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1 An experimental and theoretical investigation of the influence of surface roughness on corrosion in CO 2 environments Mohammed Al - Khateeb*, Richard Barker, Anne Neville, Harvey Thompson Institute of Functional Surfaces, School of Mechanical Engineering, Uni versity of Leeds, Leeds, LS2 9JT mnmaak@leeds .ac.uk 1. Abstract The influence of surface roughness on the rate of corrosion in CO 2 environments without film formation is investigated both experimentally and theoretically. The former measurements are obtained using the l inear p olarisation r esistance technique on a r otating c ylinder e lectrode apparatus with four different samples with roughness 0.5, 6, 20 and 34 m. Experimental measurements of corrosion rate for smooth and rough surfaces are compared against pre dictions from a modified form of a mechanistic CO 2 corrosion model [1] , where mass transfer coefficients are specified using a new correlation for rough surfaces [2] . Test conditions selected for comparison consisted of a CO 2 - saturated 1 wt.% NaCl brine at 25 o C covering pH values between 4 and 6. X65 carbon steel samples were used as working electrodes, with rotational rates varying from 1000 to 4000 rpm. Agreement between experimental and theoretical corrosion rates is good and demonstrates clearly how inc reased surface roughness accentuates corrosion rates and mass transfer coefficients, and that the latter need to be accounted when implementing theoretical models. Keywords: Electro - chemical modelling, experiments, surface roughness. 2. Introduction CO 2 corr osion of steel in oil and gas production and transportation systems is an important issue of practical and commercial interest. When CO 2 dissolves in brines it permits the formation of carbonic acid which is corrosive towards equipment made from carbon ste e l. This process is capable of affecting asset integrity, the environment and safety of personnel [3] . 2 In the context of oil and gas pipelines, the surface roughness of steel pipelines delivered to coating yards can be in the order of 20 m and may even ex ceed 50 m [4] , yet laboratory experiments in CO 2 environments tend to be with samples which are wet - ground to sub - micron surface finished (i.e. 1200 grit SiC paper). This is despite the fact that t he effect of surface roughness is believed to contribute s ignificantly towards corrosion rate s in CO 2 - containing environments [5] based on the notion that surface topography can affect the hydrodynamic and mass - transfer boundary layer which can in turn influe nce the electrochemical response of the corroding mater ial. A review of the literature reveals that surface roughness is actually believed to control the rate of mass transfer to surfaces, as opposed to the geometry of the system itself. These effects have been quantified using several empirical correlations linking the Sherwood, Reynolds and Schmidt numbers, see e.g. Poulson [6] . An assessment of t he influence of roughness on mass transfer has been conducted on geometries such as rectangular ducts [7] , pipes [8] , the rotating disk [9] and the rotating cylinde r electrode [10] , the latter of which is the focus of this study. T he general consensus of these studies is that rougher surfaces lead to an accentuation in mass - transfer and hence, the rate of material dissolution in diffusion - controlled processes . Resea rch as early as the 1930s by King and Howard [11] demonstrated that in diffusion - controlled environments, surface roughness has a major effect on enhancing the rate of metal dissolution. These observations are also supported by the work of Ibl [12] and by Brenan and Trass [13] in the 1960s who studied the effect of roughness on the diss olution of crystalline surfaces with different degrees of roughness. Their results showed that the rate of mass transfer increased fourfold with the increase in surface rough ness height from ~2.5 to 10 딀m over a Reynolds number range from 8,000 to 60,000. Given that some of the electrochemical processes on carbon steel surfaces in CO 2 environments are influenced by mass - transfer (particularly in solutions close to pH 4), and that numerous studies have shown rough surfaces can enhance mass - transfer, there is clearly a need for greater awareness and fundamental understanding of the role of surface roughening on corrosion rates in CO 2 environments and how this relates to the tran sport of species to and from the steel surface and the impact that this has on material dissolution. 3 In relation to experimentally observing the effect of surface roughness, the Rotating Cylinder Electrode (RCE) provides a convenient means of generating a uniform reaction environment for fundamental and engineering studies of corrosion and mass - transfer under open - circuit, controlled potential or constant current conditions, and is the experimental apparatus of choice for this study. The recent review by Wa lsh et al. [14] provides a comprehensive summary of the range of applications and previous experimental studies in RCE geometries. This present paper utilises the Linear Polarisation Resistance (LPR) technique in conjunction with the RCE to study the effe ct of surface roughness on carbon steel corrosion rate in CO 2 environments. It extends the recent experimental study of Al - Khateeb et al [2] , who explored the influence of surface roughness on mass - transfer through the application of the limiting current t echnique for a range of rotational speeds in NaCl solutions saturated either with N 2 or CO 2 at pH=3 and pH=4. The previous paper by Al - Khateeb et al [2] provides reliable benchmark mass - transfer data, which is used in this work to assist in the constructio n of a theoretical CO 2 corrosion prediction model for rough surfaces which are then assessed and validated against electrochemical corrosion measurements. Th ese insights into the role of surface roughness in corroding systems will improve understanding of the need to account for roughness in practical design decisions affected by material selection and pipeline thickness , for example oil and gas production and processing systems The recent review of CO 2 corrosion models by Kahyarian et al. [15] identified three categories: empirical , semi - empirical and mechanistic . Empirical/semi - empirical ones are simple models, developed when there is limited fundamental understanding of the physical phenomena, often taking the form of statistical fits based on experiment al data or correlation factors. Although these can be useful for representative conditions for which they have been designed , they should be used with caution outside their range of application. Important examples of empirical/semi - empirical models include those of de Waard & Milliams [16] and Pots et al. [17] A number of useful reviews of such models have appeared in the literature, see e.g. Olson [18] and Nesic [19] . A number of mechanistic models have been developed to provide a physical basis for corros ion rate predictions and to address the inherent limitations of empirical and semi - empirical models (see Kahyarian et al. [15] . Elementary mechanistic 4 models de - couple the main physicochemical phenomena in corrosion processes, namely mass transfer, charge transfer and chemical reactions, and use parameters which have a sound theoretical foundation. Examples of these are widely used in corrosion engineering analyses (see e.g. Sundaram et al. [20] , Han et al. [21] ) but their neglect of homogeneous chemical r eactions is a serious shortcoming. These limitations have been addressed in more comprehensive mechanistic models based on the Nernst - Planck equation for mass conservation. Nordsveen et al. [5] solved the Nernst - Planck equation to describe the time - depend ent mass transfer of species in the boundary layer using a computationally expensive multi - node approach. The computational requirements of their multi - node approach were alleviated by Remita et al. [22] who used a simplified, steady - state form of the Nord sveen et al. [5] model, which may be sufficient for practical purposes of corrosion rate estimation. Zheng et al. [1] recently proposed a novel, and much more computationally efficient 2 - node approach to predicting corrosion rates in cases of mixed CO 2 /H 2 S corrosion, which calculates the surface concentrations and corrosion rates at corroding surfaces using mass transfer coefficients and bulk concentrations of each species. This paper uses a new correlation based on the 2 - node approach for mass - transfer coe fficients in RCE environments with rough surfaces to predict corrosion rates for comparison with the experimental data gathered. The paper is organised as follows. The experimental methods and RCE experiments are first described before the corrosion modell ing strategy is described and validated. A series of experimental and computational results are then presented which explore the effect of surface roughness, RCE velocity and pH on corrosion rate at a temperature of 25 C in a CO 2 - saturated 1 wt.% NaCl brin e. Finally, conclusions are drawn . 3. Experimental Methods Four RCE samples with different surface k down to nanometer accuracy . The specificati on s of the samples are given in Table 1 where d is the diameter of the samples (m), e is the average distance from peaks to valleys on the rough surface (m) , A is the projected surface area of electrode and/or area of smooth electrode (m 2 ); A R is the real surface area of the rough electrode (m 2 ); with images of the samples and an example of the profilometry data provid ed in Figure 1. 5 Table 1 RCE surface properties of the four samples considered in this study. Sample Roughness height (e) (d/e) (A R /A S Smooth 0.5 24000 1.004 Rough 6 2000 1.108 20 600 1.219 34 353 1.234 (a⤀ (b) Figure 1 (a) Images of the RCE samples with different surfaces roughnesses of 0.5, 6, 20 and 35 m and (b) example 2D output from profilometry analysis of the 6 m roughness sample . 6 Experiments were conducted in a 1L glass cell at atmospheric pressure and 25멃. The three electrode setup shown in Figure 2 was employed for all expe riments, which comprises a working electrode (RCE sample), a reference electrode (Ag/AgCl) and a counter electrode (platinum). Electrochemical measurements were performed using an ivium compactstat connected with a computer. The tests were performed at rotational velocities between 1000 and 4000 rpm in a 1 wt.% NaCl solution saturated with carbon dioxide (CO 2 ) gas for 24 hours prior to the experiments to ensure that the system was free from oxygen. Bubbling o f gas into the electrolyte was also maintained over the duration of each experiment and temperature was controlled with the aid of a hotplate and thermocouple. The pH of the system was initially measured using a pH probe directly immersed into the electrol yte and adjusted to the desired value using sodium bicarbonate (NaHCO 3 ). The full matrix of test conditions evaluated is provided in Table 2 . Table 2 Experimental test matrix. Working Environment CO 2 pH 4 - 6 Temperature 25 C Total Pressure 1bar NaCl Concentration 1wt.% Rotation Speed 1000 4000 rpm 7 Figure 2 Schematic of RCE three electrode cell . Prior to each experiment the samples were degreased with acetone, rinsed with distilled water and then dried with compressed air before mounting onto the RCE shaft. The open circuit potential of the material was then allowed to stabilise fo r 10 minutes before starting each experiment. Following stabilisation of the OCP, i n situ corrosion rates were recorded by means of the DC linear polarisation resista nce (LPR) technique. LPR measurements were conducted by polarising the sample 넱5 mV vs. the OCP , scanning at a rate of 0.25 mV/s to obtain a polarisation resistance, R p o hm.cm 2 . LPR measurements were undertaken every 10 minutes over a total period of 3 h. In all experiments, the solution resistance, R s o hm.cm 2 ) was determined after LPR measurements were complete using electrochemical impedance spectroscopy (EIS). This consisted of polarising the sample 넵 mV vs. the OCP using a frequency range from 20 kHz to 0.1 Hz. The value of R s was subtracted from R p to produce a charge - transfer resistance, R ct o hm.cm 2 ) which was used to determine the corrosion rate behaviour with time: 8 (1) Potentiodynamic measurements were also performed on each sample at the e nd of the 3 h test. This technique was used to generate Tafel polarisation curves to determine the anodic and cathodic Tafel constants (β a c , respectively in mV/decade) and ultimately an appropriate Stern - Geary coefficient ⠀B) to enable calculation of corrosion rates from the values of R ct determined as a function of time in each experiment . Tafel polarisation curves were also collected by performing individual anodic and cathodic sweeps starting at OCP and scanning to either 250 mV or - 500 mV vs. OCP, respectively at a scan rate of 0. 5 mV /s. Only one Tafel curve ⠀either anodic or cathodic) was generated at the end of each experiment as significant polarisation can alter the surface characteristics and/or result in contamination of the test solution. From the polarisation curves produced, i t was possible to determine β a c by measuring their respective gradients over regions were linearity was observed between the applied voltage and the natural log of the measured current. The Tafel slope measurements were used to determine the Stern - Ge ary coefficient (B) , and the corrosion current density, i corr ⠀mA/cm 2 ): (2) (3) Where a and c are the coefficients which characterize the anodic and cathodic Tafel slopes of corrosion process in (V). The i corr value obtained was used i n combinatio n with Faraday’s Law and the measured values of R ct to determine the corrosion rate (CR) in mm/year (4) 9 where K is a conversion factor to obtain corrosion rate (CR) in units of mm/year (K = 3.16x10 5 ), M Fe is the molar mass of iron (55.8 g), n is the number of electrons freed in the corrosion reaction (2 electrons), is the density of steel (7.87 g/cm 3 ) and F is the Faraday constant (96,485 coulomb/mole) . Each experiment was repeated at least twice and the values of corrosion rate reported in this work reflect the average of multiple LPR measurements over both 3 hour tests complete with errors bars which indicate the maximum and minimum corrosion rates determined from the individual measurements across all experiments. 4. Corrosion Modelling The experimental measurements of the effect of surface roughness on corrosion ra te are compared with predictions based on the computationally - efficient 2 - node approach, proposed recently by Zheng et al. [1] , which calculates species concentrations at the corroding surface in a thin surface water film of thickness x by accounting for homogeneous chemical reactions, mass transfer of species and electro - chemical reactions at the corroding surface. This leads to the equation (5 where c s,j is the surface concentration of species j, N in,j is the flux of species j fr om the bulk into the surface water film, N out,j is the flux of species out of the surface water film due to the electro - chemical reactions and R j is the rate of chemical reaction of species j in the surface water film, see Figure 3. There are 7 species to be accounted for, namely CO 2 , H 2 CO 3 , , , , and Fe 2+ . 10 Figure 3 A Schematic diagram of two node model. 5. Mass Transfer Fluxes The mass transfer fluxes, N in,j are given by (6 where k m,j (m/s) and c b,j (mol/m 3 ) are the mass transfer coefficient and bulk concentration of species j respectively. Mass transfer coefficients are generally functions of the geometry and of the Reynolds, Schmidt and Sherwood numbers. Thus, the mass transfer coefficient for turbulent single phase flow inside a pipe can be calculated using the Berger and Hau [23] correlation: (7 where the Sherwood number Sh=(kd)/D, in terms of the mass - transfer coefficient, k (m/s), RCE diameter, d (m), and diffusion coefficie nt D (m 2 /s), the Reynolds number Re=(U RCE d)/ , where ν is the kinematic viscosity ( m 2 /s) and the Schmidt number Sc= ν/D. For an RCE the following Eisenberg correlation [24] has been shown to be accurate for smooth surfaces (8 11 However, for rough surfac es the recent experimental study of Al - Khateeb et al [2] proposed the following correlation for mass transfer for rough samples: (9) (10) where f c is the friction factor of RCE ; e is the average distance from peaks to valleys on the rough surface (m). The properties of the specie s depend on temperature, and are provided in Tables A.1, with reference diffusion coefficients at 20 C provided in Tabl e A.2. 6. Electrochemical Fluxes The rate of the electrochemical reactions at the metal surface depends on the surface concentrations of species involve d in electrochemical reactions and on the temperature [5] . The cathodic reactions are given by the reductio n of hydrogen, carbonic acid (via a buffering effect) and water given by respectively (11 (12 (13 It is important to stress that carbonic acid has been shown to contribute to the cathodic reaction via a buffering effect whereby it is transporte d to the steel surface and dissociates, resulting in the reaction shown in equation (11), hence there is a distinction in the pathway, but the ultimate hydrogen evolution reaction is the same. The anodic reaction is given by equation (14), although this is a simplification and is actually believed to occur through a number of complex, intermediate reactions as described by Nesic et al [25] . (14 12 Since the electrochemical reactions involve exchange of electrons, the reaction rate represents the rate at wh ich electrons are released or consumed. These exchange current densities can be calculated using the following formula: (15 where E (V) is the potential of the corroding surface and E rev (V) is the reversible potential of a specific reaction. A positiv e sign refers to the anodic reaction and a negative sign refers to a cathodic one . The exchange current densities take the general form (16 where the reference parameter values and exponents a 1 , a 2 and a 3 for each of the reactions are given in Table A .3 in the Appendix. For the hydrogen reduction reaction the total current density i H + is given by the following relationship between the charge transfer - controlled exchange current , i 0,H + , and the mass - transfer limited current , Zheng et al. [ 26] , namely (17 T he total current density for carbonic acid is calculated similarly, with (18 where (19 where f is the flow factor for the limiting current of carbonic acid which equal: 20 is the ratio of thickness of mass transfer diffusion layer to reaction layer ⠀ / 13 21 22 For spontaneous corrosion the potential , E, at the corroding surface can be found by equating the total cathodic and anodic current densities : 23 Once E is determined, the electrochemical fluxes of species can be calculated from equations (1 5 ) and (1 6 ) to yield 24 where F=96485 C/mol is Faradays constant, n i is the number of mol es of electrons created per mole of species in the i th electrochemical reaction: n i =1 for all cathodic reactions and 2 for the anodic reaction. A positive or negative sign is taken for cathodic and anodic reactions, respectively. 7. Chemical Reactions For CO 2 corrosion, the water chemistry is determined by the combined effects of carbonic acid hydration, carbonic acid dissociation, bicarbonate ion dissociation and water dissociation. These reactions are respectively 25 26 (27 (28 (29 14 The rates of each of these reactions depends on temperature, carbon dioxide partial pressure and ionic strength [5] . The reaction rate constants used here are given in Table A.4 in the Appendix. In the bulk, the equations for the 6 different specie s (CO 2 , H 2 CO 3 , , , and ) are created as follows. Firstly, CO 2 molecules are c onsumed by carbonic acid hydration: 30 a nd are forward and backword reaction rate constants of carbonic acid hydration . H 2 CO 3 is created by carbonic acid hydration and carbonic acid dissociation: - 31 and are forward and backword reaction rate constants of carbonic acid dissociation . ions are created b y carbonic acid dissociation and bicarbonate ion dissociation: 32 and are forward and backword reaction rate constants of bicarbonate ion dissociation . ions are created by bicarbonate ion dissociation: 33 OH - ions ar e created from water dissociation: 34 and are forward and backword reaction rate constants of water dissociation. H + ions are created by carbonic acid dissociation, bicarbonate ion dissociation and water dissociati o n: 15 35 Equations 30 - 35 ) for the 6 different species (CO 2 , H 2 CO 3 , , , and ) in the bulk are solved using an efficient Newton - Raphson numerical scheme implemented in Python. 8. Steady - state Corrosion Model This study investi gates the effect of surface roughness of corrosion rates in non - film - forming conditions where corrosion rate attains a steady - state. In this case it can be shown, [27] that the 2 - node model can be re - cast into the following simplified form: 36 37 38 39 40 41 42 These 7 equations are solved using a Newton - Raphson numerical scheme implemented in Python. Once (A/m 2 ) has been determined the corrosion rate in mm/year is 1.16 . 16 9. Experimental and Theoretical Results Model Va lidation The chemical solver is validated first. Solutions of equation (22⤀ - (26) for the bulk chemistry are validated against the experimental results of Meyssami et al. [28] and Tanupabrungsun et al. [29] . These are shown in Figure 4 . Agreement is excelle nt in both cases. Figure 4 Validation of bulk chemistry predictions against Meyssami et al. [28] and Tanupabrungsun et al. [29] . The corrosion rate predictions obtained here are now compared wit h the pipe flow loop experiments for smooth surfaces and corrosion model predictions for different pH values and flow speeds from Zheng et al. [1] in Figure 5 . 17 Figure 5 Comparisons between the present model predictions for pipe flow against experiments and predictions of Zheng et al. [1] at 1 bar CO , 2グC, d=0.01m, various pH values and various velocities. 18 The present corrosion model predictions agree reasonably well with both the experimental and theoretical results of Zheng et al. [1] . The prediction that corrosion rates are independent of flow sp eed for pH=6 indicates that the process is surface chemical controlled than mass transfer controlled in these cases. A new series of experimental measurements and theoretical predictions of corrosion rate for RCE systems in aqueous CO 2 solutions, with bot h smooth and rough surfaces, is now presented. For smooth RCE samples, corrosion rate values from the RCE experiments were compared with the model’s predictions by varying the solution pH and the rotational speed of the RCE. The effect of velocity was stud ied at pH=4, 5 and 6. The rotation speed started with 1000 rpm (0.628 m/s) and increased up to 4000 r pm (2.512 m/s). 19 Figure 6 Comparisons between experimental and theoretical corrosion rates at 1 bar total pressure, 25뀀C, various pH, and different rotation speeds for a smooth RCE. Figure 6 shows that for pH=4 corrosion rate increases with rotational speed, indicating that mas s transfer from the bulk is important, w hereas for the higher pH values, where the bulk concentration of H + are orders of magnitude smaller, mass transfer of H + ions is far less important. This leads to a reduction in the cathodic consumption of H + ions and a corresponding reduction in corrosion rate. Good agreement was obtained between the model predictions and the experimental 20 results for all cases considered. The average difference between the model and the experiments is about 14, 10 and 18 % for pH=4, 5 and 6 respectively. The first series of experiments on rough surfaces were carried out in static conditions and normalised based on their actual surface area determined by profilometry, as opposed to their projected area. The purpose of this analysis was to confirm that the machining of the t est samples did not modify or cold work the surface such that the corrosion rate of the material was enhanced. Tests were performed with four samples of different surface finishes of carbon steel X65 in a 1 wt.% NaCl solution saturated with carbon dioxide (CO 2 ) gas. The pH and temperature were 4 and 25°C respectively. Figure 8 presents the corrosion rate results after correcting with the true surfa ce area. It is clear that correcting for area leads to no significant change in corrosion rates across all roug h surfaces, indicating that the machining process does not influence the dissolution behaviour of the steel. Figure 8 Static corrosion rate experimental results at 1 bar total pressure, 2㖰C, various surface finish a fter correcting for total surface area . These tests were then extended to cases of turbulent flow over rough carbon steel X65 surfaces. The mass transfer coefficients used in the 2 - node corrosion model are calculated using the new correlation for mass transfer to rough RCE surfaces proposed by Al - Khateeb et al. [2] , equations ( 9) and (10). All current densities, and hence the corrosion rate, were expressed with the projected surface area. Figure 9 shows the experimental and theoretical corrosion rate results as a function of RCE 21 velocity for each of the four different roughness values. It is clear that the corrosion rate increases with the surface roughness. For example, at 3000 rpm the corrosion rate increases with 22 , 46 a nd 57 % as the surface roughness increases from d/e=2000 to d/e=600 and d/e=353 respectively . The modified 2 - node model also agrees very well for all roughness cases with average discrepancies of 10 , 6 , 5. 6 and 12.5 % in comparison with the experimen ts for the four roughness cases. The average difference between the model and the experiments is around 8 % and the maximum deviation is about1 9 % . 22 Figure 9 Comparisons between model results and experiment results at 1 bar total pressure, 25뀀C, various pH, and different rotation speed for different surface finishes. Several explanations for the effect of surface rough ness on increasing corrosion rate has been discussed in the literature. It is generally assumed that the roughness peaks disturb the viscous layer and the turbulence generated reduces the resistance to mass transfer across the concentration boundary layer and in the valleys between the roughness peaks [3 1 ] . The analysis of mass transfer intensification is based on behaviour of turbulent eddies. These eddies penetrate into a cavity on a wall 23 causing deceleration in their motion due to viscous friction with t he surface. The process of deceleration is totally non uniform . Inside these cavities, the non - uniformity causes formation of such areas where turbulence fluctuations have relatively high kinetic energies at distances from the surface which are significant ly smaller than the diffusive layer thickness calculated assuming the wall is smooth [3 2 ] . 10. Conclusion The RCE experiments have shown that surface roughness is very influential and leads to a monotonic increase in corrosion rate as surface roughness increa ses. Such accentuation of corrosion is attributed to a combined effect relating to increased surface area and enhancement of mass - transfer due to the hydrodynamic effects induced by surface roughness. The RCE experiments carried out here have shown that c oupling the well - known Eisenberg mass transfer correlation into the 2 - node corrosion model proposed by Zheng et al [1] can predict corrosion rates accurately for smooth surfaces. For rough surfaces, the Eisenberg correlation is inadequate, and should be re placed by a correlation which accounts for the effects of surface roughness. This study has shown that a new correlation for mass transfer coefficient, equations ⠀9) and ⠀10⤀, can be used within the 2 - node modelling approach to predict corrosion rates from rough surfaces accurately. In the present study the discrepancy between experimental and predicted corrosion rates is typically around 8 .5%. Such an approach may be useful for predicting corrosion rates from rough surfaces in critical infrastructure assoc iated with, for example, the oil and gas production and nuclear processing industries. Despite the success of the 2 - node model in the present context, there is a need for far greater understanding of the physical mechanisms by which roughness accelerates t he corrosion rate. Recent advances in high fidelity Computational Fluid Dynamics , Busse et al. [33], are leading to the ability to resolve the important localised flow features over rough surfaces, and will 24 further aid both the physical understanding and p redictive capability of corrosion processes over rough surfaces. A p p endix Table A.1 Species properties as a function of Temperature [3 4 ] . Density Dynamic viscosity Diffusion coefficient T ref is the reference temperature =20 C , = 1.002 kg/(m.s) Table A.2: Reference Diffusion Coefficients for Each Species in the Model Species Diffusion Coefficients (m 2 /s) Reference CO 2 1.96×10 - 9 [3 5 ] H 2 CO 3 2×10 - 9 [36 ] HCO 3 - 1.105×10 - 9 [5] CO 3 2 - 0.92×10 - 9 [36 ] H + 9.312×10 - 9 [5] OH - 5.26×10 - 9 [5] F e 2+ 0.72×10 - 9 [36 ] 25 Table A.3 Current density parameters for the cathodic and anodic reaction [5, 26, 3 7 ] . The e xchange current density is i ref a 1 a 2 a 3 T ref E rev b A/m 2 molar molar Molar KJ /mol V V 2H + +2e H 2 0.05 0.5 10 - 4 0 N/A 0 N/A 30 25 2H 2 CO 3 +2e HCO 3 - +H 2 0.018 - 0.5 10 - 5 0 N/A 1 10 - 4 50 20 2H 2 O +2e H 2 + 2OH - 0.002 0 N/A 0 N/A 0 N/A 35 25 Fe Fe 2+ +2e 0.1 ⠀RCE⤀ 1 ⠀pipe) 2 for PH4 1 for 4H5 0 for P�H5 10 - 4 1 for P CO2 b ar 0 for P CO2 �1bar 0.036 6 0 N/A 37.5 25 - 0.488 26 Table A.4 Chemical reaction rate constants The chemical constants used here are given below, together with their source references. Note: T f is the temperature in degree Fahrenheit, T absolute tem perature in Kelvin, T c is the temperature in Celsius, I is the ionic strength in molar, and p is the total pressure in psi. (molar/bar ) [3 8 ] molar 2 ) [ 39 ] molar - 1 s - 1 ) [4 0 ] [4 1 ] s - 1 ) [4 1 ] [38 ] s - 1 ) [5] (molar ) [3 8 ] s - 1 ) [5] 27 References 1. Zheng, Y. Ning, J., Brown, B., Nesic, S. 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