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Figure 10. Standard deviation of the liquid free surface heights for Cases A, B and C for all electrode rotation speeds. (wis in rad s' and the free surface standard deviation is in m).

Figure 10 Standard deviation of the liquid free surface heights for Cases A, B and C for all electrode rotation speeds. (wis in rad s' and the free surface standard deviation is in m).

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Where ¢ 1s defined as in Equation (3) and £6 =0.9130294741. Another advantage of the solution proposed by Ariel over that obtained by Cochran is that Equations (17)-(19) can be evaluated for any value of € such that O<&<o. Volgin and Davydov (Volgin and Davydov 2007) shown that the error associated to the available approximate equations to calculate the velocity field range from 1 to 0.01%. In addition, they found that to provide an accuracy of about 0.1%, the length of computational region should be approximately twice the diffusion layer thickness. To reach higher accuracy, the extension of the computational region must be increased.
Where ¢ 1s defined as in Equation (3) and £6 =0.9130294741. Another advantage of the solution proposed by Ariel over that obtained by Cochran is that Equations (17)-(19) can be evaluated for any value of € such that O<&<o. Volgin and Davydov (Volgin and Davydov 2007) shown that the error associated to the available approximate equations to calculate the velocity field range from 1 to 0.01%. In addition, they found that to provide an accuracy of about 0.1%, the length of computational region should be approximately twice the diffusion layer thickness. To reach higher accuracy, the extension of the computational region must be increased.
Figure 1. Liquid content and electrode immersion depth for the models used in this work. (a) Case A, (b) Case B and (c) Case C. The electrode is colored in gray, the liquid in the cell upper zone is colored in light blue and the liquid in the cell lower zone is colored in dark blue. 2.2.2 Governing equations
Figure 1. Liquid content and electrode immersion depth for the models used in this work. (a) Case A, (b) Case B and (c) Case C. The electrode is colored in gray, the liquid in the cell upper zone is colored in light blue and the liquid in the cell lower zone is colored in dark blue. 2.2.2 Governing equations
Figure 2. (a) Zone kinetic energy ratio for each case. (b) Normalized kinetic energy for each case. (@ is in rad s and the integral of the kinetic energy is re) inm s~).
Figure 2. (a) Zone kinetic energy ratio for each case. (b) Normalized kinetic energy for each case. (@ is in rad s and the integral of the kinetic energy is re) inm s~).
Figure 3. Streamtraces below the electrode active face for Case C with an electrode rotation speed of 25 rad s". (a) three-dimensional biphasic model. (b) Ideal model. Qualitatively, the fluid flow patterns in the vicinity of the electrode active face obtained with the three-dimensional biphasic model and the ideal model are similar; nevertheless, there is a quantitative difference between them. Early experiments on the flow over a rotating disc reported the presence of a strong instability of the laminar boundary layer due to its three-dimensionality (Gregory, Stuart et al. 1955). However, a recent experimental work related with the electrodeposition of zinc on an inverted rotating disc electrode reported that the number of spirals and their angle are exactly in accordance with the flow pattern predicted by the ideal model (Jaksic and Komnenic 1991). A detailed description of the differences between ideal and three-dimensional biphasic models will be presented in Section 3.2.3.
Figure 3. Streamtraces below the electrode active face for Case C with an electrode rotation speed of 25 rad s". (a) three-dimensional biphasic model. (b) Ideal model. Qualitatively, the fluid flow patterns in the vicinity of the electrode active face obtained with the three-dimensional biphasic model and the ideal model are similar; nevertheless, there is a quantitative difference between them. Early experiments on the flow over a rotating disc reported the presence of a strong instability of the laminar boundary layer due to its three-dimensionality (Gregory, Stuart et al. 1955). However, a recent experimental work related with the electrodeposition of zinc on an inverted rotating disc electrode reported that the number of spirals and their angle are exactly in accordance with the flow pattern predicted by the ideal model (Jaksic and Komnenic 1991). A detailed description of the differences between ideal and three-dimensional biphasic models will be presented in Section 3.2.3.
Figure 4. Liquid free surface and streamtraces arriving at the electrode edge for Case A. The streamtraces markers are colored in accordance with its velocity magnitude. (a) 25 rad s". (b) 50 rad s", (c) 75 rad s", (d) 100 rad s', (e) 125 rad s', (f) 150 rad s", (g) 175 rad s! and (h) 200 rad s!. (The velocity magnitude is in ms).
Figure 4. Liquid free surface and streamtraces arriving at the electrode edge for Case A. The streamtraces markers are colored in accordance with its velocity magnitude. (a) 25 rad s". (b) 50 rad s", (c) 75 rad s", (d) 100 rad s', (e) 125 rad s', (f) 150 rad s", (g) 175 rad s! and (h) 200 rad s!. (The velocity magnitude is in ms).
Figure 5. (Continued) Liquid free surface and streamtraces arriving at the electrode edge for Case B. The streamtraces markers are colored in accordance with its velocity magnitude. (a) 25 rad s’. (b) 50 rad st (c) 75 rad s", (d) 100 rad s', (e) 125 rad s", (f) 150 rad s", (g) 175 rad s! and (h) 200 rad s/. (The velocity magnitude is in im sv),
Figure 5. (Continued) Liquid free surface and streamtraces arriving at the electrode edge for Case B. The streamtraces markers are colored in accordance with its velocity magnitude. (a) 25 rad s’. (b) 50 rad st (c) 75 rad s", (d) 100 rad s', (e) 125 rad s", (f) 150 rad s", (g) 175 rad s! and (h) 200 rad s/. (The velocity magnitude is in im sv),
modeling of electrochemical cells with a rotating disc electrode was recognized in
modeling of electrochemical cells with a rotating disc electrode was recognized in
Figure 7. (Continued) Height of the liquid free surface for Case A. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad s", (b) 50 rad s", (c) 75 rad s", (d) 100 rad s", (e) 125 rad s", (f) 150 rad s"', (g) 175 rad s" and (h)200 rad s'. (R and Z are in m). Also, the relationship between the total liquid amount and the hollow depth of the liquid surrounding the electrode can be observed. Figures 7, 8 and 9 shows that for the highest electrode rotation speed (200 rad s"'), the hollow depth decreases as the total liquid amount increases. This is in good accordance with experimental observations and also suggests that the interaction between the electrode side wall and cell walls intensifies when the amount of liquid is smaller.
Figure 7. (Continued) Height of the liquid free surface for Case A. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad s", (b) 50 rad s", (c) 75 rad s", (d) 100 rad s", (e) 125 rad s", (f) 150 rad s"', (g) 175 rad s" and (h)200 rad s'. (R and Z are in m). Also, the relationship between the total liquid amount and the hollow depth of the liquid surrounding the electrode can be observed. Figures 7, 8 and 9 shows that for the highest electrode rotation speed (200 rad s"'), the hollow depth decreases as the total liquid amount increases. This is in good accordance with experimental observations and also suggests that the interaction between the electrode side wall and cell walls intensifies when the amount of liquid is smaller.
Figure 8. Height of the liquid free surface for Case B. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad s", (b) 50 rad s', (c) 75 rad 1, (d) 100 rad s', (e) 125 rad 5", (f) 150 rad st (g) 175 rad 5" and (h)200 rad s". (R and Z are in m).
Figure 8. Height of the liquid free surface for Case B. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad s", (b) 50 rad s', (c) 75 rad 1, (d) 100 rad s', (e) 125 rad 5", (f) 150 rad st (g) 175 rad 5" and (h)200 rad s". (R and Z are in m).
Figure 8. (Continued) Height of the liquid free surface for Case B. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad st, (b) 50 rad st, (c) 75 rad st, (d) 100 rad st, (e) 125 rad s", (f) 150 rad s"', (g) 175 rad s" and (h)200 rad s'. (R and Z are in m).
Figure 8. (Continued) Height of the liquid free surface for Case B. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad st, (b) 50 rad st, (c) 75 rad st, (d) 100 rad st, (e) 125 rad s", (f) 150 rad s"', (g) 175 rad s" and (h)200 rad s'. (R and Z are in m).
Figure 9. Height of the liquid free surface for Case C. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad s", (b) 50 rad s", (c) 75 rad s", (d) 100 rad s", (e) 125 rad s", (f) 150 rad st (g) 175 rad s? and (h)200 rad 1. (R and Z are in m).
Figure 9. Height of the liquid free surface for Case C. One half of the free surface is colored in red and the other half is colored in blue. (a) 25 rad s", (b) 50 rad s", (c) 75 rad s", (d) 100 rad s", (e) 125 rad s", (f) 150 rad st (g) 175 rad s? and (h)200 rad 1. (R and Z are in m).
However, Figure 11 shows that there are several quantitative differences. For z values lower that -1.00x10°m, the liquid velocity magnitude calculated with the three-dimensional biphasic model is higher than the predicted with the ideal model. Given that in this case the effect of the cell walls is negligible, these differences can be directly attributed to the increment of the liquid kinetic energy generated by the electrode immersed side wall, which is absent in the ideal model. differences can be directly attributed to the increment of the liquid kinetic energy
However, Figure 11 shows that there are several quantitative differences. For z values lower that -1.00x10°m, the liquid velocity magnitude calculated with the three-dimensional biphasic model is higher than the predicted with the ideal model. Given that in this case the effect of the cell walls is negligible, these differences can be directly attributed to the increment of the liquid kinetic energy generated by the electrode immersed side wall, which is absent in the ideal model. differences can be directly attributed to the increment of the liquid kinetic energy
Figure 13. Liquid velocity magnitude below the electrode for an electrode rotation speed of 200 rad s". The markers are colored in accordance with its radius magnitude (a) Ideal model, (b) Case A, (c) Case B and (d) Case C. (Z and Rare in m and the velocity magnitude is in m s").
Figure 13. Liquid velocity magnitude below the electrode for an electrode rotation speed of 200 rad s". The markers are colored in accordance with its radius magnitude (a) Ideal model, (b) Case A, (c) Case B and (d) Case C. (Z and Rare in m and the velocity magnitude is in m s").
Figure 14. Angle with respect to the horizontal of the liquid velocity vectors below the electrode for an electrode rotation speed of 200 rad s". The markers are colored in accordance with its radius magnitude. (a) ‘Wea model, (b) Case A, (c) Case B and (d) Case C. (Z and R are in m and the velocity magnitude is in ms").
Figure 14. Angle with respect to the horizontal of the liquid velocity vectors below the electrode for an electrode rotation speed of 200 rad s". The markers are colored in accordance with its radius magnitude. (a) ‘Wea model, (b) Case A, (c) Case B and (d) Case C. (Z and R are in m and the velocity magnitude is in ms").
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Chemical EngineeringMaterials ScienceMechanicsRotating Disc Electrode
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