Parametric Wall Shear Stress Characterization of the Rotating Cage Test Method

The rotating cage^1-3  is a standardized apparatus that enables investigation into the performance of corrosion inhibitors and materials under a variety of test conditions. Parameters such as fluid composition, temperature, and pressure can be varied to simulate specific user requirements, and, like other techniques such as the rotating cylinder electrode and jet impingement, testing includes the influence of flow. The method has been widely used to further understanding of corrosion, and its inhibition, in pipeline systems.^4-8  While not directly simulating pipeline flow, the method is considered comparable when the fluid dynamic shear stress on the test coupons is equal to that on the pipe wall. The objective of this paper is to present a parametric characterization of the wall shear stress on the test coupons over the full range of rotating cage conditions.

Previous validation studies have been performed to gain confidence in the computational fluid dynamics (CFD) method:

• 1.

  Comparison⁹  of the magnitude and variation of shear stress over the surface of the coupon predicted by the CFD calculation to the measurements of Deslouis, et al.¹⁰ 

• 2.

  Comparison⁹  of the variation of shear stress over the coupon surfaces with the corrosion pattern observed on test specimens.

• 3.

  Comparison¹¹  of the vortex depths predicted by the CFD calculation to those observed in laboratory measurements.

• 4.

  The discovery of CFD solutions both with and without a vortex for a given rotational speed and operating condition and observation of the same behavior in the laboratory.¹¹ 

These validation studies have provided confidence in the results of the CFD calculations. The methodology for performing such calculations will be described in the section, Model Description.

Kumar, et al.,¹²  performed CFD calculations and experimental studies for the ASTM G170 rotating cage and found that the variation in predicted shear stress correlated reasonably well with the corrosion pattern observed in the experiments. They reported wall shear stresses close to those in the literature.^9-11 

As corrosion occurs only in the presence of water, the rotating cage requires considerable water to prevent the coupons being partially exposed to either gas or oil, drawn toward the cage by the vortex; such an occurrence invalidates the test. Experimentally, the authors have found that for a system containing only water and gas, 5 L of water are required to prevent the vortex reaching the cage for conventional test speeds up to 1,000 rpm, and this is what was modeled in the present work.

The ASTM-standardized^1-3  geometry of the rotating cage was used throughout this work. The full 3D apparatus was modeled. The cage itself consisted of two Teflon^† (polytetrafluoroethylene [PTFE]) disks, 80 mm in diameter and 6 mm in thickness, held together with a 32 mm-diameter PTFE cylinder that was 76 mm in length. These disks held eight test coupons that were each 76 mm long, 19 mm wide, and 3 mm thick. The distance between opposite coupons (center-to-center) was 68 mm. Each PTFE disk included a single hole 10 mm in diameter, intended to increase the fluid shear stress on the inside surfaces of the coupons.²  The location of coupon number one was radially adjacent to the disk hole, with coupon number increasing in a clockwise fashion when seen from above. The rotating cage was spun in a counter-clock-wise direction. A schematic of this arrangement is provided in Figure 1, which is also the model used in the simulations. The cage was mounted toward the bottom end of a spindle, 6 mm from the bottom of the test vessel. The spindle was connected to a motor mounted on the vessel lid. The test vessel was typically 150 mm in diameter, with an internal volume of approximately 8 L.

The simulated test vessel contained 5.0 L of liquid and operated at atmospheric pressure and with a downward pull of gravity. For the purpose of this study, the following temperatures were simulated: 4°C, 20°C, and 70°C, representing a wide range of operating temperatures for oil transmission pipelines. Three fluids were simulated: water; Mixed Sweet Blend (MSB), a light sweet crude; and Western Canada Select (WCS), representative of diluted bitumen (dilbit). Data for the crudes were obtained from the publicly available crudemonitor.ca,¹³  where the reported oil properties were averaged from between January 1, 2009 and March 31, 2013. Given the similarity between the sodium chloride brine used in rotating cage testing⁷  and the physical properties of pure water, the two can be used interchangeably with little error. The viscosity and density values used in this work are provided in Table 1.

The standardized atmospheric rotating cage method (acrylic vessel, 8 L volume) is capable of spinning the specimen cage at up to 1,000 rpm. This was therefore used as the upper limit for the calculations. The study considered rotational speeds of 350, 500, 850, and 1,000 rpm.

It is assumed the fluid behavior can be approximated by a pseudo-steady-state result. The validity of this assumption has been described in detail in Senior, et al.¹¹  It is also assumed there are no mechanical vibrations (the system is perfectly coaxial and balanced) and fluid dynamic forces do not result in the vibration of the apparatus.

Generally, the physical behavior of this two-phase, free surface flow is dependent upon liquid viscosity and density, and the surface tension of the liquid/gas interface, which themselves are dependent upon temperature. Variation of the surface tension in the CFD calculations has shown no effect on the rotating cage flow, so only liquid viscosity and density are considered. The gas space was filled with air at the operating temperature and pressure.

The ANSYS-CFX^®†, version 16.0, software package was used to create the model. The model used a rotating frame of reference (moving at the angular velocity of the rotating cage with respect to the laboratory reference frame about the centerline of the shaft), in which the cage and shaft are stationary and the container wall and floor are moving. The computational grid, representing the full geometry of the rotating cage, consisted of approximately 1.5 million hexahedral volumes. Computation and time constraints necessitated the use of a single grid of 1.5 million volumes for the parametric study, so this resource needed to be optimized. The grid refinement work focused on local regions with complex flow patterns and steep gradients. Previous experience with this flow calculation, and its success in matching measurements of wall shear stress and the shape of the air/water interface,^9-11  informed where to concentrate the computational volumes. The grid was refined at the cage to account for the complex geometry and correspondingly complex flow (such as the wake region between coupons where there are small vortices, shown in Runstedtler, et al.⁹ ), leaving enough volume to capture the gas/liquid interface as well as secondary flow vortices (discussed in Senior, et al.¹¹ ) in the space between the cage and container wall. The coupon surfaces, being the location of primary interest, were represented by a mesh having an average element area of 1.85 mm². Special care was taken to resolve the fluid dynamic boundary layer at these surfaces, as will be described below.

The range of simulated liquids results in the flow regime spanning from laminar (for dilbit) to turbulent (for water and light crude oil). Turbulent flow was modeled with the standard shear-stress transport (SST) turbulence model, which is considered appropriate for this rotating flow.¹⁴  All surfaces were modeled as smooth walls with the no-slip condition, i.e., fluid velocity is zero relative to the coupon surface. The coupon roughness effect will be discussed in the Results section.

The wall shear stress is given by: where μ is the fluid dynamic viscosity, u is the flow velocity relative and parallel to the wall, the coordinate, y, is the perpendicular distance from the wall, and the derivative is evaluated at the wall. The wall shear stress is inherently determined as part of the CFD calculation.

For the turbulent simulations with water and light oil, the model used the automatic near-wall treatment, as implemented in the software for the SST turbulence model, to accommodate the wide range of boundary layers at all surfaces including the coupon surfaces. The automatic near-wall treatment switches between resolving the near-wall boundary layer when the wall-adjacent grid points are inside the viscous sublayer and using logarithmic wall functions when the wall-adjacent grid points are outside the viscous sublayer. The scaled, or dimensionless, wall distance, y⁺, as implemented in the software, for the closest grid point to the coupon surface is provided in Table 2. The automatic near-wall treatment is necessary for this parametric study and for the individual cases themselves. Table 2 shows that the average y⁺ value varies by approximately two orders of magnitude over the range of cases. The logarithmic wall functions are necessary because it would not be feasible to resolve the very thin boundary layer for high-Reynolds-number flows, as the computational mesh would need to be extremely large. On the other hand, the computational mesh must also accommodate the lower-Reynolds-number flows, for which the wall-adjacent nodes are within the viscous sublayer (approximately y⁺ < 10), where using logarithmic wall functions would be incorrect. Moreover, Table 2 shows that, even within a given case, the maximum and minimum y⁺ values vary widely, meaning the automatic near-wall treatment is required not just over a range of cases but for the individual cases themselves. This wide variation in y⁺ is a result of the complex flow past the coupons (shown in Runstedtler, et al.⁹ ). The software documentation recommends that, when using the logarithmic wall functions, wall-adjacent grid points should have y⁺ < 200. The near-wall grid spacing was adjusted to accommodate the requirements of the automatic near-wall treatment. Table 2 shows the result of this work.

The normalized residual target was 10⁻⁶ and the normalized conservation target was 10⁻³. Note that the rotating cage simulations display iteration-to-iteration variability (as reported in Senior, et al.¹¹ ), indicating that the flow is not completely steady-state. The Results section will discuss this further.

The spatial variation of coupon wall shear stress has been described in previous work,^9-11  including examining the effect of the holes in the PTFE disks.¹¹  High wall shear stress could lead to mechanical removal of a corrosion product layer or inhibitor film. Figure 2 provides an example of this variation for 20°C water at 850 rpm. The high shear stress imprint on the inner face caused by flow through the hole in the PTFE disk is apparent, as are high shear stress regions on the outer face and leading edge. Figure 3 provides the variation over all coupon surfaces for all of the water and light oil cases in the form of cumulative distributions. A single characteristic value of shear stress, the area-weighted average, as well as the peak shear stress, have been of primary interest, although other statistics could be extracted from the CFD data. This paper characterizes both the area-weighted average and peak shear stresses.

Senior, et al.,¹¹  reported iteration-to-iteration variability of approximately 10% in the steady-state solver for the case of water at 20°C and 500 rpm, indicating that there is likely temporal variability as well. Runstedtler, et al.,⁹  performed transient simulations and displayed the temporal variability of wall shear stress—both in the area-weighted average and via plots of the spatial distribution.

The area-weighted average wall shear stress values are provided in Table 3. The cases cover a wide range of operating conditions, i.e., Reynolds numbers, as will be shown in Figure 4. The values in Table 3 represent data from a single results file and therefore do not account for the temporal variability in the turbulent regime, previously estimated to be ±10%.^9,11  Note that the area-weighted average is not the same as the 50th percentile (or median) wall shear stress of Figure 3.

The wall shear stress, τ, depends on the following dimensional parameters. where ρ is the fluid density, μ is the dynamic viscosity, r is the radius of the cage given by half the 68-mm distance between opposite coupons (center-to-center) as described in the Model Description section, ε is the surface roughness height, and v is the coupon tangential velocity, given by: where F is rotational frequency in revolutions per minute (rpm). According to dimensional analysis, the relationship in dimensionless form is expressed as: where Re = 2rvρ/μ is the rotating cage Reynolds number, C_f = τ/ρv² is the shear stress coefficient, and ε/r is the relative roughness. Typical coupon surface roughness, ε, was applied to the highest Reynolds number simulations (where roughness would have the greatest effect) and the roughness was found to have negligible effect. Therefore,

This result is plotted in Figure 4. It is emphasized that this result is only applicable to the standard rotating cage as described in this paper.⁽¹⁾

The exact Reynolds number boundary (or range) between laminar and turbulent flows is unknown for the rotating cage class of flow. It is possible that this boundary lies at or around Re = 10⁴, between the dilbit flows and the water and light oil flows, and this possibility seems to be supported by the alignment of the data in Figure 4. There is scatter in the turbulent data that is comparable to the previously estimated temporal variability of ±10%.

In previous work,¹¹  after fitting the data set for water and light oil simulations, it was concluded that, for water and light oil, the average wall shear stress was best fit by a power relationship of the form: with an adjusted R² value of 0.995. This relationship is plotted in Figure 4 and it seems to fit the water and light oil data well. The best fit for the present data is:

There are not significant differences between Equations (6) and (7) so it is recommended to continue using Equation (6) for water and light oil (turbulent) flows (as Equation [6] may already be in use). For dilbit (laminar) flows, the average wall shear stress is best fit by:

In order to use the rotating cage method to simulate pipeline flow in the laboratory, it is necessary to determine the Reynolds number and relative roughness for the real pipe, followed by the friction factor, typically from the Moody chart, and subsequently the wall shear stress on the pipe surface. This procedure can be found in introductory engineering fluid dynamics books, such as Mechanics of Fluids.¹⁵  Having the wall shear stress in the pipe, in order to determine the experimental rotational frequency, F (in rpm), required to replicate the pipe wall shear stress, it is necessary to solve for it via Equations (6) or (8), and Equation (3).

Alternatively, as both the density and dynamic viscosity of water are functions of temperature, Equation (6) can be approximated by: where T is temperature in Celsius and F is the rotational frequency in rpm. This equation is only valid for water between 0°C and 100°C at atmospheric pressure, using the cage geometry described above.

The dimensional analysis applies to any wall shear stress, not just the area-weighted average. Table 4 provides the 90th percentile wall shear stresses obtained from the cumulative distribution functions of Figure 3, and Figure 5 provides the nondimensional characterization of these shear stresses.

For water and light oil (turbulent) flows, the 90th percentile wall shear stress is best fit by:

For dilbit (laminar) flows, the 90th percentile wall shear stress is best fit by:

Using Figure 2 as a guide, the user may then correlate the highest wall shear stresses to observed localized corrosion.

The calculation of average wall shear stress is pertinent primarily when using mass loss to determine average corrosion rate. The wall shear stress value is important because high wall shear stress may lead to mechanical removal of a corrosion product layer or inhibitor film. Wall shear stress, τ, is also related to the mass transfer coefficient, k, that describes the flux of reactive species to the metal surface. The expected form of this relationship for the rotating cage is given by Sh = Sh(Re, Sc), where Sh = 2rk/D is the rotating cage Sherwood number, Re is the rotating cage Reynolds number defined earlier, Sc = μ/(ρD) is the Schmidt number, and D is the mass diffusivity of the reactive species. For example, Silverman¹⁶  provides a relationship between wall shear stress and mass transfer coefficient, τ = ρkv Sc^2/3, for the rotating cylinder electrode, by using the Chilton-Colburn analogy between momentum and mass transfer. It is straightforward to recast this equation in the form, Sh = Re × C_f × Sc^1/3, which has the expected form because, as shown in this paper, C_f = C_f(Re). Therefore, one could obtain the Sherwood number and, consequently, the mass transfer coefficient, k, from the expressions for shear stress coefficient, C_f, given in this paper, provided one accepts the Chilton-Colburn analogy for the rotating cage. Assessing the validity of the Chilton-Colburn analogy for the rotating cage is beyond the scope of this work.

As demonstrated in Figure 2, a small percentage of the coupon surface area will experience significantly higher wall shear stresses than the average, particularly around the blunt leading edge. This has also been observed in other modeling studies.¹⁷  Due to this variability, the determination of localized corrosion rates, for instance through laser profilometry, ought to be correlated against the 90th percentile wall shear stress, calculated through the proposed formulae, rather than the area-weighted average wall shear stress.

A second important caveat for use of the rotating cage is that the method only replicates the corrosion phenomena of single-phase flow. For multi-phase fluids, such as slug flow in oil and gas production tubing, pipeline corrosion phenomena are complex. The turbulent mixing zones both leading and trailing the water slug, can result in transient high wall shear stresses. It is not yet clear whether the wall shear stresses are sufficiently high to mechanically remove corrosion products from the pipe surface, or whether these observed stresses are experimentally limited.^18-20  Bubble impingement and collapse appear to play a significant role in the degradation of corrosion products;^20-22  entrained solids are typically neglected from experimental work but may also play a supporting role in localized corrosion; the reversal in the direction of wall shear stress between liquid and gas slugs may induce corrosion product fatigue.²³  None of these phenomena can be replicated with a rotating cage and efforts at using this method to evaluate corrosion inhibitor performance will not be reliable.

The wall shear stress values determined by simulation in the current work lie well within the range of wall shear stresses that can be experimentally tested using the rotating disk and cylinder electrodes, which permit in situ measurements and a significantly smaller quantity of test fluid. This has additional benefits, such as the ease with which the experimental apparatus can be deoxygenated, and greater uniformity of wall shear stress on the electrode surface.

• •

  A parametric computational fluid dynamics study of the standardized rotating cage was conducted, investigating the influence of rotational velocity and temperature on coupon wall shear stresses, for water, light crude oil, and diluted bitumen. The study covered the range of operating conditions achievable by the rotating cage apparatus. The methodology for extracting coupon shear stresses from the CFD data was presented. Equations were derived from the simulation results that characterize both the area-weighted average and 90th percentile wall shear stresses and that cover the full range from laminar to turbulent flow.

This work was supported by the Program of Energy Research and Development (PERD) of the Government of Canada.