Volume 1 Paper 3

Superposition of diffusion and chemical reaction controlled limiting currents - Application to Corrosion

Srdjan Nesic¤, B.F.M. Pots#, John Postlethwaite* and Nicolas Thevenot¤

¤ Institute for Energy Technology (IFE), P.O.Box 40, N-2007 Kjeller, Norway, e-mail:
# Koninklijke/Shell-Laboratorium, Amsterdam (Shell Research B.V.), P.O.Box 38000, 1030 BN Amsterdam, The Netherlands, e-mail:
* IFE, on sabbatical leave from the University of Saskatchewan, Saskatoon, Canada, e-mail:
¤ Institute for Energy Technology (IFE), P.O.Box 40, N-2007 Kjeller, Norway

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It was observed experimentally that a chemical reaction limiting current can be affected by flow.  In the present study a new more general expression than the one found in literature was derived for the superposition of the diffusion and chemical reaction controlled limiting currents.  It was found that their interaction in the case of CO2 corrosion is significant at temperatures lower than 40°C and velocities  higher than 1 m/s  when the mass transfer layer is of the similar thickness as the reaction layer.


The corrosion of steel in water containing dissolved CO2 gas is a topic of considerable interest with practical applications and substantial economic impact in the oil and gas production and transportation industry.1  When dissolved in water, the CO2 is hydrated to give carbonic acid:


This weak, partly dissociated acid is responsible for high corrosion rates of steel in water CO2 solutions.  The electrochemistry of CO2 corrosion is still not certain although a number of good studies exist in this field.2-8 One of the simplest assumptions is that the dominant cathodic reaction is the reduction of hydrogen ions, where the hydrogen ions are supplied by dissociation of carbonic acid:




The other possibility is the direct reduction of carbonic acid:2


When conducting potentiodynamic sweeps on steel in CO2 solutions, it is difficult to identify a pure Tafel region for the cathodic reaction as a limiting current is reached for relatively small overpotentials.  The origin of this limiting current has been investigated 3,4 previously and is the topic of the present study.



Experiments were conducted at atmospheric pressure in a glass cell. Gas (CO2 or N2) was continuously bubbled through the cell. A three electrode set-up (Figure 1) was used in all electrochemical experiments.  A rotating cylinder electrode with a speed control unit (0-5000 rpm - 5000 rpm for our cylinder  corresponds to a peripheral velocity of 2.61 m/s, a shear stress of 25 Pa., and a Reynolds number of 26175) was used as the working electrode. A concentric platinum ring was used as a counter electrode.  A saturated Ag/AgCl reference electrode was used externally connected to the cell via a Luggin capillary and a porous wooden plug.  The speed of rotation of the working electrode was controlled with the aid of a stroboscope. The pH was followed with an electrode directly immersed into the electrolyte. The temperature was followed with a Pt-100 probe which also served as an input for the temperature regulating system - a  hot plate combined with a magnetic stirrer.  Oxygen concentration was monitored with an Orbisphere oxygen meter.  The concentration of Fe++ was measured occasionally using a photospectrometric method.  The concentration of CO2 in the water was also measured in selected experiments. Electrochemical measurements were made with a Gamry Instruments Inc. potentiostat connected with a PC 486/25 computer. 


A typical construction carbon steel St52 was tested (corresponding to ASTM A537 Grade 1).  Chemical composition of the steel is given in Table 1.  The working electrode was machined from the parent material into a cylinder 10 mm in diameter and 10 mm long.  The exposed area of the specimen was 3.14 cm2.

Table 1. Chemical composition of the St52 steel used for the working electrode (mass%)





















Figure 1.  Schematic of the experimental test cell: 1-reference electrode, 2-gas in, 3-gas out, 4-Luggin capillary, 5-platinum counter electrode, 6-rotating cylinder, 7-temperature probe, 8-pH electrode, 9-working electrode.


The glass cell was filled with 3 litres of electrolyte: distilled water + 1 mass% NaCl. In different experiments CO2 or N2 gas were bubbled through the electrolyte (min. 60 min.) in order to saturate or deaerate the solution.  Monitoring of pH and O2 concentration was used to judge when the solution was in equilibrium.  When needed, HCl or NaHCO3 were added to adjust the pH. The temperature was set and maintained with an accuracy of 1oC in all experiments.

Before each polarisation experiment, the steel working electrode surface was polished with 500 and 1000 grit silicon carbide paper, washed with alcohol, mounted on the specimen holders and immersed into the electrolyte.  The free corrosion potential was followed immediately after immersion.  Depending on the conditions, the potential stabilised within ±1 mV in 1 to 10 min. 

The cathodic and anodic sweeps were conducted separately starting from the free corrosion potential. Typical scanning rate used was 0.1-0.2 mV/s.  The cathodic sweeps were sometimes repeated by sweeping in the opposite direction, without significant difference in the result.  In each experiment the anodic sweeps were conducted only once for a single working electrode specimen and a given electrolyte (starting from the free corrosion potential) since they altered the specimen surface and contaminated the electrolyte with significant amounts of dissolved iron (Fe++>3 ppm).  Typically the Fe++ concentration was kept below 1 ppm. 

Table 2. Experimental conditions

Test solution

water + 1 mass% NaCl

Test material

low carbon steel: St52




1 bar N2 or CO2




<1 ppm

Dissolved oxygen

<20 ppb


static - 10000 rpm

Test duration

0.5 hours

Sweep rate

0.1 - 0.2 mV/s

Potentiodynamic sweep

from -600 to +200 mV vs. Eoc

IR compensation



When conducting cathodic potentiodynamic sweeps in strong acids, limiting currents found are clearly flow dependent (Figure 2).  It was shown previously9 that the rate of the hydrogen evolution reaction in the limiting current region proceeds only as fast as the hydrogen ions can diffuse from the bulk to the surface.

Figure 2.  Potentiodynamic sweep conducted in HCl solution at pH 4 purged with N2 , t=22 °C, 3% NaCl, using a rotating cylinder electrode d=1 cm.

Figure 3. Potentiodynamic sweep conducted in a CO2 solution at pH 4, t=22 °C, 3% NaCl, using a rotating cylinder electrode d=1 cm.

In CO2 solutions it was found3 that the current limitation partly comes from a slow chemical step preceding the charge transfer step (see also Figure 3).  It was assumed that the slow CO2 hydration step (1) preceding the direct reduction of carbonic acid (5) is the cause for the observed limiting currents. 

In the present study limiting currents were measured over the range of 500 - 10000 rpm in both HCl and CO2 solutions using potentiodynamic sweeps.  The correction was made for the contribution of the direct water reduction and the resulting limiting currents as a function of rotation speed are shown in  Figure 4.  The gap between the two curves which exists over the whole velocity range confirms the assumption of Schmitt and Rothman3  and Eriksrud and Søntvedt4 that there is a flow independent component of the limiting current in CO2  solutions which is probably controlled by a chemical step: the hydration of CO2 into H2CO3.

If we assume that in CO2 solution at pH 4, both the H+ ions and H2CO3 are reduced at the surface, then at a given flow rate the limiting current for a CO2 solution can be separated into two components.  The first component is related to the diffusion of H+ ions from the bulk (the same as in HCl solutions).  The other flow independent (chemical reaction controlled) component which comes from H2CO3 is actually the gap between the two curves.  Since the gap increases with rotation speed, it is hypothesised that the chemical reaction limiting current is also affected by the flow.  This assumption will be analysed below.

Figure 4. Limiting currents for a CO2 and a HCl solution at pH4, t=22°C measured potentiostatically using a rotating cylinder electrode d=1 cm.  


Means for calculating the magnitude of a pure chemical reaction limiting current were first proposed by Vetter:10


This equation was later successfully used to explain observed limiting currents in CO2 solutions (glass-cell experiments).5, 11  However, it was recently reported12 that by using (6), limiting currents measured in loop experiments were underpredicted especially at higher velocities (>1m/s).  Inspection of Vetter’s10 derivation showed that (6) is strictly valid only for stagnant solutions when the thickness of the so-called “reaction layer” is much smaller than the thickness of the “diffusion layer”.  In that case the reported discrepancy12 can be explained by assuming that at higher velocities the thickness of diffusion layer was reduced and at some point became comparable to the thickness of the reaction layer.  This concept is illustrated on Figure 5 where the calculated thickness of the two boundary layers are compared.

Figure 5. Thickness of the boundary layers for pipe flow, t=20°C, pCO2=1 bar, dp=0.1m.

Figure 6. Boundary layer thickness ratio as a function of velocity and temperature for pipe flow, pCO2=1 bar, dp=0.1m.

The thickness of the mass transfer (diffusion) layer shown in Figure 5 is estimated by using the relation:


Here, D is the diffusion coefficient for carbonic acid and km is the mass transfer coefficient for straight pipe flow calculated using the correlation ofBerger and Hau13  The thickness of the chemical reaction layer  is calculated using the relation derived by Vetter10 for a first order chemical reaction:


where is the rate of carbonic acid dehydration (described in more detail below in the text).  From Figure 5 it can be seen that under given conditions at 4 m/s the two boundary layers are of the same thickness.  The ratio  is a strong function of temperature as shown in Figure 6.  It is clear that for lower temperatures  already for velocities larger than 1 m/s. Thus a more general expression than (6) is needed for the reaction-controlled limiting current which accounts for any  ratio and covers a wider range of applications.  

In order to derive such an expression, we will assume here the following sequence of reactions in the limiting current region:

               slow chemical reaction              (9)

         partial electrode reaction(10)

If we further assume that reaction (9) is a first order chemical reaction, the rate of change of H2CO3 concentration is:


We can assume that the concentration of dissolved CO2 is constant for all practical purposes and denote the rate of hydration with vo =const..  For the sake of simplicity we can drop the subscript so (11)  becomes:


At equilibrium v=0 , hence:


where  is the equilibrium concentration of H2CO3. Substitution of (13) into (12) gives:


Here u is the nondimensional concentration of H2CO3 .  It is the gradient of u (concentration) at the metal surface that will give us the desired chemical reaction limiting current.

To obtain the concentration profile the steady state mass balance (Fick’s second law) for the case of an accompanying homogeneous chemical reaction has to be solved:


For a steady state case .  We can further assume that the diffusion coefficient is independent of concentration:and that there are no temperature gradients so .  Finally, by substituting v from (14)  into (15), the nondimensional steady state mass balance is obtained:


The boundary conditions are:

at the metal surface, in the limiting current case, the concentration of H2CO3 is approaching zero, so for

                             Þ                                                        (17)

for the bulk of the fluid due to turbulence the fluid is well mixed so there are no concentration gradients and we can assume that all reactions are in equilibrium, so for:

                 Þ                                                         (18)

Here we have assumed that the edge of the mass transfer boundary layer at  is the point where everything is well mixed and all reactions are in equilibrium.  At this stage the present derivation departs from the one in Vetter’s10 book.  Vetter 10 assumes that the fluid is well mixed and in equilibrium only for  that is “very far” from the metal surface.  This is a good assumption for stagnant solutions or laminar flow, however one can imagine that for a high enough velocity and turbulent flow the thickness of mass transfer layer  is of the same order of magnitude as the reaction layer which we are calculating.  Of course by setting  the present derivation follows the one in Vetter’s10  book.

Integration of (16) with the boundary conditions (17) and (18) yields the nondimensional concentration profile:


We are interested in the limiting current which is:


When   is returned to (20) we obtain:


The original Vetter’s10 expression (6) is now recovered, however corrected with the multiplier here called “flow factor”:


which takes into account the effect of flow on the chemical reaction limiting current. 

Figure 7. Flow factor as a function of velocity and temperature for pipe flow, dp=0.1 m, pCO2=1 bar.

Assuming a stagnant solution, so the flow factor f=1 and the solution reduces to the one derived by Vetter.10  The sensitivity of the flow factor to velocity and temperature is illustrated in Figure 7.

As a rule of thumb in CO2 applications one can say that this correction is important for  temperatures lower than 40°C and velocities  higher than 1 m/s  when the mass transfer layer is of the similar thickness as the reaction layer.

Another way of looking at the superposition of the diffusion and reaction limiting currents is to express it in terms of a pure diffusion limiting current corrected for the presence of a rate limiting chemical reaction 14 .  By using (7) and (8) together with (21) it is obtained:


Finally, the derived equations can be compared with the measured limiting currents shown in Figure 4.  The result with and without the derived correction is shown in Figure 8.  Although in the measured velocity range the effect is not large it is clear that the flow factor improves the agreement of the measurements and the theory. 

Figure 8. Comparison of the model prediction and experimental results.  Conditions the same as in Figure 4.  The points represent measurements, the lines are the model:
  red solid line - mass transfer limiting current (Eisenberg et al.15),
  black dotted line - mass transfer + chemical reaction limiting current (equation 6),
  black solid line - mass transfer + corrected chemical reaction limiting current (equation 21).


It was observed experimentally that a chemical reaction limiting current can be affected by flow.  A new more general expression was derived for the superposition of the diffusion and chemical reaction controlled limiting currents.  It was found that the their interaction in the case of CO2 corrosion is significant at temperatures lower than 40°C and velocities  higher than 1 m/s  when the mass transfer layer is of the similar thickness as the reaction layer.


c concentration, mol/m3;
  equilibrium concentration, mol/m3;
D                  diffusion coefficient, m2/s ;
f  flow factor;
Faraday constant (96490 C/equiv.);
chemical reaction limiting current density, A/m2 ;
forward reaction rate (CO2hydration reaction), 1/s ;
backward reaction rate (H2CO3  dehydration reaction), 1/s ;
km mass transfer coefficient, m/s ;
u nondimensional concentration;
v chemical reaction rate, mol/(s m3);
x distance from the metal surface, m ;
thickness of the mass transfer (diffusion) layer, m;
thickness of the chemical reaction layer, m;
ratio of the diffusion and reaction layers;


1. A. Dugstad, L. Lunde and S. Nesic, “Control of Internal Corrosion in Multi-Phase Oil and Gas Pipelines”, Proceedings of the conference Prevention of Pipeline Corrosion, Gulf Publishing Co., 1994.

2. C. deWaard and D. E. Milliams, Corrosion, 31 (1975): p.131.

3. G. Schmitt and B. Rothman, Werkstoffe und Korrosion, 28 (1977): p.816.

4. E. Eriksrud and T. Søntvedt, "Effect of Flow on CO2 Corrosion Rates in Real and Synthetic Formation Waters", Advances in CO2 Corrosion, Vol. 1. Proceedings of the Corrosion /83 Symposium on CO2 Corrosion in the Oil and Gas Industry, Editors: R. H. Hausler, H. P. Goddard , p.20, (Houston, TX: NACE, 1984).

5. T. Hurlen, S. Gunvaldsen, R. Tunold, F. Blaker and P. G. Lunde, J. Electroanal. Chem., 180 (1984): p. 511.

6. L. G. S. Gray, B. G. Anderson, M. J. Danysh, P. G. Tremaine, “Mechanism of Carbon Steel Corrosion in Brines Containing Dissolved Carbon Dioxide at pH 4”, Corrosion/89, paper no. 464, (Houston, TX: NACE International, 1989).

7.  L. G. S. Gray, B. G. Anderson,  M. J. Danysh and P. R. Tremaine, "Effect of pH and Temperature on the Mechanism of Carbon Steel Corrosion by Aqueous Carbon Dioxide", Corrosion/90, paper no. 40, (Houston, TX: NACE International, 1990).

8. M. R. Bonis and J. L. Crolet, “Basics of the Prediction of the Risks of CO2 Corrosion in Oil and Gas Wells”, Corrosion/89, paper no. 466, (Houston, TX: NACE International, 1989).

9. M. Stern,  J.Electrochem. Soc., 102 (1955): p.609.

10. K. J. Vetter, Electrochemical Kinetics, Theoretical Aspects, Sections 1, 2, and 3 of Electrochemical Kinetics: Theoretical and Experimental Aspects, translation from German, (New York: Academic Press, 1967): pp.235-250.

11. S. Nesic, J. Postlethwaite and S. Olsen,An Electrochemical Model for Prediction of CO2 Corrosion”, Corrosion/95, paper no. 131, (Houston, TX: NACE International, 1995).

12. S. Nesic, G. Th. Solvi, and J. Enerhaug,Comparison of the Rotating Cylinder and Pipe Flow Tests for Flow Sensitive CO2 Corrosion”, Corrosion/95, paper no. 130, (Houston, TX: NACE International, 1995).

13. F. P. Berger and K.-F. F.-L Hau, Int. J. Heat Mass Transfer, 20 (1977): p.1185.

14. B. F. M. Pots,Mechanistic Models for the Prediction of CO2 Corrosion Rates under Multi-Phase Flow Conditions”, Corrosion/95, paper no. 137, (Houston, TX: NACE International, 1995).

15. M. Eisenberg, C. W. Tobias and C. R. Wilke, J. Electrochem. Soc., 101 (1954): p. 306.