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Mass transfer inside conical cavities under transverse laminar flow
Chemical Engineering and Processing 44 (2005) 1306–1311
Mass transfer inside conical cavities under transverse laminar flow M.M. Zaki a , I. Nirdosh a,∗ , G.H. Sedahmed b a
Department of Chemical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1 b Department of Chemical Engineering, Alexandria University, Alexandria, Egypt Received 4 February 2005; received in revised form 23 May 2005; accepted 23 May 2005 Available online 18 July 2005
Abstract Rates of mass transfer inside conical cavities machined in the wall of a vertical rectangular duct were measured by the electrochemical technique. Variables studied were solution flow rate, physical properties of the solution and cavity apex angle. The mass transfer coefficient inside the cavity was found to be less than the flat surface value, and decreased with increasing the cavity apex angle. Mass transfer data inside the cavity were correlated by the equation: Sh = 0.549Sc0.33 Re0.33 (h/d)−0.524 Practical implications of the present results in determining the rate of diffusion controlled processes which may take place inside cavities, such as corrosion, electropolishing, electrochemical machining, electroplating, electroless plating, etching, etc. have been highlighted. © 2005 Elsevier B.V. All rights reserved. Keywords: Mass transfer; Conical cavity; Forced convection
1. Introduction Study of heat and mass transfer inside cavities machined in ducts with transverse flow is of a considerable importance in engineering and medicine as shown by the following examples. Stevenson [1] suggested that clot formation in flowing blood (thrombogenesis) might be explained by a study of mass transfer from open cavities that may exist in the artery wall. Surface reactions at the cavity wall may produce certain materials, which remain trapped in the vortex that exist in the cavity. A high concentration of these materials could lead to thrombogenesis. Cavities filled with liquids are used in cooling equipments, such as nuclear reactors, transformer cores, turbine bodies and internal combustion engines by virtue of the strong natural convection inside these cavities, which are called thermosyphons [2,3]. Fabrication and surface finishing of such cavities may be carried out by diffusion controlled processes, such as electrochemical machining, electroform∗
Corresponding author. Tel.: +1 807 343 8343; fax: +1 807 343 8928. E-mail address: [email protected] (I. Nirdosh).
0255-2701/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2005.05.002
ing, electroplating, electropolishing, etching and electroless (catalytic) plating. Transverse forced convection is one of the tools, which can be used to enhance the rate of such processes [4]. The present work is of relevance to local corrosion of cavities, which may exist at a rupture in internally coated tubes or ducts or a break in a protective oxide film (erosion–corrosion). Corrosion of steel within the pH range 4–10 is known to controlled by the diffusion of dissolved oxygen to the corroding surface [5]. Also the present study is relevant to pitting corrosion, which occurs to passive metals and alloys, such as stainless steel and aluminum in seawater where chloride ions initiate a pit at a weak point in the oxide film. The rate of pit growth depends on the rate of mass transfer between the solution inside the pit and the outside flowing solution [6,7]. The aim of the present work is to study the forced convection mass transfer behaviour of conical cavities under transverse laminar flow conditions. To this end, an electrochemical technique which involves measuring the limiting current of the cathodic reduction of K3 Fe(CN)6 was used.
M.M. Zaki et al. / Chemical Engineering and Processing 44 (2005) 1306–1311
Apart from the accuracy of the technique it has the advantage that the surface of the cavity and its dimensions remain unaltered during experiment. The present study would make it possible the prediction of the rate of the aforementioned diffusion controlled processes used in fabricating conical cavities. The study may be useful also in predicting the rate of diffusion controlled corrosion of metallic surfaces containing conical pits and cavities. It is hoped that, by virtue of the analogy between heat and mass transfer, the present study would serve in predicting the rate of heat transfer from conical cavities to transversely flowing solutions under laminar flow. Previous forced convection mass transfer studies in large cavities have dealt with hemispherical [8], cubical [9] and semicylinderical cavities [10]. Mass transfer inside conical cavities has been only studied under natural convection [11]. Georgiadou et al. [12], and Shin and Economou [13] studied the effect of fluid flow on the rate of mass transfer at the bottom of small rectangular cavities in relation to etching of masked surfaces, which is used in printed circuit fabrication.
2. Theory Consider a short mass transfer flat section placed in the fully developed region of a rectangular duct under laminar flow conditions (Fig. 1 ). For such a case the steady state unidirectional convective mass transfer equation has the form: Vx
∂C ∂2 C =D 2 ∂x ∂y
(1)
In the region near the wall the variation of Vx with y is approximated by the linear equation [14] Vx =
6Vy s
(2)
Substituting for Vx in (1) ∂2 C 6Vy ∂C =D 2 s ∂x ∂y
(3)
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Solving the above equation for the pertinent boundary conditions leads to D 2V 0.33 (4) Kx = 0.893 3Dxs The above equation predicts the local mass transfer coefficient at any distance x from the leading edge of the flat mass transfer surface. Introducing the equivalent duct diameter de , where: de =
4bs 2(b + s)
(5)
If the plate width (b) is large compared to the distance between the plates (s), the equivalent diameter can be approximated by de ≈ 2s
(6)
Introducing the solution kinematic viscosity into both the numerator and denominator of the group (2V/3Ds), the following equation is obtained de ν de 0.33 (7) Shx = 1.23 V ν D x de 0.33 (8) Shx = 1.23 ReSc x 1 L Shx dx (9) Shav = L 0 de 0.33 Shav = 1.85 ScRe (10) L If the flat mass transfer surface is replaced by a conical cavity, Eq. (10) is no longer valid owing to the change in the hydrodynamic conditions, such as hydrodynamic boundary layer separation and the formation of secondary flow inside the cavity [12]. In view of this, the present experimental study was conducted to obtain a mass transfer correlation for the conical cavity similar to Eq. (10). It is plausible to use cavity mouth diameter as a characteristic length in calculating Sh and Re in view of the fact that a concentration boundary layer is built along the cavity mouth, the ratio of mouth diameter to cavity depth (d/h) will replace (de /L) in Eq. (10).
3. Experimental technique
Fig. 1. Schematic representation of the location of the present mass transfer in the rectangular duct: (- - -) diffusion layer and (—) hydrodynamic boundary layer.
The apparatus (Fig. 2) consisted of 30 L plexiglass storage tank, plastic centrifugal pump and the cell. The cell consisted of a plexiglass rectangular duct of cross-section measuring 2 cm × 8 cm, and 294 cm height. Cavities were fixed in the duct wall at a distance 70de from the duct bottom to obtain fully developed flow. Nickel plated copper conical cavities of the mouth diameters 0.51, 0.71, 1.23, 2.13 and 4.59 cm were used, in all cases cavity vertical depth was fixed at 0.61 cm. Cavity apex angles, θ, were 45◦ , 60◦ , 90◦ , 120◦ and 150◦ . A
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Fig. 2. The apparatus: (1) storage tank, (2) centrifugal pump, (3) bypass, (4) rotameter, (5) rectangular cell, (6) cavity [cathode], (7) stainless steel sheet [anode] and (8) dc power supply.
Fig. 3. A typical conical cavity.
(which contained an opening sufficient to accommodate the cavity) by screw threads so that the cavity mouth was flush with the internal duct wall. Solution leakage was avoided by using rubber O-rings between the outer duct wall and the cavity holder. The cavity which acted as a cathode, was fed by electrical current through a 2 mm copper wire brazed to the copper cavity bottom and passed through the plastic holder. A vertical rectangular stainless steel plate anode measuring 8 cm × 30 cm was fixed flush to the duct wall opposite to the cavity cathode. In addition to cavities, rates of mass transfer were also measured at a flat nickel disc of 1.5 cm diameter placed at the same location as the cavities flush to the duct wall. The distance between the cavity mouth and the anode was 2 cm. The high anode area compared to the cavity area made it possible to use the anode as a reference electrode against which cathode potential was measured thus obviating the need for an external reference electrode which may disturb the flow in the duct. The cavity cathode was connected to the negative pole of 10 V dc power supply while the stainless steel anode was connected to the positive pole of the dc power supply. A multirange ammeter was connected in series with the cell to measure the current, the cell voltage was measured with a digital voltmeter connected in parallel with the cell. The cavity zone was followed by an outlet section of 42 cm height (13de ). Before each run, the cavity cathode was treated as mentioned elsewhere [15]. The electrolyte (20 L) was circulated between the storage tank and the cell by a centrifugal pump. The flow rate was controlled by a bypass and was measured by a rotameter. The solution was freed from dissolved oxygen before and during runs by bubbling N2 gas in the storage tank solution. The solution was composed of 0.025 M K3 Fe(CN)6 and 0.025 M K4 Fe(CN)6 dissolved in a large excess of NaOH as a supporting electrolyte. Three different NaOH concentrations were used, namely, 1, 2 and 4 M. All solutions were prepared from A.R. grade chemicals and distilled water. Temperature was 25 ± 1 ◦ C during all experiments. Current–voltage data from which the limiting current was obtained were recorded by increasing the current stepwise and measuring the corresponding cell voltage. The physical properties of the solution (ρ, µ, D) needed for data correlation were taken from Refs. [15,16].
Table 1 Characteristics of the conical cavities used in the present work Apex angle, θ (◦ )
Cavity mouth diameter, d (cm)
(h/d)
45 60 90 120 150
0.51 0.71 1.23 2.13 4.59
1.22 0.87 0.5 0.29 0.13
4. Results and discussion Fig. 4 shows a typical current–voltage curve from which the limiting current was determined. The mass transfer coefficient was calculated from the limiting current using the equation [15]:
Cavity depth (h) = 0.61 cm.
K= typical cavity is shown in Fig. 3. Table 1 shows the aspect ratio (h/d) of the cavities. The nickel plated copper cavities were glued with epoxy resin to plastic cavity machined in a plastic holder. The plastic holder was fixed to the external duct wall
I ZFAC
(11)
Fig. 5 shows that the flat surface data, which were obtained using an active circular area of 1.5 cm diameter lie above the prediction of Eq. (10) which represents mass transfer at a flat
M.M. Zaki et al. / Chemical Engineering and Processing 44 (2005) 1306–1311
Fig. 4. Typical polarization curves obtained at different solution velocities. Cavity apex angle = 45 ◦ .
plate under fully developed laminar flow. The discrepancy between the present data for a flat circular disc and that for a flat plate (Eq. (10)) may be attributed to the difference in the length of the mass transfer surface in the flow direction between the disc and the plate. The length of the mass transfer surface at the flat plate in the flow direction is uniform across its width whereas for the flat disc this length ranges from zero at the disc periphery to disc diameter at the disc center. Since the diffusion layer thickness increases with increasing the length of the mass transfer surface in the direction of flow in the mass transfer entry region [14], it follows that for a given solution velocity the average diffusion layer thickness at the flat disc is less than that at a flat plate of a length equal to the disc diameter. A similar finding was reported in case of natural convection where the mass transfer coefficient at a vertical disc was found to be higher than that at a vertical plate of a height equal to the disc diameter [17,18]. Fig. 6 shows the effect of solution velocity and cavity apex angle on the mass transfer coefficient inside the cavity, the mass transfer coefficient increases with increasing solution velocity and decreases with increasing apex angle, i.e. for a given solution velocity the mass transfer coefficient decreases with increasing cavity size. This result is consistent with the earlier results obtained with hemispherical and cubical cavities [8,9]. Table 2 shows also that the mass transfer coefficient
Fig. 5. Comparison of the present data for the flat disc with the model prediction (Eq. (10)).
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Fig. 6. Effect of cavity apex angle on the mass transfer coefficient.
inside the cavity is lower than that at the flat surface of a diameter equal to the diameter of the cavity mouth, the larger the cavity size the higher the deviation of the mass transfer coefficient from the flat surface value. Although no attempt has been made to visualize the flow pattern inside conical cavities, it is plausible to assume, in view of previous studies of flow behaviour in relatively deep cavities [19–23], that as a result of boundary layer separation at the edge of the cavity a circulating eddy (secondary flow) is formed inside the cavity, the larger the size of the cavity the larger the size of the circulating eddy and the slower its motion. The resistance to mass transfer from the external flowing solution to the cavity wall is determined by two diffusion layers, a diffusion layer at the cavity wall whose thickness is determined by the rotation speed of the recirculating eddy and a diffusion layer at the interface between the cavity solution and the solution outside the cavity. As the cavity apex angle increases, the length of the surface across which diffusion takes place increases Table 2 Comparison of the mass transfer coefficient inside cavities of different diameters with the value at a flat surface of a diameter equal to the cavity diameter Cavity mouth diameter (cm)
0.51 0.71 1.23 2.13 4.59 0.51 0.71 1.23 2.13 4.59 0.51 0.71 1.23 2.13 4.59
Solution velocity in the duct (cm/s)
2.4 2.4 2.4 2.4 2.4 4.7 4.7 4.7 4.7 4.7 6.3 6.3 6.3 6.3 6.3
Cavity depth (h) = 0.61 cm, Sc = 1585.
Mass transfer coefficient (×104 cm/s) Cavity
Flat disc of equal diameter
4.23 3.15 2.77 2.62 2.54 5.17 3.77 3.46 3.35 3.27 5.71 4.09 3.80 3.73 3.63
7.40 6.62 5.52 4.59 3.56 9.32 8.34 6.95 5.79 4.48 10.30 9.18 7.65 6.37 4.93
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Appendix A. Nomenclature
A b C d de
Fig. 7. Overall mass transfer correlation at conical cavities.
both at the cavity mouth and at the inner side of the cavity. Accordingly, the average diffusion layer thickness increases with increasing cavity apex angle. The decrease in the mass transfer coefficient with increasing cavity apex angle may be also accounted for by the formation of a large slowly rotating eddy inside the cavity whose ability to enhance the rate of mass transfer at the cavity mouth and walls is low. Mass transfer data inside the cavity were correlated for the conditions 1585 < Sc < 5676; 113 < Re < 1643 and 0.13 < (h/d) < 1.22 by the equation (Fig. 7): Sh = 0.549Sc0.33 Re0.33 (h/d)−0.524
(12)
with an average deviation of ±9%. In calculating Sh and Re in the above equation, the cavity diameter (d) was used as a characteristic length.
5. Conclusions The rate of mass transfer between a fully developed laminar flow and conical cavity machined in the wall of a rectangular duct is less than that at a flat surface having the same diameter as the cavity mouth. The larger the apex angle of the cavity the lower the mass transfer coefficient. The present finding corroborates our earlier finding with hemispherical and cubical cavities where the rate of mass transfer inside the cavity was less than that at the corresponding flat surface. The proposed dimensionless equation can be used to predict the rate of metal finishing inside conical cavities under transverse laminar flow.
Acknowledgements This research was funded by the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Mr. Ed Drotar of the Science Workshop for making the apparatus and the pumping station, and to Mr. Kailash Bhatia, the Mechanical Engineering Technologist, for making the cavities.
D F h I K Kx L Re s Sc Sh V Vx x y Z
active cavity surface area width of the rectangular duct ferricyanide concentration cavity mouth diameter equivalent diameter of the duct (de = 4 × crosssection/wetted perimeter) diffusivity of ferricyanide ion Faraday’s constant cavity depth limiting current mass transfer coefficient local mass transfer coefficient active length of the flat mass transfer surface Reynolds number (Vde /ν) for the flat surface and (Vd/ν) for the cavity distance between the two electrodes (channel thickness) Schmidt number (µ/ρD) Sherwood number (Kde /D) for the flat surface and (Kd/D) for the cavity average solution velocity point liquid velocity in the x direction distance in the direction of flow distance perpendicular to the direction of flow number of electrons involved in the reaction
Greek letters θ cavity apex angle µ solution viscosity ν solution kinematic viscosity ρ solution density
References [1] J.F. Stevenson, Flow in a tube with a circumferential wall cavity, J. Appl. Mech. 40 (1973) 355–361. [2] D. Japkise, Advances in thermosyphon technology, in: J.P. Hartnett, T.F. Irvine Jr. (Eds.), Advance Heat Transfer, vol. 9, Academic Press, New York, 1973, pp. 1–111. [3] D. Japkise, E.R. Winter, Single phase transport processes in the open thermosyphone, Int. J. Heat Mass Transfer 14 (1971) 427–441. [4] C. Bryce, D. Berk, Mass transfer on the etch rate of GaAs in flow systems, Chem. Eng. Commun. 191 (2004) 189–207. [5] K.J. Kennelley, R.H. Hausler, in: D.C. Silverman (Ed.), FlowInduced Corrosion, NACE, Houston, 1991. [6] T.R. Beck, Effect of hydrodynamics on pitting, Corrosion 33 (1977) 9–13. [7] J.R. Galvele, Transport processes and the mechanism of pitting of metals, J. Electrochem. Soc. 123 (1976) 464–474. [8] M.M. Zaki, I. Nirdosh, G.H. Sedahmed, Forced convection mass transfer inside large hemispherical cavities under laminar flow conditions, Chem. Eng. Commun. 159 (1997) 161–171. [9] M.M. Zaki, I. Nirdosh, G.H. Sedahmed, Effect of laminar flow on the rate of mass transfer inside cubical cavities, Chem. Eng. Technol. 20 (1997) 199–202.
M.M. Zaki et al. / Chemical Engineering and Processing 44 (2005) 1306–1311 [10] J.K. Aggarwal, L. Talbot L, Electrochemical measurements of mass transfer in semi-cylinderical hollows, Int J. Heat Mass Transfer 22 (1979) 61–75. [11] G.H. Sedahmed GH, A.M. Ahmed, Natural convection mass transfer in slender conical cavities, Ind. Eng. Chem. Res. 29 (1990) 1728–1731. [12] M. Georgiadou, R. Mohr, R.C. Alkire, Local mass transport in twodimensional cavities in laminar shear flow, J. Electrochem. Soc. 147 (2000) 3021–3028. [13] C.B. Shin, D.J. Economou, Forced and natural convection effects on the shape evolution of cavities during wet chemical etching, J. Electrochem. Soc. 138 (1991) 527–538. [14] D.J. Pickett, Electrochemical Reactor Design, Elsevier, NY, 1977. [15] J.R. Selman, C.W. Tobias, Mass transfer by the limiting current technique, Adv. Chem. Eng. 10 (1978) 211–318. [16] J.R. Bourne, P. P. Dell’Ava, O. Dossenbach, T. Post, Densities, viscosities and diffusivities in aqueous sodium hydroxide–potassium ferri and ferrocyanide solution, J. Chem. Eng. Data 30 (1985) 160–163.
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[17] G.H. Sedahmed, I. Nirdosh, Free convection mass transfer at horizontal cylinders with active ends, Int. Commun. Heat Mass Transfer 17 (1990) 355–366. [18] I. Nirdosh, G.H. Sedahmed, A mass transfer study of the electropolishing of vertical discs under free convection, J. Appl. Electrochem. 21 (1990) 651–653. [19] H.W. Townes, R.H. Sabersky, Experiments on the flow over a rough surface, Int. J. Heat Mass Transfer 9 (1966) 729–738. [20] F. Pan, A. Acrivos, Steady flows in rectangular cavities, J. Fluid Mech. 28 (1967) 643–655. [21] O.R. Burggraf, Analytical, Numerical studies of the structure of steady separated flows, J. Fluid Mech. 24 (1966) 113– 151. [22] J.J.L. Higdon, Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities, J. Fluid Mech. 159 (1985) 195–226. [23] C. Shen, J.M. Floryan, Low Reynolds number flow over cavities, Phys. Fluids 28 (1985) 3191–3207.
Mass transfer inside conical cavities under transverse laminar flow M.M. Zaki a , I. Nirdosh a,∗ , G.H. Sedahmed b a
Department of Chemical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1 b Department of Chemical Engineering, Alexandria University, Alexandria, Egypt Received 4 February 2005; received in revised form 23 May 2005; accepted 23 May 2005 Available online 18 July 2005
Abstract Rates of mass transfer inside conical cavities machined in the wall of a vertical rectangular duct were measured by the electrochemical technique. Variables studied were solution flow rate, physical properties of the solution and cavity apex angle. The mass transfer coefficient inside the cavity was found to be less than the flat surface value, and decreased with increasing the cavity apex angle. Mass transfer data inside the cavity were correlated by the equation: Sh = 0.549Sc0.33 Re0.33 (h/d)−0.524 Practical implications of the present results in determining the rate of diffusion controlled processes which may take place inside cavities, such as corrosion, electropolishing, electrochemical machining, electroplating, electroless plating, etching, etc. have been highlighted. © 2005 Elsevier B.V. All rights reserved. Keywords: Mass transfer; Conical cavity; Forced convection
1. Introduction Study of heat and mass transfer inside cavities machined in ducts with transverse flow is of a considerable importance in engineering and medicine as shown by the following examples. Stevenson [1] suggested that clot formation in flowing blood (thrombogenesis) might be explained by a study of mass transfer from open cavities that may exist in the artery wall. Surface reactions at the cavity wall may produce certain materials, which remain trapped in the vortex that exist in the cavity. A high concentration of these materials could lead to thrombogenesis. Cavities filled with liquids are used in cooling equipments, such as nuclear reactors, transformer cores, turbine bodies and internal combustion engines by virtue of the strong natural convection inside these cavities, which are called thermosyphons [2,3]. Fabrication and surface finishing of such cavities may be carried out by diffusion controlled processes, such as electrochemical machining, electroform∗
Corresponding author. Tel.: +1 807 343 8343; fax: +1 807 343 8928. E-mail address: [email protected] (I. Nirdosh).
0255-2701/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2005.05.002
ing, electroplating, electropolishing, etching and electroless (catalytic) plating. Transverse forced convection is one of the tools, which can be used to enhance the rate of such processes [4]. The present work is of relevance to local corrosion of cavities, which may exist at a rupture in internally coated tubes or ducts or a break in a protective oxide film (erosion–corrosion). Corrosion of steel within the pH range 4–10 is known to controlled by the diffusion of dissolved oxygen to the corroding surface [5]. Also the present study is relevant to pitting corrosion, which occurs to passive metals and alloys, such as stainless steel and aluminum in seawater where chloride ions initiate a pit at a weak point in the oxide film. The rate of pit growth depends on the rate of mass transfer between the solution inside the pit and the outside flowing solution [6,7]. The aim of the present work is to study the forced convection mass transfer behaviour of conical cavities under transverse laminar flow conditions. To this end, an electrochemical technique which involves measuring the limiting current of the cathodic reduction of K3 Fe(CN)6 was used.
M.M. Zaki et al. / Chemical Engineering and Processing 44 (2005) 1306–1311
Apart from the accuracy of the technique it has the advantage that the surface of the cavity and its dimensions remain unaltered during experiment. The present study would make it possible the prediction of the rate of the aforementioned diffusion controlled processes used in fabricating conical cavities. The study may be useful also in predicting the rate of diffusion controlled corrosion of metallic surfaces containing conical pits and cavities. It is hoped that, by virtue of the analogy between heat and mass transfer, the present study would serve in predicting the rate of heat transfer from conical cavities to transversely flowing solutions under laminar flow. Previous forced convection mass transfer studies in large cavities have dealt with hemispherical [8], cubical [9] and semicylinderical cavities [10]. Mass transfer inside conical cavities has been only studied under natural convection [11]. Georgiadou et al. [12], and Shin and Economou [13] studied the effect of fluid flow on the rate of mass transfer at the bottom of small rectangular cavities in relation to etching of masked surfaces, which is used in printed circuit fabrication.
2. Theory Consider a short mass transfer flat section placed in the fully developed region of a rectangular duct under laminar flow conditions (Fig. 1 ). For such a case the steady state unidirectional convective mass transfer equation has the form: Vx
∂C ∂2 C =D 2 ∂x ∂y
(1)
In the region near the wall the variation of Vx with y is approximated by the linear equation [14] Vx =
6Vy s
(2)
Substituting for Vx in (1) ∂2 C 6Vy ∂C =D 2 s ∂x ∂y
(3)
1307
Solving the above equation for the pertinent boundary conditions leads to D 2V 0.33 (4) Kx = 0.893 3Dxs The above equation predicts the local mass transfer coefficient at any distance x from the leading edge of the flat mass transfer surface. Introducing the equivalent duct diameter de , where: de =
4bs 2(b + s)
(5)
If the plate width (b) is large compared to the distance between the plates (s), the equivalent diameter can be approximated by de ≈ 2s
(6)
Introducing the solution kinematic viscosity into both the numerator and denominator of the group (2V/3Ds), the following equation is obtained de ν de 0.33 (7) Shx = 1.23 V ν D x de 0.33 (8) Shx = 1.23 ReSc x 1 L Shx dx (9) Shav = L 0 de 0.33 Shav = 1.85 ScRe (10) L If the flat mass transfer surface is replaced by a conical cavity, Eq. (10) is no longer valid owing to the change in the hydrodynamic conditions, such as hydrodynamic boundary layer separation and the formation of secondary flow inside the cavity [12]. In view of this, the present experimental study was conducted to obtain a mass transfer correlation for the conical cavity similar to Eq. (10). It is plausible to use cavity mouth diameter as a characteristic length in calculating Sh and Re in view of the fact that a concentration boundary layer is built along the cavity mouth, the ratio of mouth diameter to cavity depth (d/h) will replace (de /L) in Eq. (10).
3. Experimental technique
Fig. 1. Schematic representation of the location of the present mass transfer in the rectangular duct: (- - -) diffusion layer and (—) hydrodynamic boundary layer.
The apparatus (Fig. 2) consisted of 30 L plexiglass storage tank, plastic centrifugal pump and the cell. The cell consisted of a plexiglass rectangular duct of cross-section measuring 2 cm × 8 cm, and 294 cm height. Cavities were fixed in the duct wall at a distance 70de from the duct bottom to obtain fully developed flow. Nickel plated copper conical cavities of the mouth diameters 0.51, 0.71, 1.23, 2.13 and 4.59 cm were used, in all cases cavity vertical depth was fixed at 0.61 cm. Cavity apex angles, θ, were 45◦ , 60◦ , 90◦ , 120◦ and 150◦ . A
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M.M. Zaki et al. / Chemical Engineering and Processing 44 (2005) 1306–1311
Fig. 2. The apparatus: (1) storage tank, (2) centrifugal pump, (3) bypass, (4) rotameter, (5) rectangular cell, (6) cavity [cathode], (7) stainless steel sheet [anode] and (8) dc power supply.
Fig. 3. A typical conical cavity.
(which contained an opening sufficient to accommodate the cavity) by screw threads so that the cavity mouth was flush with the internal duct wall. Solution leakage was avoided by using rubber O-rings between the outer duct wall and the cavity holder. The cavity which acted as a cathode, was fed by electrical current through a 2 mm copper wire brazed to the copper cavity bottom and passed through the plastic holder. A vertical rectangular stainless steel plate anode measuring 8 cm × 30 cm was fixed flush to the duct wall opposite to the cavity cathode. In addition to cavities, rates of mass transfer were also measured at a flat nickel disc of 1.5 cm diameter placed at the same location as the cavities flush to the duct wall. The distance between the cavity mouth and the anode was 2 cm. The high anode area compared to the cavity area made it possible to use the anode as a reference electrode against which cathode potential was measured thus obviating the need for an external reference electrode which may disturb the flow in the duct. The cavity cathode was connected to the negative pole of 10 V dc power supply while the stainless steel anode was connected to the positive pole of the dc power supply. A multirange ammeter was connected in series with the cell to measure the current, the cell voltage was measured with a digital voltmeter connected in parallel with the cell. The cavity zone was followed by an outlet section of 42 cm height (13de ). Before each run, the cavity cathode was treated as mentioned elsewhere [15]. The electrolyte (20 L) was circulated between the storage tank and the cell by a centrifugal pump. The flow rate was controlled by a bypass and was measured by a rotameter. The solution was freed from dissolved oxygen before and during runs by bubbling N2 gas in the storage tank solution. The solution was composed of 0.025 M K3 Fe(CN)6 and 0.025 M K4 Fe(CN)6 dissolved in a large excess of NaOH as a supporting electrolyte. Three different NaOH concentrations were used, namely, 1, 2 and 4 M. All solutions were prepared from A.R. grade chemicals and distilled water. Temperature was 25 ± 1 ◦ C during all experiments. Current–voltage data from which the limiting current was obtained were recorded by increasing the current stepwise and measuring the corresponding cell voltage. The physical properties of the solution (ρ, µ, D) needed for data correlation were taken from Refs. [15,16].
Table 1 Characteristics of the conical cavities used in the present work Apex angle, θ (◦ )
Cavity mouth diameter, d (cm)
(h/d)
45 60 90 120 150
0.51 0.71 1.23 2.13 4.59
1.22 0.87 0.5 0.29 0.13
4. Results and discussion Fig. 4 shows a typical current–voltage curve from which the limiting current was determined. The mass transfer coefficient was calculated from the limiting current using the equation [15]:
Cavity depth (h) = 0.61 cm.
K= typical cavity is shown in Fig. 3. Table 1 shows the aspect ratio (h/d) of the cavities. The nickel plated copper cavities were glued with epoxy resin to plastic cavity machined in a plastic holder. The plastic holder was fixed to the external duct wall
I ZFAC
(11)
Fig. 5 shows that the flat surface data, which were obtained using an active circular area of 1.5 cm diameter lie above the prediction of Eq. (10) which represents mass transfer at a flat
M.M. Zaki et al. / Chemical Engineering and Processing 44 (2005) 1306–1311
Fig. 4. Typical polarization curves obtained at different solution velocities. Cavity apex angle = 45 ◦ .
plate under fully developed laminar flow. The discrepancy between the present data for a flat circular disc and that for a flat plate (Eq. (10)) may be attributed to the difference in the length of the mass transfer surface in the flow direction between the disc and the plate. The length of the mass transfer surface at the flat plate in the flow direction is uniform across its width whereas for the flat disc this length ranges from zero at the disc periphery to disc diameter at the disc center. Since the diffusion layer thickness increases with increasing the length of the mass transfer surface in the direction of flow in the mass transfer entry region [14], it follows that for a given solution velocity the average diffusion layer thickness at the flat disc is less than that at a flat plate of a length equal to the disc diameter. A similar finding was reported in case of natural convection where the mass transfer coefficient at a vertical disc was found to be higher than that at a vertical plate of a height equal to the disc diameter [17,18]. Fig. 6 shows the effect of solution velocity and cavity apex angle on the mass transfer coefficient inside the cavity, the mass transfer coefficient increases with increasing solution velocity and decreases with increasing apex angle, i.e. for a given solution velocity the mass transfer coefficient decreases with increasing cavity size. This result is consistent with the earlier results obtained with hemispherical and cubical cavities [8,9]. Table 2 shows also that the mass transfer coefficient
Fig. 5. Comparison of the present data for the flat disc with the model prediction (Eq. (10)).
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Fig. 6. Effect of cavity apex angle on the mass transfer coefficient.
inside the cavity is lower than that at the flat surface of a diameter equal to the diameter of the cavity mouth, the larger the cavity size the higher the deviation of the mass transfer coefficient from the flat surface value. Although no attempt has been made to visualize the flow pattern inside conical cavities, it is plausible to assume, in view of previous studies of flow behaviour in relatively deep cavities [19–23], that as a result of boundary layer separation at the edge of the cavity a circulating eddy (secondary flow) is formed inside the cavity, the larger the size of the cavity the larger the size of the circulating eddy and the slower its motion. The resistance to mass transfer from the external flowing solution to the cavity wall is determined by two diffusion layers, a diffusion layer at the cavity wall whose thickness is determined by the rotation speed of the recirculating eddy and a diffusion layer at the interface between the cavity solution and the solution outside the cavity. As the cavity apex angle increases, the length of the surface across which diffusion takes place increases Table 2 Comparison of the mass transfer coefficient inside cavities of different diameters with the value at a flat surface of a diameter equal to the cavity diameter Cavity mouth diameter (cm)
0.51 0.71 1.23 2.13 4.59 0.51 0.71 1.23 2.13 4.59 0.51 0.71 1.23 2.13 4.59
Solution velocity in the duct (cm/s)
2.4 2.4 2.4 2.4 2.4 4.7 4.7 4.7 4.7 4.7 6.3 6.3 6.3 6.3 6.3
Cavity depth (h) = 0.61 cm, Sc = 1585.
Mass transfer coefficient (×104 cm/s) Cavity
Flat disc of equal diameter
4.23 3.15 2.77 2.62 2.54 5.17 3.77 3.46 3.35 3.27 5.71 4.09 3.80 3.73 3.63
7.40 6.62 5.52 4.59 3.56 9.32 8.34 6.95 5.79 4.48 10.30 9.18 7.65 6.37 4.93
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Appendix A. Nomenclature
A b C d de
Fig. 7. Overall mass transfer correlation at conical cavities.
both at the cavity mouth and at the inner side of the cavity. Accordingly, the average diffusion layer thickness increases with increasing cavity apex angle. The decrease in the mass transfer coefficient with increasing cavity apex angle may be also accounted for by the formation of a large slowly rotating eddy inside the cavity whose ability to enhance the rate of mass transfer at the cavity mouth and walls is low. Mass transfer data inside the cavity were correlated for the conditions 1585 < Sc < 5676; 113 < Re < 1643 and 0.13 < (h/d) < 1.22 by the equation (Fig. 7): Sh = 0.549Sc0.33 Re0.33 (h/d)−0.524
(12)
with an average deviation of ±9%. In calculating Sh and Re in the above equation, the cavity diameter (d) was used as a characteristic length.
5. Conclusions The rate of mass transfer between a fully developed laminar flow and conical cavity machined in the wall of a rectangular duct is less than that at a flat surface having the same diameter as the cavity mouth. The larger the apex angle of the cavity the lower the mass transfer coefficient. The present finding corroborates our earlier finding with hemispherical and cubical cavities where the rate of mass transfer inside the cavity was less than that at the corresponding flat surface. The proposed dimensionless equation can be used to predict the rate of metal finishing inside conical cavities under transverse laminar flow.
Acknowledgements This research was funded by the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Mr. Ed Drotar of the Science Workshop for making the apparatus and the pumping station, and to Mr. Kailash Bhatia, the Mechanical Engineering Technologist, for making the cavities.
D F h I K Kx L Re s Sc Sh V Vx x y Z
active cavity surface area width of the rectangular duct ferricyanide concentration cavity mouth diameter equivalent diameter of the duct (de = 4 × crosssection/wetted perimeter) diffusivity of ferricyanide ion Faraday’s constant cavity depth limiting current mass transfer coefficient local mass transfer coefficient active length of the flat mass transfer surface Reynolds number (Vde /ν) for the flat surface and (Vd/ν) for the cavity distance between the two electrodes (channel thickness) Schmidt number (µ/ρD) Sherwood number (Kde /D) for the flat surface and (Kd/D) for the cavity average solution velocity point liquid velocity in the x direction distance in the direction of flow distance perpendicular to the direction of flow number of electrons involved in the reaction
Greek letters θ cavity apex angle µ solution viscosity ν solution kinematic viscosity ρ solution density
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