s e m b l y and t h e c a t h o d e . T h e p i c k - u p b r u s h m o n i t o r e d p o t e n t i a l readings ( o n a digital m i l l i v o l t m e t e r ) b e t w e e n an H g - H g S O 4 (MMS) r e f e r e n c e e l e c t r o d e and t h e cylindrical c a t h o d e while eliminating t h e p o t e n t i a l d r o p across t h e b r u s h a s s e m b l y w h i c h resulted f r o m a c o n t a c t resistance g e n e r a t e d b e t w e e n t h e b r u s h e s and t h e c o m m u t a t o r d u r i n g r o t a t i o n .
2.2. Experimental procedure F o r d i f f u s i o n - c o n t r o l l e d m a s s t r a n s f e r d e t e r m i n a t i o n s t h e acidified CuSO4 s y s t e m was selected. A relatively dilute 0 . 0 1 4 M CuSO4 solution w i t h 1.5 M H 2 S O 4 as t h e s u p p o r t i n g e l e c t r o l y t e was e m p l o y e d at 22 °C for all test runs. A 50 1 b a t c h o f solution was p r e p a r e d using d e i o n i z e d w a t e r and t h e c o p p e r ion c o n t e n t o f 8 8 9 p p m was m e a s u r e d t o within +10 p p m using an a t o m i c a b s o r p t i o n s p e c t r o p h o t o m e t e r . T h e t e m p e r a t u r e in t h e c a t h o l y t e c h a m b e r was c o n t r o l l e d b y a t h e r m o s t a t i c b a t h t o within + 0.5 °C. T h e s m o o t h t e s t c y l i n d e r was a b r a d e d w i t h fine e m e r y p a p e r , w a s h e d in a c o m m e r c i a l cleanser and t h e n rinsed in water. It was t h e n d e g r e a s e d b y cleaning w i t h 1 , 1 , 1 - t r i c h l o r o e t h a n e following w h i c h it was again rinsed t h o r o u g h l y w i t h d e i o n i z e d water. A f t e r this, it was p l a c e d in a b e a k e r containing 1.5 M H 2 SO4 for a f e w seconds. It was rinsed o n c e again in d e i o n i z e d w a t e r and was t h e n a t t a c h e d to t h e r o t o r r e a d y f o r t h e s t a r t o f a run. T h e s a m e p r o c e d u r e was a d o p t e d f o r t h e e x p e r i m e n t a l e l e c t r o d e s L1 - L6 e x c e p t t h a t t h e y w e r e n o t a b r a d e d w i t h fine e m e r y p a p e r . T h e t e s t s o l u t i o n was r e p l a c e d w i t h a fresh 1 1 s o l u t i o n b e t w e e n successive runs and the large v o l u m e o f p r e p a r e d e l e c t r o l y t e was e x p e c t e d to e l i m i n a t e variations in m e t a l c o n c e n t r a t i o n . T h e c o p p e r ion c o n c e n t r a t i o n o f t h e s p e n t s o l u t i o n was r e c h e c k e d a f t e r every c y c l e o f o p e r a t i o n . During m a s s t r a n s f e r runs a soluble c o p p e r a n o d e , w h i c h was c o n c e n t r i c w i t h t h e c a t h o d e , was used. A t o t h e r t i m e s t h e divided cell was r e t a i n e d f o r certain c o n s t a n t - p o t e n t i a l runs. A t t h e start o f a r u n t h e c y l i n d e r was p r e p l a t e d at 300 rev m i n -1 for up to 10 m i n at a relatively low p o t e n t i a l ( - - 6 5 0 mV) relative to an MMS r e f e r e n c e e l e c t r o d e c o n n e c t e d via a 1.0 M Na2SO4 salt bridge s o l u t i o n . P o l a r i z a t i o n curves w e r e o b t a i n e d b y e m p l o y i n g a scan r a t e o f 150 m V m i n -1 f r o m a s w e e p g e n e r a t o r . H y d r o g e n gas e v o l u t i o n c o m m e n c e d a f t e r a b o u t 3 0 0 s and t h e run was t e r m i n a t e d s h o r t l y a f t e r w a r d s .
33 24 runs each were carried out for the sm oot h cylinder and for the electrodes L1 - L4, and 12 each were p e r f o r m e d for t he electrodes L5 and L6. The polarization scans were similar for bot h t he sm oot h and the rough cylinders. As a check on the limiting current values obtained, potentials within the limiting current plateau range were instantaneously impressed on the cathode [21] and steady state current readings were achieved within a bo u t 30 s. The pick-up brush was invaluable in helping to pinpoint a required potential value relative to the MMS electrode connect ed via the 1.0 M Na2 SO4 solution. Th e readings from bot h m e t hods of measurement were in satisfactory agreement.
2.3. Production of cylinders The geometrical similarity o f the grooves was considered to be of p a r a m o u n t importance. Therefore, although the segment lying between two adjacent grooves constituted the roughness element, it was in fact the grooves which were of prime interest as far as mass transfer to the surface was concerned. One s m o o t h and six V-grooved cylinders were made. The sm oot h test cylinder, o f stainless steel material, was 6.3 cm long and 3.0 cm in diameter as were the o t h e r cylinders before the knurling process. Two knurling wheels were employed, both of the straight type. Both had 90 ° threads at a fixed distance to the wheel axis. Th e first had ten grooves per 1.5 cm of wheel circumference while the o t h e r had ten smaller-sized grooves per 1.0 cm of wheel circumference. The process o f knurling proceeded by fixing a length of stainless steel in a collet head on a lathe. The cylinder was s m oot h ed down to 3.0 cm in diameter. A knurling wheel was horizontally positioned on the machine such that its axis coincided with that of the cylinder. The cutting edge was then placed so that it just t o u c h e d the solid surface. A certain d e p t h was then fed into each test piece by slowly rotating t he cylinder and running t he wheel to and fro under pressure along the entire length. Subsequently, bot h the wheels and the grooves made f r om them were examined and their dimensions were measured using a travelling microscope. The accuracy of t he measurements was within 0.01 mm. Ten readings were taken each time and the average values are presented in Appendix A, Table A1. The importance attached to the inert end-caps for rotating cylinders has been d e m o n s t r a t e d [22]. T op and b o t t o m end-caps with fitting b o t t o m screws were made from polytetrafluoroethylene.
3. Experimental results Limiting current measurements were made for the s m o o t h as well as the grooved cylinders (L1 - L6) at constant electrolyte conditions. The rotational speed was varied from 200 t o 2500 rev min -1. The raw data from the experiments are presented as limiting current versus r o tatio n speed on log-log plots in Figs. 1 and 2. Sometimes in the past,
34
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Fig. 1. A double logarithmic plot of I against rotational speed for the smooth cylinder and cylinders L1 - L4: e, smooth cylinder; ×, L1; •, L2; ©, L3; +, L4.
this mode of data presentation has been employed in explaining the angular velocity dependence on transfer rate. The velocity power law (eqn. (2)) is considered to be very simplistic, especially for artificially rough or roughening surfaces as observed in dissolution experiments. There are one or two reasons for suggesting this. The first problem lies in the fact that it fails to take account of the true active surface area. A quick glance at Figs. 1 - 4 shows instantly that, prior to area correction, there is liable to be a distortion in the interpretation of the mass transfer enhancement capability o f one roughness height compared with another. The major objection to this approach will be enunciated later in this discussion. It has been noted that these test runs mark the first phase in a series of experiments. These will include two-dimensional roughness elements o f the wire-wound type and three-dimensional elements, an example of which is pyramid-type roughness. It was convenient to begin with the relatively simple V-grooves. With the aid of a travelling microscope, it was not difficult to obtain satisfactory estimates of groove dimensions and hence the effective surface area for each cylinder. Illustrative calculations and other cylinder parameters are shown in Appendix A. From the groove depth and surface area values given in Appendix A, Table A2, it was possible to calculate the limiting current density IL for
35
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2.0
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200
I
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500
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1000
2000
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ROTATIONAL SPEED (rev min-')
Fig. 2. A double logarithmic plot of I against rotational speed for cylinders L1 (x), L3 (o), L5 ([]) and L6 (v).
various Reynolds numbers using the effective diameter of each cylinder. During earlier studies in which CuSO4 in aqueous solutions was em pl oyed [1, 15], the value of the diffusion coefficient D at the stated solution conditions was given as 5.2 × 10 -6 cm 2 s-1. In the wider literature, it has been reported th at the D values for this particular system varied over a threefold range [23]. These workers have been able to show that the use of the simple Levich diffusion equation in correlating data from rotating-disk measurements is inadequate. T h e y proposed the usage o f the m i xed-cont rol Levich equation which takes into account bot h the chemical and the diffusional contributions which determine the disk current. The new corrected D value o f 6.303 × 10 -6 cm 2 s-] has been substituted in the calculation of the Sherwood n u mbe r in this paper. The Schmidt number, which is also affected by D, was held constant at 1825. 4. Discussion
4.1. Limiting current values The Schmidt n u m b e r of 1825 falls well within the range covered by the most extensive study y e t reported for mass transfer t o a rotating cylinder under t u r b u l e n t flow conditions. In this study, Eisenberg et al. [8, 9] varied
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37 Sc from 835 to 11 500 by using several diffusing species. In a technique similar to that employed by Sherwood and Ryan [24], and also by Kappesser et al. [20], it has been assumed that kL varies as Sc °'3s6, as determined by Eisenberg et al. [8, 9]. The dimensionless quantity Sh/Sc °'3s6 is plotted against Re in Figs. 3 and 4. Equation (1) for a smooth rotating cylinder is plotted in Fig. 3 and the agreement with the experimental data points is quite satisfactory. A simple least~squares curve fit confirmed the agreement with the correlation coefficient and the Reynolds number exponent. For L1 - L6, Fig. 1 shows that the maximum variation in the value of I at 2400 rev min -1 is 32.6% and exists between L1 and L4. However, the variation in kL (Fig. 3) at the corresponding value of Re is below 10%. Furthermore, Figs. 2 and 4 show a comparison between L5 and L6 and between L1 and L3. The average groove depths of L5 and L6 lie between those of the other two, noting of course that the former have a greater number of grooves. In the linear range of the graphs, I for L1 varies by 24.55% from the value for L6 at 2400 rev min -1 whereas the kL variation in the Re range is about 7.0%. For L1 and L4 the enhancement above smooth cylinder coefficients is depicted by value in data set A minus value in data set B in Fig. 3. This represents an average increase of 80%. The data points in the linear portion of Figs. 3 and 4 all lie within a 10% band. The effect of the ratio p / e appears to be negligible with respect to L1 - L4 and L5 - L6. We envisage a qualitative model of the cylinder surface in which, at low rotational speeds, the surface eddies within one groove mix with the turbulent eddies in the main body of fluid. This occurs because the scales o f the eddies are constantly changing. At higher Reynolds numbers approaching 20 000 it appears from Figs. 3 and 4 that the scale of the small eddies assumes a more constant value relative to the groove depth. Also, the rotation of the eddies within the grooves will necessarily have to be maintained by m o m e n t u m exchange with the main stream. It is assumed that this state, which can be termed "saturation", explains the apparent behaviour of L1 L6 at high Reynolds numbers and could be viewed as a " p s e u d o s m o o t h " condition. However, the closer the grooves are to each other as in L4 - L6, the greater are the chances for surface mixing, i.e. the distance y decreases and eddies in adjacent grooves may begin to interact. In the limit, of course, x=p. Ultimately, if the two-dimensional character of the roughness element gives way to a three-dimensional type (e.g. pyramidal), it can safely be assumed that the fluid distribution at the surface will be altered. The case of x = p, together with the three-dimensional situation, will shortly be investigated. The correlation of the linear portion of the L plots will be discussed in Section 4.2.
4.2. Mass transfer and its velocity dependence In view of the velocity power law (eqn. (2)) it has seemed convenient in the past to correlate rough surface data in this form where, in general, n ~ 1.0.
38 It was implied here that kL had a direct relation to angular velocity. However, it is known that when a fluid flows past a smooth surface some amount of drag is created within the turbulent region such that the velocity U0 of the fluid, just beyond a spurious boundary layer is less than the velocity U in the main stream [21]. This is also true for a rotating cylinder. It has similarly been established that the drag on the fluid by a hydrodynamically smooth boundary surface is a function of Reynolds number only [25]. By extension, when mass (or heat) transfer processes are considered, it is the velocity gradient which derives from the drag coefficient or shear stress that is of prime importance. This is because the mixing of the surface eddies are a direct result of the velocity gradients [ 26]. For a smooth rotating cylinder the quantitative m a s s - h e a t - m o m e n t u m analogy has been confirmed for turbulent flow. Sherwood and Ryan [24] showed that some theoretical justification exists for this analogy by obtaining relatively good agreement between predicted results and experimental data. For rough surfaces the shear stress is greater at the wall than for a smooth surface. It is therefore safe to predict that U0 close to a rough cylinder (as in this case} is lower than that near a smooth cylinder. Roald and Beck [ 18] have suggested that it is possible that the difference in velocity between the surface of the cylinder and the fluid just outside the boundary layer is given by the expression
Uo o: constant × U n
(3)
where U is the rate of agitation. This makes it unlikely that the exponent in the power law (n ~ 1.0) is credible. In addition, for hydrodynamically rough surfaces the drag or friction factor is independent of the Reynolds number. We are making no immediate contention that the m a s s - h e a t - m o m e n t u m analogy holds for all rough cylinders, i.e. that there exists a quantitative analogy between mass flux and the friction coefficient. It is believed, however, that the velocity distribution near the solid surface determines both the mass flux and the velocity gradients from which friction coefficients are derived. Indeed, Makrides and Hackerman [16] discovered that their experimental data from discrete roughness measurements were in limited agreement with the friction data of Theodorsen and Regier [7]. Some other workers [16, 17] have utilized the Theodorsen-Regier equation
(f/2)1/2 - 1.25 + 5.76 log
(4)
and it has been implied that the drag is a function of the relative roughness e/d. It is therefore conceivable that experimental data from mass transfer to a rough cylinder may be o f assistance in the understanding of the velocity distribution at the solid surface. The graphs in Fig. 3 for L1 at 8000 < Re < 2 0 0 0 0 and L2 at 8000 < Re < 10 000 have slopes of 1.169 and 1.036 respectively. L3 and L4 establish a more consistent flow pattern almost as early as the lowest value of Re con-
39 sidered. During the " s a t u r a t i o n " state, all six cylinders have an average slope of 0.544. As an illustration, L1 was correlated in the linear range as Sh -
8c0.356
0.567Re °'sTs
(5)
Now, let us consider cylinder L1. The implication that, for 8000 < Re 20 0 0 0 , k L ~ U 1"169 and, for 20 000 < Re < 100 000, k L cc U 0.s78 leads to our next observation. It is believed that in fact kL cc U00.s78 in the whole range 8000 - 100000. If we consider once again a simple model of the rough surface, it can be assumed that in the lower speed range the velocity gradients, which establish the drag coefficient at the wall, extend beyond the grooves into the main stream. However, when a " p s e u d o s m o o t h " state exists, the surface conditions are altered in one respect. Although the mass transfer enhancement caused by the rolling eddies within the grooves is maintained, the velocity distribution may be different. This suggests that kL may in fact be dependent on a " f r i c t i o n " velocity just beyond an arbitrary boundary from the wall (expressed as kL oc Uo") whereas, in contrast, the drag exerted on the turbulent flow in the bulk solution is not directly proportional to U0" but is related to Uo m". From the correlation obtained, n = 0.578 for the two-dimensional roughness studied here. Hence, if m n = 1.169, m ~ 2. It is apparent that, since the velocity of a fluid gradually diminishes as a wall is approached, the velocity U0 ought to be lower near a rough wall than near a smooth wall as a result of the increase in drag. Hence the earlier claims of a direct relationship between kL (or k) and U may not be an adequate explanation of the actual physical process especially in a fully rough regime. A direct quantitative comparison of mass transfer data and friction factor values may require a derivation of the latter from experiments for the particular rough surface of interest. A theoretical analysis of such complex flow distributions has not yet been achieved. It is most likely that the drag exerted by sand grain, pyramid-type and V-groove roughnesses will probably differ. As Gabe and Walsh [15] discovered, the measured JD values for pyramid-type roughness were consistently higher than the values predicted using eqn. (4). Some support for our hypothesis and observations may have been inadvertently derived from the experiments of Roald and Beck [18] on metal dissolution. They obtained an exponent o f 0.71 for their smooth rotating cylinders, but owing to a stirring action at the surface caused by gas bubbles the dissolution rate was found to be proportional to the speed of rotation to the power 1.4, i.e. k o~ U 1.4. They proposed a simple analysis which gave an approximate shearing rate d V / d x given by dV cc constant × U ~'42 dx
(6)
The explanation of this behaviour would be that, although the shearing effect of the gas bubbles does not appreciably retard fluid motion near the
40 surface, the velocity distribution has departed from that for a smooth cylinder. Therefore, as the gas bubbles would be expected to detach from the solid surface and mix into the bulk stream, the velocity close to the wall ought not to depart significantly from that for a smooth cylinder. Hence the dissolution rate is in fact proportional to the "friction" velocity to the power 0.7 or 0.71 while the shear stress is related to a velocity exponent of 1.4 or 1.42. It should be noted that this analysis applies only to a rotating cylinder and may in fact not hold for other systems such as conduits, channels, pipes or annuli. Finally, it seems worthwhile to attempt to elucidate the surface conditions during mass transfer enhancement. In diffusion-controlled reactions the reacting species (in this case, copper ions) are discharged and deposited on their arrival at the solid-electrolyte interface. Under this limiting condition the reacting-ion concentration at the surface is essentially zero. This gives the maximum cathode efficiency possible {100%) when there are no side reactions and suggests that enhancement can only be brought about by increasing the mass flux. In view of the expression [ 12] IL =
zFDCb
(7)
which is appropriate to the presence of an excess of a supporting electrolyte, it is conceivable that the enhancement brought about by small-scale eddies at the surface or by mixing from gas evolution is a direct result of a decrease in the magnitude of 6, i.e. the diffusion layer thickness. Owing to the proximity of the small-scale turbulence eddies to the cylindrical wall, the surface is assumed to be continuously scoured. This scouring action may be the possible cause of a reduction in the diffusion layer thickness, thus leading to a lower diffusional resistance to approaching ions. Otherwise, IL can only be increased by raising the value of D or Cb. The dynamic viscosity is a function of Cb while D is inversely proportional to the viscosity. Furthermore an increase can be achieved by raising the temperature, because D is also a function of temperature. A corollary to this was suggested by Owen and T h o m s o n [27] in their study of heat transfer across rough surfaces. This states that, irrespective of how much the heat capacity of a fluid beyond the surface is raised through the turbulence promoted by rough elements, heat cannot be transferred at a higher rate than conduction into the solid surface allows. The main difference between diffusion-controlled mass transfer and heat transfer at a solid wall is that for the former there exists a diffusion layer through which matter has to be transferred. The thickness of this diffusion layer invariably depends on the conditions at the solid surface. This paper serves as an introduction to an ongoing research programme in which other types of roughness will be investigated and their effect on mass transfer characterized.
41 5. Conclusions (1) By use o f V-grooved cylinders a degree o f roughness can be i n d u c e d in a s y s t e m a t i c m a n n e r , with changes in surface area and friction f a c t o r being exploited. (2) The i m p o r t a n c e o f area c o r r e c t i o n f o r r o u g h cylinders has been d e m o n s t r a t e d t h r o u g h a mass transfer correlation. (3) The mass transfer c o e f f i c i e n t t o a r o u g h c y l i n d e r has been f o u n d t o be p r o p o r t i o n a l to the surface " f r i c t i o n " velocity u n d e r " s a t u r a t e d " conditions. (4) The friction f a c t o r c o n s t a n t shows a m a r k e d increase f r o m t h a t for a s m o o t h cylinder; its precise value d e p e n d s on t h e degree o f roughness induced.
Acknowledgments The a u t h o r s wish t o t h a n k Professor I. A. Menzies f o r the provision o f l a b o r a t o r y facilities, T h e a u t h o r s also gratefully a c k n o w l e d g e financial assistance (to P.A.M.) f r o m t h e C o m m o n w e a l t h Scholarship C o m m i s s i o n in the U.K.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
D. R. Gabe and D. J, Robinson, Trans. Inst. Met. Finish., 48 (1970) 35; 49 (1971) 17. D. R. Gabe and F. C. Walsh, J. Appl. Electrochern., 13 (1983) 3. A. E. Bergles, Prog. HeatMass Transfer, 1 (1969) 331. A. E. Bergles, Appl. Mech. Rev., 26 (1973) 675. E. Brunner, Z. Phys. Chem. (Leipzig), 47 (1904) 56. G.I. Taylor, Philos. Trans. R. Soc. London, Ser. A, 223 (1923) 289. T. Theodorsen and A. Regier, Rep. 793, 1944 (National Advisory Committee for Aeronautics). M. Eisenberg, C. W. Tobias and C. R. Wilke, J. Electrochern. Soc., 101 (1954) 306. M. Eisenberg, C. W. Tobias and C. R. Wilke, Chem. Eng. Prog., Syrnp. Set., 51 (1954) 1. U. Landau, AIChE Syrnp. Ser., 77 (1981) 75. D. R. Gabe, J. Appl. Electrochern., 4 (1974)91. N. Ibl and K. Schadegg, J. Electrochern. Soc., 114 (1967) 54. B. E. Conway and J. O'M. Bockris, Proc. R. Soc. London, Ser. A, 248 (1958) 394. F. S. Holland, Chem. Ind. (London), (1978)453;Br. Patent 1,505,736, 1978. D. R. Gabe and F. C. Walsh, J. Appl. Electrochern., 14 (1984) 555,565. A. C. Makrides and N. Hackerman, J. Electrochern. Soc., 105 (1958) 156. C. V. King and P. L. Howard, Ind. Eng. Chem., 29 (1937) 75. B. Roald and W. Beck, J. Electrochern. Soc., 98 (1951} 277. V. G. Levich, Physicochernical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. R. Kappesser, I. Cornet and R. Greif, J. Electrochern. Soc., 118 (1971) 1957. J. R. Selman and C. W. Tobias,Adv. Chem. Eng., 10 (1978) 211. J. Pang and I. M. Ritchie, Electrochirn. Acta, 26 (1981} 1345. T. I. Quickenden and X. Jiang, Electrochirn. Acta, 29 (1984) 693.
42 24 25 26 27
T. K. Sherwood and J. M. Ryan, C h e m . Eng. Sci., 11 (1959) 81. S. B. Verma and J. E. Cermak, Int. J. H e a t Mass Transfer, 1 7 (1974) 567. T. K. Sherwood, Ind. Eng. C h e m . , 4 2 (1950) 2077. P. R. Owen and W. R. Thomson, J. F l u i d M e c h . , 15 (1963) 3.
A
Appendix
T h e f o r m o f t h e g r o o v e d c y l i n d e r is s h o w n s c h e m a t i c a l l y i n Fig. A 1 . T h e g r o o v e s w e r e m a c h i n e d i n a r e g u l a r p a t t e r n : L1 - L 4 h a d 6 3 g r o o v e s a n d L 5 a n d L 6 h a d 92 g r o o v e s , a n d e a c h g r o o v e h a d a r i g h t - a n g l e a p e x . B y c o n s i d e r i n g t h e c y l i n d e r a n d o n e g r o o v e (Fig. A 2 ( a ) ) t h e p i t c h is d e f i n e d as x + y . F o r 0.1 c m ~ x ~ 0 . 1 5 c m , ( x - - a ) / x ~ 0.5% a n d h e n c e x ~ a. T h e v a l u e s o f a a n d e w e r e m e a s u r e d u s i n g a g r a d u a t e d m i c r o s c o p e a n d are g i v e n in T a b l e A 1 . T h e g r o o v e p r o f i l e i t s e l f c a n b e r e g a r d e d as t h e s e g m e n t o f a circle o f r a d i u s r {Fig. A 2 ( b ) ) f r o m w h i c h it c a n b e s e e n t h a t a
I
tan 0 2e TABLE A1 Specification of grooved cylinders Cylinder
Number of grooves N
L1 L2 L3 L4 L5 L6
63 63 63 63 92 92
Groove depth e
D i a m e t e r d s - - 2e
(era)
(cm)
0.0174 0.0262 0.0404 0.0540 0.0216 0.0335
2.965 2.948 2.919 2.892 2.957 2.933
TABLE A2 ~tablishmentoftruetotalsurfaceareas Cylinder
a = 2e
r = a12 ~/2
Pitch p
y = p - - 2e
A = 2Nr
A s = Ny
(cm)
(cm)
(cm)
(cm)
× 6.3 (cm 2)
× 6.3 (cm 2 )
Total surface area A R
(era 2 )
Smooth S . . . L1 0.0348 0.0246 L2 0.0524 0.0371 L3 0.0808 0.0571 L4 0.108 0.0763 L5 0.0432 0.0305 L6 0.067 0.0474
. 0.1496 0.1496 0.1496 0.1496 0.1024 0.1024
.
. 45.564 38.579 27.307 16.511 34.312 20.518
--
0.1148 0.0972 0.0688 0.0416 0.0592 0.0354
19.527 29.450 45.326 60.567 35.356 54.946
59.38 65.091 68.029 72.633 77.078 69.668 75.464
43
Fig. A1. End view of a typical rough cylinder.
(a)
(b)
Fig. A2. Schematic diagrams of (a) a unit segment of a grooved cylinder and (b) a groove to establish the true areas.
w h e r e 0 = 45 °. Thus a = 2e and r = e × 21/2. T h e t o t a l active c i r c u m f e r e n c e is t h u s (2r + y ) N , and the surface area o f the c y l i n d e r is (2r + y)lN where N is the n u m b e r o f grooves and I is t h e length o f the cylinder. Values are given in Table A2.
A p p e n d i x B: N o m e n c l a t u r e As AR Cb ds D
f/2 F I
c a t h o d e area ( s m o o t h cylinder) (cm 2) c a t h o d e area (rough c y l i n d e r ) (cm 2 ) m e t a l ion c o n c e n t r a t i o n in bulk solution (mol cm -3) d i a m e t e r o f s m o o t h r o t a t i n g c y l i n d e r (cm) d i f f u s i o n c o e f f i c i e n t (cm 2 s -1) dimensionless friction f a c t o r 96 500 A s m o l -~, F a r a d a y c o n s t a n t limiting cell c u r r e n t (A)
44 IL JD k kb m n p Re Sc Sh U Uo x z
limiting current density (mA cm -2) ShSc c, dimensionless mass transport factor specific rate constant (cm s -1) IL/ZFCb, mass transfer coefficient (cm s-x) exponent exponent pitch (roughness spacing) (cm) Ud/v, Reynolds number y/D, Schmidt number kLd/D, Sherwood number peripheral velocity of cylinder (cm s -i) "friction" velocity (cm s-1) groove width (cm) valence change
Greek symbols diffusion layer thickness (cm) e groove depth (characteristic roughness parameter) (cm) 77 dynamic viscosity (g cm -1 s-1) v p/~?, kinematic viscosity (cm 2 s-x) p fluid density (g cm -3)
29
A STUDY OF ROUGHNESS AND MASS T R A N S F E R ENHANCEMENT FOR THE R O T A T I N G CYLINDER E L E C T R O D E * P. A. MAKANJUOLA and D. R. GABE Department of Materials Engineering and Design, University of Technology, Loughborough, Leics. LE11 3TU (Gt. Britain) (Received March 29, 1984; in revised form June 21, 1984)
Summary It is well established that enhanced mass transfer occurs at a rotating. cylinder electrode operated in the t ur bu l ent regime when cathodically deposited metal grows in a rough p o w d e r y or dendritic manner. In the absence o f quantitative data for the degree of roughness and t he consequent increase in the friction factor, such e nha nc em ent has been expressed empirically as an increase in the power e x p o n e n t of a simple mass transfer relationship K L = constant × U n where n is normally a b o u t 0.7 in t ur bul ent flow for a s m o o t h electrode. By using cylinder electrodes o f known induced roughness, which can be characterized by a true surface area, t o p o g r a p h y etc., it is possible to distinguish between the effects of increased surface area and microturbulence, whence it becomes possible to recognize and devise optimal m et hods of increasing the surface roughness in order to maximize mass transfer. For t he V-grooved cylinders investigated, the saturated or m a x i m u m enhanced mass transfer was correlated according to the pow er law relationship where n varied between 0.51 and 0.58. For six such cylinders the average increase in mass transfer at a high velocity, or degree o f enhancement, was 80% relative to a s m o o t h cylinder.
1. I n t r o d u c t i o n The work r epor t e d here is a cont i nuat i on of earlier studies concerned with the d e v e l o p m e n t of dendritic and powder electrodeposition o f metals on rotating cylinder electrodes [1, 2], which are known to allow enhanced mass transfer to develop during operation. Wide-ranging m et hods have been used in attemp ts to p r o m o t e higher mass or heat transfer rates or to reduce friction coefficients in flow systems. It is apparent t hat t he greatest strides have been achieved in t he field of heat transfer as t he t w o major reviews by Bergles [ 3 , 4 ] show. Some o f the m a ny augmentation techniques discussed *Paper presented at a meeting on Engineering Aspects of Electrochemistry, Loughborough, Gt. Britain, April 17, 1984. 0376-4583/85/$3.30
© Elsevier Sequoia/Printed in The Netherlands
30 for improving forced convection heat transfer include surface vibration, fluid vibration, mechanical aids, swirl flow devices, rough surfaces and surface tension devices. It is worth mentioning that only a handful of these methods are in any serious contention for intensification purposes, and two of these are considered in this paper. They come under the headings of mechanical aids (mechanical agitation of the cylinder electrode) and rough surfaces. The first known report on a rotating cylinder was by Brunner [5] whose observations and conclusions were generally qualitative in nature. Following Brunner, the established configuration was two concentric cylinders with the inner one rotating and the outer stationary, and Taylor [6] was the first to study fluid stability within the annular gap formed by such coaxial cylinders. Thereafter, Theodorsen and Regier [7 ] published friction data based on both smooth and rough inner rotating cylinders. In 1954, Eisenberg e t al. [8, 9] conducted the most thorough study to date on mass transfer to an inner rotating cylinder electrode with the outer cylinder serving as the counterelectrode. Their data for turbulent flow regimes were correlated by Sh = 0.079Re°'7°Sc °'a56
(1)
where Sh, Re and Sc are the Sherwood, Reynolds and Schmidt numbers respectively. A comparison of the above correlation with the friction results obtained by Theodorsen and Regier [7] for a smooth cylinder confirmed quite adequately the analogy between mass and m o m e n t u m transfer for this system and also confirmed that the critical dimension is the diameter of the inner cylinder. The technique for mass transfer measurement employed by Eisenberg et al. [8] was the electrochemical limiting current m e t h o d utilizing the ferriferrocyanide redox couple in alkaline solutions. Certain electrodeposition reaction systems can also be employed. The limiting current technique and its m a n y advantages over conventional mass and heat transfer measurement methods have been discussed by Landau [ 10] amongst others. A concise review of the rotating cylinder electrode was produced by Gabe [11]. Among many other topics, he discussed the emerging interest in the use of such an agitation cell in metal-finishing processes and for electrowinning metals from solutions of increasing dilution. The rotating cylinder causes fluid instability at quite low Reynolds numbers {below 200) and hence achieves high turbulence at moderate operating speeds. This characteristic is a useful attribute for maximizing electrodeposition rates while recovering metals from effluent solutions. Laminar flow conditions will not be considered for this reason. It is well established that a rough granular deposit results during the approach to and within the limiting current range when copper deposition at the cathode is the chosen system. Ibl and Schadegg [12] and Conway and Bockris [13] have studied this effect. Such a dendritic deposit is known to cause an enhancement in mass transfer coefficients as reported by Holland [14] and Gabe and Walsh [15]. Makrides and Hackerman [16] and King and Howard [17] observed a similar phenomenon during metal dissolution experiments. Such reactions have been proved to be
31 diffusion controlled [18]. The increase in dissolution rates was accompanied by a gradual surface roughening which resulted in a form of discrete roughness. Holland [14] proposed a simple velocity power law K (or I) = constant × U n
(2)
where n = 0.7 for a smooth rotating cylinder and n -+ 1.0 for a roughened cylinder. As we shall attempt to show, this power law has certain limitations. Makrides and Hackerman [16] and other workers have attributed the increase in mass transfer to an increase in the reaction surface area. However, as Levich [19] pointed out, if the average height o f the protuberances exceeded that of the hydrodynamic viscous sublayer, microturbulence will be expected at the solid-liquid interface. These irregular roughnesses are understandably difficult to characterize. Bergles [3, 4] suggested the use of artificially roughened surfaces produced by machining techniques, an example of which is the work of Kappesser et al. [20] who resorted to knurled pyramid-type roughness elements. As pointed out by Bergles [3], because of the variety of geometrical configurations it has not been possible to adopt a unified theory or a broad correlation for the effects of surface roughness on single-phase heat transfer; neither has it been achieved for mass transfer. In the present study we attempt to separate the effects due to interface microturbulence from any enlargement of the surface area. Various roughness types will eventually be tested during the duration of this research in order to understand what effects the critical roughness dimension, i.e. the size or height, and the orientation of the element have on an enhancement factor which is a function of these variables. Of these, only one is reported here, i.e. V-grooves normal to the direction of fluid flow. All the data will be obtained within a controlled environment to ensure that all the results are directly comparable.
2. Experimental details 2.1. The ro rating cylinder electrode rig The present rig has been described in detail elsewhere [15]. The general structure remains unaltered although a few adjustments have been made to accommodate the experiments reported here. In its original state the squaresectioned catholyte compartment was separated from the anolyte section by four cation exchange membranes, one on each side. The removable 1 1 catholyte chamber could be positioned in a 20 1 bath containing the supporting electrolyte. In this divided cell, lead alloy anodes {anodized to lead dioxide during use) were placed along the walls of the bath. In this form the cell is excellent for batch concentration decay experiments or for fast constant-potential mass transfer runs. However, for constant-concentration test runs a cylindrical copper anode of 8.0 cm diameter was fabricated. This was positioned inside the catholyte compartment such t h a t it could easily be
32 lifted o u t b e t w e e n successive runs. This could be d e s c r i b e d as an u n d i v i d e d cell w h e r e b y t h e electrical c o n t a c t s to t h e lead d i o x i d e a n o d e s w e r e disconn e c t e d while t h e adjoining b a t h served o n l y as a t e m p e r a t u r e c o n t r o l . This w h o l e a s s e m b l y c o u l d be raised vertically to enclose a test cylinder, w h i c h in t u r n c o u l d be u n s c r e w e d f r o m a rigid r o t o r s t e m b e t w e e n runs. T h e r o t o r was driven b y a 1/8 hp S e r v o m e x C o n t r o l s m o t o r . An auxiliary t a c h o m e t e r was s i t u a t e d e x t e r n a l l y to m a i n t a i n a c h e c k o n t h e r o t a t i o n a l speed. Electrical s u p p l y was c o n t r o l l e d using a Chemical E l e c t r o n i c s 20 V, 50 A m o d e l p o t e n t i o s t a t and fed t h r o u g h t w o silver-filled g r a p h i t e b r u s h e s to t h e c a t h o d e . An a d d i t i o n a l f e a t u r e was a p o t e n t i a l p i c k - u p b r u s h in c o n t a c t with a silver-plated c o p p e r ring. T h e ring was situated b e t w e e n t h e b r u s h a s s e m b l y and t h e c a t h o d e . T h e p i c k - u p b r u s h m o n i t o r e d p o t e n t i a l readings ( o n a digital m i l l i v o l t m e t e r ) b e t w e e n an H g - H g S O 4 (MMS) r e f e r e n c e e l e c t r o d e and t h e cylindrical c a t h o d e while eliminating t h e p o t e n t i a l d r o p across t h e b r u s h a s s e m b l y w h i c h resulted f r o m a c o n t a c t resistance g e n e r a t e d b e t w e e n t h e b r u s h e s and t h e c o m m u t a t o r d u r i n g r o t a t i o n .
2.2. Experimental procedure F o r d i f f u s i o n - c o n t r o l l e d m a s s t r a n s f e r d e t e r m i n a t i o n s t h e acidified CuSO4 s y s t e m was selected. A relatively dilute 0 . 0 1 4 M CuSO4 solution w i t h 1.5 M H 2 S O 4 as t h e s u p p o r t i n g e l e c t r o l y t e was e m p l o y e d at 22 °C for all test runs. A 50 1 b a t c h o f solution was p r e p a r e d using d e i o n i z e d w a t e r and t h e c o p p e r ion c o n t e n t o f 8 8 9 p p m was m e a s u r e d t o within +10 p p m using an a t o m i c a b s o r p t i o n s p e c t r o p h o t o m e t e r . T h e t e m p e r a t u r e in t h e c a t h o l y t e c h a m b e r was c o n t r o l l e d b y a t h e r m o s t a t i c b a t h t o within + 0.5 °C. T h e s m o o t h t e s t c y l i n d e r was a b r a d e d w i t h fine e m e r y p a p e r , w a s h e d in a c o m m e r c i a l cleanser and t h e n rinsed in water. It was t h e n d e g r e a s e d b y cleaning w i t h 1 , 1 , 1 - t r i c h l o r o e t h a n e following w h i c h it was again rinsed t h o r o u g h l y w i t h d e i o n i z e d water. A f t e r this, it was p l a c e d in a b e a k e r containing 1.5 M H 2 SO4 for a f e w seconds. It was rinsed o n c e again in d e i o n i z e d w a t e r and was t h e n a t t a c h e d to t h e r o t o r r e a d y f o r t h e s t a r t o f a run. T h e s a m e p r o c e d u r e was a d o p t e d f o r t h e e x p e r i m e n t a l e l e c t r o d e s L1 - L6 e x c e p t t h a t t h e y w e r e n o t a b r a d e d w i t h fine e m e r y p a p e r . T h e t e s t s o l u t i o n was r e p l a c e d w i t h a fresh 1 1 s o l u t i o n b e t w e e n successive runs and the large v o l u m e o f p r e p a r e d e l e c t r o l y t e was e x p e c t e d to e l i m i n a t e variations in m e t a l c o n c e n t r a t i o n . T h e c o p p e r ion c o n c e n t r a t i o n o f t h e s p e n t s o l u t i o n was r e c h e c k e d a f t e r every c y c l e o f o p e r a t i o n . During m a s s t r a n s f e r runs a soluble c o p p e r a n o d e , w h i c h was c o n c e n t r i c w i t h t h e c a t h o d e , was used. A t o t h e r t i m e s t h e divided cell was r e t a i n e d f o r certain c o n s t a n t - p o t e n t i a l runs. A t t h e start o f a r u n t h e c y l i n d e r was p r e p l a t e d at 300 rev m i n -1 for up to 10 m i n at a relatively low p o t e n t i a l ( - - 6 5 0 mV) relative to an MMS r e f e r e n c e e l e c t r o d e c o n n e c t e d via a 1.0 M Na2SO4 salt bridge s o l u t i o n . P o l a r i z a t i o n curves w e r e o b t a i n e d b y e m p l o y i n g a scan r a t e o f 150 m V m i n -1 f r o m a s w e e p g e n e r a t o r . H y d r o g e n gas e v o l u t i o n c o m m e n c e d a f t e r a b o u t 3 0 0 s and t h e run was t e r m i n a t e d s h o r t l y a f t e r w a r d s .
33 24 runs each were carried out for the sm oot h cylinder and for the electrodes L1 - L4, and 12 each were p e r f o r m e d for t he electrodes L5 and L6. The polarization scans were similar for bot h t he sm oot h and the rough cylinders. As a check on the limiting current values obtained, potentials within the limiting current plateau range were instantaneously impressed on the cathode [21] and steady state current readings were achieved within a bo u t 30 s. The pick-up brush was invaluable in helping to pinpoint a required potential value relative to the MMS electrode connect ed via the 1.0 M Na2 SO4 solution. Th e readings from bot h m e t hods of measurement were in satisfactory agreement.
2.3. Production of cylinders The geometrical similarity o f the grooves was considered to be of p a r a m o u n t importance. Therefore, although the segment lying between two adjacent grooves constituted the roughness element, it was in fact the grooves which were of prime interest as far as mass transfer to the surface was concerned. One s m o o t h and six V-grooved cylinders were made. The sm oot h test cylinder, o f stainless steel material, was 6.3 cm long and 3.0 cm in diameter as were the o t h e r cylinders before the knurling process. Two knurling wheels were employed, both of the straight type. Both had 90 ° threads at a fixed distance to the wheel axis. Th e first had ten grooves per 1.5 cm of wheel circumference while the o t h e r had ten smaller-sized grooves per 1.0 cm of wheel circumference. The process o f knurling proceeded by fixing a length of stainless steel in a collet head on a lathe. The cylinder was s m oot h ed down to 3.0 cm in diameter. A knurling wheel was horizontally positioned on the machine such that its axis coincided with that of the cylinder. The cutting edge was then placed so that it just t o u c h e d the solid surface. A certain d e p t h was then fed into each test piece by slowly rotating t he cylinder and running t he wheel to and fro under pressure along the entire length. Subsequently, bot h the wheels and the grooves made f r om them were examined and their dimensions were measured using a travelling microscope. The accuracy of t he measurements was within 0.01 mm. Ten readings were taken each time and the average values are presented in Appendix A, Table A1. The importance attached to the inert end-caps for rotating cylinders has been d e m o n s t r a t e d [22]. T op and b o t t o m end-caps with fitting b o t t o m screws were made from polytetrafluoroethylene.
3. Experimental results Limiting current measurements were made for the s m o o t h as well as the grooved cylinders (L1 - L6) at constant electrolyte conditions. The rotational speed was varied from 200 t o 2500 rev min -1. The raw data from the experiments are presented as limiting current versus r o tatio n speed on log-log plots in Figs. 1 and 2. Sometimes in the past,
34
÷÷4t"q,÷
~.
2.0
+e
•
OA~
4, ÷ ÷
0
~- 1.0
0 A X
0 4, X
0 0
O
oO 8a~XXX ~ XX XX x
x
o~x O ~. X
b. X • • Oo ° O o
o O O
X O o
_=, 0.S
0,2
100
X
•
i
i
200
I
i
I
i
i
I
I
500 1000 ROTATE)NAL SPEED(rev rnin"}
I
i
2000
Fig. 1. A double logarithmic plot of I against rotational speed for the smooth cylinder and cylinders L1 - L4: e, smooth cylinder; ×, L1; •, L2; ©, L3; +, L4.
this mode of data presentation has been employed in explaining the angular velocity dependence on transfer rate. The velocity power law (eqn. (2)) is considered to be very simplistic, especially for artificially rough or roughening surfaces as observed in dissolution experiments. There are one or two reasons for suggesting this. The first problem lies in the fact that it fails to take account of the true active surface area. A quick glance at Figs. 1 - 4 shows instantly that, prior to area correction, there is liable to be a distortion in the interpretation of the mass transfer enhancement capability o f one roughness height compared with another. The major objection to this approach will be enunciated later in this discussion. It has been noted that these test runs mark the first phase in a series of experiments. These will include two-dimensional roughness elements o f the wire-wound type and three-dimensional elements, an example of which is pyramid-type roughness. It was convenient to begin with the relatively simple V-grooves. With the aid of a travelling microscope, it was not difficult to obtain satisfactory estimates of groove dimensions and hence the effective surface area for each cylinder. Illustrative calculations and other cylinder parameters are shown in Appendix A. From the groove depth and surface area values given in Appendix A, Table A2, it was possible to calculate the limiting current density IL for
35
3.0
2.0
v
v o 0 v
0
~ 0 o
~o
K~x xx
o 0 0 Xx X X
0 X
X
x
x
to 0
2
x
0.5
0.2
i
100
200
I
i
i
500
i
i
i
i
i
i
1000
2000
i
ROTATIONAL SPEED (rev min-')
Fig. 2. A double logarithmic plot of I against rotational speed for cylinders L1 (x), L3 (o), L5 ([]) and L6 (v).
various Reynolds numbers using the effective diameter of each cylinder. During earlier studies in which CuSO4 in aqueous solutions was em pl oyed [1, 15], the value of the diffusion coefficient D at the stated solution conditions was given as 5.2 × 10 -6 cm 2 s-1. In the wider literature, it has been reported th at the D values for this particular system varied over a threefold range [23]. These workers have been able to show that the use of the simple Levich diffusion equation in correlating data from rotating-disk measurements is inadequate. T h e y proposed the usage o f the m i xed-cont rol Levich equation which takes into account bot h the chemical and the diffusional contributions which determine the disk current. The new corrected D value o f 6.303 × 10 -6 cm 2 s-] has been substituted in the calculation of the Sherwood n u mbe r in this paper. The Schmidt number, which is also affected by D, was held constant at 1825. 4. Discussion
4.1. Limiting current values The Schmidt n u m b e r of 1825 falls well within the range covered by the most extensive study y e t reported for mass transfer t o a rotating cylinder under t u r b u l e n t flow conditions. In this study, Eisenberg et al. [8, 9] varied
36
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37 Sc from 835 to 11 500 by using several diffusing species. In a technique similar to that employed by Sherwood and Ryan [24], and also by Kappesser et al. [20], it has been assumed that kL varies as Sc °'3s6, as determined by Eisenberg et al. [8, 9]. The dimensionless quantity Sh/Sc °'3s6 is plotted against Re in Figs. 3 and 4. Equation (1) for a smooth rotating cylinder is plotted in Fig. 3 and the agreement with the experimental data points is quite satisfactory. A simple least~squares curve fit confirmed the agreement with the correlation coefficient and the Reynolds number exponent. For L1 - L6, Fig. 1 shows that the maximum variation in the value of I at 2400 rev min -1 is 32.6% and exists between L1 and L4. However, the variation in kL (Fig. 3) at the corresponding value of Re is below 10%. Furthermore, Figs. 2 and 4 show a comparison between L5 and L6 and between L1 and L3. The average groove depths of L5 and L6 lie between those of the other two, noting of course that the former have a greater number of grooves. In the linear range of the graphs, I for L1 varies by 24.55% from the value for L6 at 2400 rev min -1 whereas the kL variation in the Re range is about 7.0%. For L1 and L4 the enhancement above smooth cylinder coefficients is depicted by value in data set A minus value in data set B in Fig. 3. This represents an average increase of 80%. The data points in the linear portion of Figs. 3 and 4 all lie within a 10% band. The effect of the ratio p / e appears to be negligible with respect to L1 - L4 and L5 - L6. We envisage a qualitative model of the cylinder surface in which, at low rotational speeds, the surface eddies within one groove mix with the turbulent eddies in the main body of fluid. This occurs because the scales o f the eddies are constantly changing. At higher Reynolds numbers approaching 20 000 it appears from Figs. 3 and 4 that the scale of the small eddies assumes a more constant value relative to the groove depth. Also, the rotation of the eddies within the grooves will necessarily have to be maintained by m o m e n t u m exchange with the main stream. It is assumed that this state, which can be termed "saturation", explains the apparent behaviour of L1 L6 at high Reynolds numbers and could be viewed as a " p s e u d o s m o o t h " condition. However, the closer the grooves are to each other as in L4 - L6, the greater are the chances for surface mixing, i.e. the distance y decreases and eddies in adjacent grooves may begin to interact. In the limit, of course, x=p. Ultimately, if the two-dimensional character of the roughness element gives way to a three-dimensional type (e.g. pyramidal), it can safely be assumed that the fluid distribution at the surface will be altered. The case of x = p, together with the three-dimensional situation, will shortly be investigated. The correlation of the linear portion of the L plots will be discussed in Section 4.2.
4.2. Mass transfer and its velocity dependence In view of the velocity power law (eqn. (2)) it has seemed convenient in the past to correlate rough surface data in this form where, in general, n ~ 1.0.
38 It was implied here that kL had a direct relation to angular velocity. However, it is known that when a fluid flows past a smooth surface some amount of drag is created within the turbulent region such that the velocity U0 of the fluid, just beyond a spurious boundary layer is less than the velocity U in the main stream [21]. This is also true for a rotating cylinder. It has similarly been established that the drag on the fluid by a hydrodynamically smooth boundary surface is a function of Reynolds number only [25]. By extension, when mass (or heat) transfer processes are considered, it is the velocity gradient which derives from the drag coefficient or shear stress that is of prime importance. This is because the mixing of the surface eddies are a direct result of the velocity gradients [ 26]. For a smooth rotating cylinder the quantitative m a s s - h e a t - m o m e n t u m analogy has been confirmed for turbulent flow. Sherwood and Ryan [24] showed that some theoretical justification exists for this analogy by obtaining relatively good agreement between predicted results and experimental data. For rough surfaces the shear stress is greater at the wall than for a smooth surface. It is therefore safe to predict that U0 close to a rough cylinder (as in this case} is lower than that near a smooth cylinder. Roald and Beck [ 18] have suggested that it is possible that the difference in velocity between the surface of the cylinder and the fluid just outside the boundary layer is given by the expression
Uo o: constant × U n
(3)
where U is the rate of agitation. This makes it unlikely that the exponent in the power law (n ~ 1.0) is credible. In addition, for hydrodynamically rough surfaces the drag or friction factor is independent of the Reynolds number. We are making no immediate contention that the m a s s - h e a t - m o m e n t u m analogy holds for all rough cylinders, i.e. that there exists a quantitative analogy between mass flux and the friction coefficient. It is believed, however, that the velocity distribution near the solid surface determines both the mass flux and the velocity gradients from which friction coefficients are derived. Indeed, Makrides and Hackerman [16] discovered that their experimental data from discrete roughness measurements were in limited agreement with the friction data of Theodorsen and Regier [7]. Some other workers [16, 17] have utilized the Theodorsen-Regier equation
(f/2)1/2 - 1.25 + 5.76 log
(4)
and it has been implied that the drag is a function of the relative roughness e/d. It is therefore conceivable that experimental data from mass transfer to a rough cylinder may be o f assistance in the understanding of the velocity distribution at the solid surface. The graphs in Fig. 3 for L1 at 8000 < Re < 2 0 0 0 0 and L2 at 8000 < Re < 10 000 have slopes of 1.169 and 1.036 respectively. L3 and L4 establish a more consistent flow pattern almost as early as the lowest value of Re con-
39 sidered. During the " s a t u r a t i o n " state, all six cylinders have an average slope of 0.544. As an illustration, L1 was correlated in the linear range as Sh -
8c0.356
0.567Re °'sTs
(5)
Now, let us consider cylinder L1. The implication that, for 8000 < Re 20 0 0 0 , k L ~ U 1"169 and, for 20 000 < Re < 100 000, k L cc U 0.s78 leads to our next observation. It is believed that in fact kL cc U00.s78 in the whole range 8000 - 100000. If we consider once again a simple model of the rough surface, it can be assumed that in the lower speed range the velocity gradients, which establish the drag coefficient at the wall, extend beyond the grooves into the main stream. However, when a " p s e u d o s m o o t h " state exists, the surface conditions are altered in one respect. Although the mass transfer enhancement caused by the rolling eddies within the grooves is maintained, the velocity distribution may be different. This suggests that kL may in fact be dependent on a " f r i c t i o n " velocity just beyond an arbitrary boundary from the wall (expressed as kL oc Uo") whereas, in contrast, the drag exerted on the turbulent flow in the bulk solution is not directly proportional to U0" but is related to Uo m". From the correlation obtained, n = 0.578 for the two-dimensional roughness studied here. Hence, if m n = 1.169, m ~ 2. It is apparent that, since the velocity of a fluid gradually diminishes as a wall is approached, the velocity U0 ought to be lower near a rough wall than near a smooth wall as a result of the increase in drag. Hence the earlier claims of a direct relationship between kL (or k) and U may not be an adequate explanation of the actual physical process especially in a fully rough regime. A direct quantitative comparison of mass transfer data and friction factor values may require a derivation of the latter from experiments for the particular rough surface of interest. A theoretical analysis of such complex flow distributions has not yet been achieved. It is most likely that the drag exerted by sand grain, pyramid-type and V-groove roughnesses will probably differ. As Gabe and Walsh [15] discovered, the measured JD values for pyramid-type roughness were consistently higher than the values predicted using eqn. (4). Some support for our hypothesis and observations may have been inadvertently derived from the experiments of Roald and Beck [18] on metal dissolution. They obtained an exponent o f 0.71 for their smooth rotating cylinders, but owing to a stirring action at the surface caused by gas bubbles the dissolution rate was found to be proportional to the speed of rotation to the power 1.4, i.e. k o~ U 1.4. They proposed a simple analysis which gave an approximate shearing rate d V / d x given by dV cc constant × U ~'42 dx
(6)
The explanation of this behaviour would be that, although the shearing effect of the gas bubbles does not appreciably retard fluid motion near the
40 surface, the velocity distribution has departed from that for a smooth cylinder. Therefore, as the gas bubbles would be expected to detach from the solid surface and mix into the bulk stream, the velocity close to the wall ought not to depart significantly from that for a smooth cylinder. Hence the dissolution rate is in fact proportional to the "friction" velocity to the power 0.7 or 0.71 while the shear stress is related to a velocity exponent of 1.4 or 1.42. It should be noted that this analysis applies only to a rotating cylinder and may in fact not hold for other systems such as conduits, channels, pipes or annuli. Finally, it seems worthwhile to attempt to elucidate the surface conditions during mass transfer enhancement. In diffusion-controlled reactions the reacting species (in this case, copper ions) are discharged and deposited on their arrival at the solid-electrolyte interface. Under this limiting condition the reacting-ion concentration at the surface is essentially zero. This gives the maximum cathode efficiency possible {100%) when there are no side reactions and suggests that enhancement can only be brought about by increasing the mass flux. In view of the expression [ 12] IL =
zFDCb
(7)
which is appropriate to the presence of an excess of a supporting electrolyte, it is conceivable that the enhancement brought about by small-scale eddies at the surface or by mixing from gas evolution is a direct result of a decrease in the magnitude of 6, i.e. the diffusion layer thickness. Owing to the proximity of the small-scale turbulence eddies to the cylindrical wall, the surface is assumed to be continuously scoured. This scouring action may be the possible cause of a reduction in the diffusion layer thickness, thus leading to a lower diffusional resistance to approaching ions. Otherwise, IL can only be increased by raising the value of D or Cb. The dynamic viscosity is a function of Cb while D is inversely proportional to the viscosity. Furthermore an increase can be achieved by raising the temperature, because D is also a function of temperature. A corollary to this was suggested by Owen and T h o m s o n [27] in their study of heat transfer across rough surfaces. This states that, irrespective of how much the heat capacity of a fluid beyond the surface is raised through the turbulence promoted by rough elements, heat cannot be transferred at a higher rate than conduction into the solid surface allows. The main difference between diffusion-controlled mass transfer and heat transfer at a solid wall is that for the former there exists a diffusion layer through which matter has to be transferred. The thickness of this diffusion layer invariably depends on the conditions at the solid surface. This paper serves as an introduction to an ongoing research programme in which other types of roughness will be investigated and their effect on mass transfer characterized.
41 5. Conclusions (1) By use o f V-grooved cylinders a degree o f roughness can be i n d u c e d in a s y s t e m a t i c m a n n e r , with changes in surface area and friction f a c t o r being exploited. (2) The i m p o r t a n c e o f area c o r r e c t i o n f o r r o u g h cylinders has been d e m o n s t r a t e d t h r o u g h a mass transfer correlation. (3) The mass transfer c o e f f i c i e n t t o a r o u g h c y l i n d e r has been f o u n d t o be p r o p o r t i o n a l to the surface " f r i c t i o n " velocity u n d e r " s a t u r a t e d " conditions. (4) The friction f a c t o r c o n s t a n t shows a m a r k e d increase f r o m t h a t for a s m o o t h cylinder; its precise value d e p e n d s on t h e degree o f roughness induced.
Acknowledgments The a u t h o r s wish t o t h a n k Professor I. A. Menzies f o r the provision o f l a b o r a t o r y facilities, T h e a u t h o r s also gratefully a c k n o w l e d g e financial assistance (to P.A.M.) f r o m t h e C o m m o n w e a l t h Scholarship C o m m i s s i o n in the U.K.
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
D. R. Gabe and D. J, Robinson, Trans. Inst. Met. Finish., 48 (1970) 35; 49 (1971) 17. D. R. Gabe and F. C. Walsh, J. Appl. Electrochern., 13 (1983) 3. A. E. Bergles, Prog. HeatMass Transfer, 1 (1969) 331. A. E. Bergles, Appl. Mech. Rev., 26 (1973) 675. E. Brunner, Z. Phys. Chem. (Leipzig), 47 (1904) 56. G.I. Taylor, Philos. Trans. R. Soc. London, Ser. A, 223 (1923) 289. T. Theodorsen and A. Regier, Rep. 793, 1944 (National Advisory Committee for Aeronautics). M. Eisenberg, C. W. Tobias and C. R. Wilke, J. Electrochern. Soc., 101 (1954) 306. M. Eisenberg, C. W. Tobias and C. R. Wilke, Chem. Eng. Prog., Syrnp. Set., 51 (1954) 1. U. Landau, AIChE Syrnp. Ser., 77 (1981) 75. D. R. Gabe, J. Appl. Electrochern., 4 (1974)91. N. Ibl and K. Schadegg, J. Electrochern. Soc., 114 (1967) 54. B. E. Conway and J. O'M. Bockris, Proc. R. Soc. London, Ser. A, 248 (1958) 394. F. S. Holland, Chem. Ind. (London), (1978)453;Br. Patent 1,505,736, 1978. D. R. Gabe and F. C. Walsh, J. Appl. Electrochern., 14 (1984) 555,565. A. C. Makrides and N. Hackerman, J. Electrochern. Soc., 105 (1958) 156. C. V. King and P. L. Howard, Ind. Eng. Chem., 29 (1937) 75. B. Roald and W. Beck, J. Electrochern. Soc., 98 (1951} 277. V. G. Levich, Physicochernical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. R. Kappesser, I. Cornet and R. Greif, J. Electrochern. Soc., 118 (1971) 1957. J. R. Selman and C. W. Tobias,Adv. Chem. Eng., 10 (1978) 211. J. Pang and I. M. Ritchie, Electrochirn. Acta, 26 (1981} 1345. T. I. Quickenden and X. Jiang, Electrochirn. Acta, 29 (1984) 693.
42 24 25 26 27
T. K. Sherwood and J. M. Ryan, C h e m . Eng. Sci., 11 (1959) 81. S. B. Verma and J. E. Cermak, Int. J. H e a t Mass Transfer, 1 7 (1974) 567. T. K. Sherwood, Ind. Eng. C h e m . , 4 2 (1950) 2077. P. R. Owen and W. R. Thomson, J. F l u i d M e c h . , 15 (1963) 3.
A
Appendix
T h e f o r m o f t h e g r o o v e d c y l i n d e r is s h o w n s c h e m a t i c a l l y i n Fig. A 1 . T h e g r o o v e s w e r e m a c h i n e d i n a r e g u l a r p a t t e r n : L1 - L 4 h a d 6 3 g r o o v e s a n d L 5 a n d L 6 h a d 92 g r o o v e s , a n d e a c h g r o o v e h a d a r i g h t - a n g l e a p e x . B y c o n s i d e r i n g t h e c y l i n d e r a n d o n e g r o o v e (Fig. A 2 ( a ) ) t h e p i t c h is d e f i n e d as x + y . F o r 0.1 c m ~ x ~ 0 . 1 5 c m , ( x - - a ) / x ~ 0.5% a n d h e n c e x ~ a. T h e v a l u e s o f a a n d e w e r e m e a s u r e d u s i n g a g r a d u a t e d m i c r o s c o p e a n d are g i v e n in T a b l e A 1 . T h e g r o o v e p r o f i l e i t s e l f c a n b e r e g a r d e d as t h e s e g m e n t o f a circle o f r a d i u s r {Fig. A 2 ( b ) ) f r o m w h i c h it c a n b e s e e n t h a t a
I
tan 0 2e TABLE A1 Specification of grooved cylinders Cylinder
Number of grooves N
L1 L2 L3 L4 L5 L6
63 63 63 63 92 92
Groove depth e
D i a m e t e r d s - - 2e
(era)
(cm)
0.0174 0.0262 0.0404 0.0540 0.0216 0.0335
2.965 2.948 2.919 2.892 2.957 2.933
TABLE A2 ~tablishmentoftruetotalsurfaceareas Cylinder
a = 2e
r = a12 ~/2
Pitch p
y = p - - 2e
A = 2Nr
A s = Ny
(cm)
(cm)
(cm)
(cm)
× 6.3 (cm 2)
× 6.3 (cm 2 )
Total surface area A R
(era 2 )
Smooth S . . . L1 0.0348 0.0246 L2 0.0524 0.0371 L3 0.0808 0.0571 L4 0.108 0.0763 L5 0.0432 0.0305 L6 0.067 0.0474
. 0.1496 0.1496 0.1496 0.1496 0.1024 0.1024
.
. 45.564 38.579 27.307 16.511 34.312 20.518
--
0.1148 0.0972 0.0688 0.0416 0.0592 0.0354
19.527 29.450 45.326 60.567 35.356 54.946
59.38 65.091 68.029 72.633 77.078 69.668 75.464
43
Fig. A1. End view of a typical rough cylinder.
(a)
(b)
Fig. A2. Schematic diagrams of (a) a unit segment of a grooved cylinder and (b) a groove to establish the true areas.
w h e r e 0 = 45 °. Thus a = 2e and r = e × 21/2. T h e t o t a l active c i r c u m f e r e n c e is t h u s (2r + y ) N , and the surface area o f the c y l i n d e r is (2r + y)lN where N is the n u m b e r o f grooves and I is t h e length o f the cylinder. Values are given in Table A2.
A p p e n d i x B: N o m e n c l a t u r e As AR Cb ds D
f/2 F I
c a t h o d e area ( s m o o t h cylinder) (cm 2) c a t h o d e area (rough c y l i n d e r ) (cm 2 ) m e t a l ion c o n c e n t r a t i o n in bulk solution (mol cm -3) d i a m e t e r o f s m o o t h r o t a t i n g c y l i n d e r (cm) d i f f u s i o n c o e f f i c i e n t (cm 2 s -1) dimensionless friction f a c t o r 96 500 A s m o l -~, F a r a d a y c o n s t a n t limiting cell c u r r e n t (A)
44 IL JD k kb m n p Re Sc Sh U Uo x z
limiting current density (mA cm -2) ShSc c, dimensionless mass transport factor specific rate constant (cm s -1) IL/ZFCb, mass transfer coefficient (cm s-x) exponent exponent pitch (roughness spacing) (cm) Ud/v, Reynolds number y/D, Schmidt number kLd/D, Sherwood number peripheral velocity of cylinder (cm s -i) "friction" velocity (cm s-1) groove width (cm) valence change
Greek symbols diffusion layer thickness (cm) e groove depth (characteristic roughness parameter) (cm) 77 dynamic viscosity (g cm -1 s-1) v p/~?, kinematic viscosity (cm 2 s-x) p fluid density (g cm -3)