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David C. Silverman |
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Hydrodynamics of a Smooth Cylinder Understanding the physics of the rotating cylinder electrode requires understanding the concept of the boundary layer. When a fluid moves past a stationary, hydraulically smooth surface, there is an implicit assumption that the fluid is stationary at that surface. That assumption is often called "no slip at the wall". In many flow configurations and especially when turbulent flow conditions prevail, the velocity profile shows an increase from 0 at the solid-fluid interface to the free stream value across a relatively short distance into the fluid. Though in the rotating cylinder the surface is moving, the fluid adjacent to that surface moves at the same velocity as the cylinder so, in effect, the fluid is stationary relative to the surface and fulfills the no slip at the wall criterion. The boundary layer thickness is independent of position on the surface meaning that the velocity profile, momentum transfer, and wall shear stress are independent of position. The rotating cylinder electrode as traditionally constructed cannot examine situations where such uniformity does not exist without some modification. Mass transfer requires a concentration gradient between the surface and fluid bulk. Such a gradient implies that the concentration changes across some small distance between the surface and the bulk. This region is called the concentration or mass transfer boundary layer. For the large Schmidt Numbers normally encountered in liquids the fully developed mass transfer boundary layer for hydraulically smooth surfaces is much thinner than the fully developed hydrodynamic boundary layer. This relationship is shown in this figure Corrosion being limited by the rate of mass transfer has usually been assumed when the slope of the line calculated from the measured mass transfer rate versus velocity, both in logarithmic format, is approximately 0.7 (Eisenberg, M., Tobias, C. W., and Wilke, C. R., J. Electrochem. Society, 101(6), 306 (1954)). This exponent was derived using the Chilton-Colburn analogy (Chilton, T. H. and Colburn, A. P., Ind. Eng. Chem., 26, 1183 (1934)). The curvature of the friction factor vs. Reynolds number relationship was fit over the range of Reynolods number of 103 to 105 in linear format in log-log coordinates by:  (1)This equation led to the commonly used expression relating the Sherwood number to the Reynolds number of:  (2)But, this relationship is not the only relationship that has been developed for the rotating cylinder electrode. Equation (1) is actually a linearization of a curved function. What may be thought to be a deviation from the 0.7 power as caused by a departure from complete mass transfer control may be nothing more than deviations caused by the limitations of the predictive power of the "linearized" equation. Understanding the predictive limitations is important for making appropriate conclusions about mass transfer control. Early measurements of drag coefficients on rotating cylinders over the range of Reynolds numbers between about 103 and 106 resulted in the following relationship between the friction factor and Reynolds number:  (3)(Theodorsen, T. and Regier, A., "Experiments on Drag of Revolving Disks, Cylinders,and Streamline Rods at High Speeds" Nat. Advisory Comm. Aeronaut., p. 367, Report No. 793, U.S. Government Printing Office, Washington, D.C., 1945.) This equation itself was derived as a best fit to friction factor data plotted as 
vs. .
The relationship between friction factor and Reynolds number implied by this equation and the
data are actually curved on log-log coordinates. From the Chilton-Colburn analogy the
Sherwood number or mass transfer coefficient can be related to the friction factor by: (4)where the exponent c-1 has the value of -0.644 to -0.69 depending on author and study and St is the Stanton number. Equation (4) is substituted into equation (3) to obtain the Sherwood number versus Reynolds number relationship from the friction factor. The resulting equation would require an iterative calculation of the Sherwood number because the Sherwood number (mass transfer coefficient) would appear on both sides of the equation. Equation (2) is one approximation of this relationship. The desire of a linear approximation has led to a number of Sherwood number vs. Reynolds number relationships being developed for the hydraulically smooth rotating cylinder. The following table is a compilation of a number of the relationships. CYLEXPERT uses this type of relationship for the rotating cylinder electrode portion of the calculation.
This figure  (328k))
by iteratively solving equation (3) followed by using
equation (4) to estimate the quantity Sh/Scc. The area denoting mass transfer
control in the diagram found in Method B for assessing mass transfer control in
CYLEXPERT is the area surrounding all of these curves. As discussed in
Assessing Mass Transfer Control or in
Using CYLEXPERT-a step by step procedure
the actual points measured in any system should be placed on a plot
such as this one to more fully assess the degree of mass transfer control.
Though the Sherwood number vs. Reynolds number relationships are clustered fairly closely together, the relative difference between the largest and smallest estimates at a Reynolds number of 100 is about a factor of 3 and the difference at a Reynolds number of 106 is about a factor of 2. While equation (3) does provide a reasonable representation of the relationship and a reasonable estimate of the mass transfer coefficient, the estimate is just that, an estimate. As shown in the figure (328k))
could actually provide a slightly better prediction at higher Reynolds numbers
but the difference still lies within the extremes.
The implication of this discussion is that if equation (3) or any of the equations
in the above table are used and measured data points do not lay on it but are still
within the extremes of the variability, then mass transfer control is likely a dominant
contributor to the mechanism.
3. Assessing Mass Transfer Control To determine if mass transfer controls the corrosion rate, one might measure the actual corrosion rate as function of rotation rate either electrochemically or by mass loss. Then, one of two methods could be used to characterize the mass transfer influence on the corrosion process. In all cases, if the data points or slope are similar to those expected from the literature-derived relationship for the rotating cylinder, mass transfer probably controls the corrosion rate. These methods are presented in CYLEXPERT and are also found in the section on Using CYLEXPERT-a step by step procedure.Method #1 This method is more qualitative and may be less accurate.
Method #2 This method is more quantitative and could be more accurate but requires more knowledge of the physical properties of the environment.
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David C. Silverman, Ph.D. - Primary Consultant |
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