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David C. Silverman |
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Wall Shear Stress vs. Mass Transfer Lively discussions have always been a part of choosing the "appropriate" laboratory methodology to model velocity sensitive corrosion or using the appropriate factor to derive the operating conditions in the laboratory. Two approaches that have received attention are use of equality of wall shear stress and equality of mass transfer coefficients between the rotating cylinder electrode and the geometry being modeled. For fully developed turbulent flow, the two methodologies are linked because of the relationship that exists between the friction factor and the mass transfer coefficient (Silverman, D. C., Corrosion, 40(5), 220 (1984)1 (467k)).
Equations (6) and (7) provide insight into the linkage which is derived directly from
the analogy: (4)Since  (6)substituting equation (6) into equation (4) results in  (7)What equation (7) means is that though the shear stress at the wall and the mass transfer coefficient are two distinct quantities, they are related when there are attached boundary layers. Boundary layer separation may lead to other conclusions. The implication is that using either measure should result in somewhat related predictions as long as the hydrodynamics (e.g. turbulent flow, fully developed boundary layers, no detachment of the boundary layer, and minimal surface roughness) and the corrosion mechanism (e.g. degree of mass transfer control of corrosion) are the same in both the modeled and modeling configurations. This linkage suggests that these quantities should not be treated as completely unrelated entities. The approach of using the mass transfer coefficients to relate flow effects between geometrical configurations has been shown to provide reasonable predictions if corrosion is controlled by mass transfer and surface roughness is absent (Silverman, D. C., Corrosion, 44(1), 42 (1988)1 (517k)).
Some justification for being able to use equality of mass transfer coefficients is as follows. The mass transfer boundary layer for large Schmidt numbers should be much thinner than the hydrodynamic boundary layer. The concentration of the rate limiting species would become the free stream concentration (concentration gradient disappears) while the velocity is still less than the free stream velocity (velocity gradient still exists). The wall shear stress has been related to the velocity profile within the boundary layer. Since that velocity profile drives mass transfer between the fluid and the surface, the wall shear stress and mass transfer coefficient are interrelated. This thought process led to the hypothesis that the friction velocity (  )
and not the free stream velocity may be the appropriate scaling velocity when trying
to relate geometries handling fluids of high Schmidt number
(Silverman, D. C., Corrosion, 44(1), 42 (1988)1 (517k)).
A related question is: If the mass transfer coefficients are set equal among geometries and one has fully developed turbulent flow, fully developed hydrodynamic and mass transfer boundary layers, and a hydraulically smooth wall surface, would the wall shear stresses be similar? Equation (7) suggests that they would probably not be equal but they may have a similar order of magnitude. This similarity was investigated for the rotating cylinder electrode, the wall jet region of the impinging jet, and the pipe. The calculation was performed by assuming rotation rates of 100 to 104 RPM for a 1.91 cm cylinder and calculating the corresponding mass transfer coefficient and wall shear stress for the rotating cylinder electrode. Then, those mass transfer coefficients were assumed as the average mass transfer coefficient over the distance of 3 to 5 times the 0.254 cm nozzle diameter in the wall jet region of impinging flow and for a 5.08 cm diameter pipe to calculate the expected wall shear stresses for those geometries. Care was taken to ensure that the correlations used were reasonably valid over the respective ranges in Reynolds numbers. Equations (2) and (3) were used for the friction factor and Sherwood number versus Reynolds number relationships for the rotating cylinder for simplicity. Appropriate equations were used for the wall jet region of the impinging jet (Giralt, F. and Trass, O., Canad. J. of Chem. Eng., 53, 505 (1975), ibid., 54, 148 (1976) and the pipe (Berger, F. P., Hau, K.-F. F.-L., Int. J. Heat Mass Transfer, 20, 1185, (1977)). The wall shear stresses are comparable (within a factor of 5) but not the same for the three geometries. This figure )
shows the results. The linearity on a log-log scale is not surprising because of the
nature of the boundary layer correlations. The plot suggests that if one has
hydraulically smooth walls, fully established turbulent flow, and fully established
boundary layers, one cannot categorically rule out either using equal wall shear stress
or equal mass transfer coefficients to set the operating conditions in the laboratory so
that a mass transfer affected corrosion mechanism can be modeled in the laboratory.
Violating any of the above provisos can negate the relationship.
Requiring similarity in wall shear stress or similarity in mass transfer coefficients have both been proposed to establish flow conditions (rpm) within the rotating cylinder electrode that could enable predictions of velocity sensitive corrosion mechanisms in other geometries. Which approach is most appropriate is a subject of debate. The two approaches are linked because of the relationship that exists between the friction factor and the mass transfer coefficient indicated in equation (7) above. One possible way to satisfy similarity in both criteria simultaneously might be to create experimental conditions under which the wall shear stress and the mass transfer coefficient are both approximately equal between the field and experimental configurations. Equation (7) provides the background for the approach. If conditions are established in the same environment so that both the mass transfer coefficient and fluid velocity are set equal simultaneously in the two geometries then the shear stress at the wall would also be equal in the two geometries. This concept means that if the rotating cylinder electrode could be operated so that both its mass transfer coefficient and peripheral velocity are the same as the mass transfer coefficient and mean velocity in the pipe, the possible mass transfer affected corrosion mechanism might be modeled independent of whether equality of wall shear stress or equality of mass transfer coefficient is the most appropriate approach. Following is how this method can be applied to using the rotating cylinder electrode to model corrosion in a straight pipe. The pipe and rotating cylinder were assumed to be hydraulically smooth. Equations were presented to estimate the relationship between the velocities in the rotating cylinder electrode and pipe that would create equal mass transfer coefficients. These equations were derived using the log-linear correlation of the Sherwood number (mass transfer coefficient) vs. Reynolds number data in these two geometries. By setting the velocity of the rotating cylinder equal to that of the pipe, the equations can be rearranged so that the rotating cylinder electrode diameter can be estimated as a function of the pipe diameter. The mass transfer coefficients are equal when these equations are satisfied. This figure )
shows the cylinder diameters as functions of pipe velocity for
pipe diameters of 5 cm, 10 cm, 20 cm, 50 cm, and 100 cm. This figure
)
shows the corresponding rotation rates (in rpm). The viscosity,
density, and Schmidt numbers were assumed to be 0.01 centipoise, 1 g/cm3, and 1000
respectively. Two implicit boundaries are the Reynolds number limits of the correlations
used in equations (4) and (5), from less than 1000 to greater than 105 for
the cylinder and 2300 to greater than 106 for the pipe. The lines in the figures
have been drawn so that all pipe Reynolds numbers would be turbulent. Not all of the
dimensions and rotation rates suggested in the figures can be achieved in the laboratory.
Limitations may exist on the electrode diameter because of the configuration of the
rotating cylinder apparatus and on rotation rates because of the drive mechanism.
The lines with arrows in both figures have been drawn to show what range of values
of cylinder diameter, pipe diameter, and pipe velocity would enable cylinder rotation
rates between 10 rpm and 104 rpm and cylinder diameters between 1 and 10 cm.
The results imply that some combinations of pipe diameter, pipe velocity, rotating
cylinder diameter, and rotation rate exist so that the mass transfer coefficient
and the wall shear stress can be similar simultaneously. Additional information can be
found in D. C. Silverman, "Conditions for Similarity of Mass-Transfer Coefficients
and Fluid Shear Stresses Between the Rotating Cylinder Electrode and Pipe",
Corrosion, Vol.61, No.6, p. 515, 2005.
1 (403k)
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David C. Silverman, Ph.D. - Primary Consultant |