In this study, we want to measure the dimensionless numbers characteristic of this flow and the corrosion rate by computer programming using the software Jupiter (Python). This software is a web-based, interactive computing notebook environment. It edits and runs human-readable documents while describing the data analysis. This study is made by modelling the flow using dimensionless numbers and programming the characteristic equations of this flow within a particular geometry. To do this, we vary the Reynolds and Peclet numbers according to the velocity parameter U of the fluid. The Sherwood number is studied as a function of the mass transfer coefficient parameter K. The Schmidt number is constant for this study. The corrosion rate is measured by the Sherwood number, which varies with the Reynolds number. Experimentally, the order of the velocity of the vaporized fluid is between 0.1 to 2 m∙s⁻¹, and the order of the mass transfer coefficient is between 0.1 to 1 m∙s⁻¹.

Our study system, as shown in Figure 1 , involves a fluid, specifically vaporized crude oil, moving within a pipe that is 50 meters high and 8 meters in diameter. The velocity of the fluid adjacent to the wall is zero. Viscous forces make the fluid slow down as it approaches the walls upon entering the pipe. Consequently, a velocity gradient is established in the boundary layer. The thickness of this boundary layer increases as the fluid progresses through the pipe, near a distance denoted as $L_h$, referred to as the hydrodynamic entrance length. Typically, for a pipe, $L_h$ is approximately 70 times the diameter [14] . This scenario results in a parabolic velocity profile.

Advection refers to the transport of properties such as humidity, temperature, pollution, or corrosivity by a fluid during its movement. Diffusion, on the other hand, involves the migration of sometimes corrosive chemical species into a liquid or gaseous medium.

In the absence of detachment, advection occurs in the general direction imposed by the conditions at infinity along the current line, deviating only slightly from the geometry of the obstacle. Under these conditions, a representative timescale for the advective transport of momentum over the distance $L_h$ can be expressed as follows [16] :

$T_a=\frac{L_h}{U_{\infty }}$ (1)

where $U_{\infty }$ is the flow velocity, $T_a$ is the advection timescale, $L_h$ is the hydrodynamic entrance length.

The presence of a fixed obstacle in a flow of viscous fluid results in a deficit or “well” of momentum for the moving medium. This condition imposed on the wall diffuses beyond the entire field with a diffusivity that is essentially the kinematic viscosity ν of the fluid. If we denote by δ the orthogonal transverse distance to the wall (the thickness of the hydrodynamic boundary layer) characteristic of this diffusion at the fluid edge of this obstacle, then for a distance equal to the hydrodynamic entry distance $L_h$, the timescale corresponding to the diffusive transfer of momentum is given by [16] :

$T_d\approx \frac{{\delta }^2}{\nu }$ (2)

Advection and diffusion occur on the same time scale, which is given in [16] :

$\frac{\delta }{L_h}\approx \sqrt{\frac{\nu }{U_{\infty }\cdot L_h}}$ (3)

With the described phenomenology, it can be demonstrated that the relative thickness decreases as the characteristic Reynolds number of the flow increases.

$\frac{\delta }{L_h}$ decreases as $R_e^{\frac{-1}{2}}$, where

$Re=\frac{U_{\infty }\cdot L_h}{\nu }$ (4)

This represents a novel interpretation of the Reynolds number as a dimensionless variable comparing diffusive and advective transport times occurring over the same distance [16] . Previous studies have demonstrated the importance of the relationship between the Reynolds, Sherwood, Peclet, and Schmidt numbers. Even when vaporized, crude oil is continually emulsified with water. For aqueous electrolytes, the kinematic viscosity is approximately $\nu \approx {10}^{-6}{\text{m}}^2\cdot {\text{s}}^{-1}$ and the mass diffusion coefficient is approximately $D\approx {10}^{-9}{\text{m}}^2\cdot {\text{s}}^{-1}$ [14] . The distillation column diameter is approximatively L = 8 m.

Consequently, the dimensionless variables relevant to this study can be expressed as follows:

$Sc=\frac{\nu }{D}={10}^3$ (5)

$Re=\frac{U\cdot L}{\upsilon }=8\times {10}^6\cdot U$ (6)

$Sh=\frac{K\cdot L}{D}=8\times {10}^9\cdot K$ (7)

$Pe=Re\cdot Sc=8\times {10}^9\cdot U$ (8)

The constant flow parameters are given in Table 1 :

Hydrodynamic mass transfers are treated by presenting corrosion rates in terms of dimensionless parameters. The number of variables involved in mass transfer problems is quite large. The Reynolds and Schmidt numbers provide the dependence between velocity and limit current under laminar and turbulent conditions. Utilizing dimensional and dimensionless variable analyses in mass transfer measurements is highly beneficial for obtaining information on induced flow corrosion control [3] . The corrosion rate can be predicted without testing if certain parameters are known. For example, for flow in a smooth tube, the correlation is [3] .

$Sh=0.023\cdot R{e}^{0.8}\cdot S{c}^{0.33}$ (9)

But, for this study Sc = 10³, Equation (8) can be rewritten as follows:

$Sh=0.2247\cdot R{e}^{0.8}$ (10)

The Sherwood number quantifies the evolution of corrosion; indeed, the corrosion rate evolves by half compared to the Sherwood number as a function of the Reynolds number [3] .

Consequently, the corrosion rate can be evaluated as follows:

$C=0.11235\cdot R{e}^{0.8}$ (11)

Figure 2 summarizes the methodology used to characterize the fluid flow and to measure the corrosion rate in the flow of the fluid in the atmospheric distillation column.

Equation (6) shows in Figure 3 that the Reynolds number increases significantly and in proportion to the flow velocity. This translates into a critical or even a supercritical regime in this study.

Equation (7) indicates in Figure 4 that the Sherwood number varies significantly and in proportion to the mass transfer coefficient.

Equation (8) shows through Figure 5 that the Peclet number increases with the fluid velocity.

The Schmidt number is a dimensionless number representing the ratio between momentum diffusivity ν (or kinematic viscosity) and mass diffusivity D. It is employed to characterize the flow of fluids where viscosity and material transfer occur simultaneously. For this study, its value is 1000.

Equation (10) shows, in Figure 6 , the evolution of the Sherwood number which translates the ions transport from the electrolytes to the metal and Equation (11) models the measurement of corrosion rate using dimensionless numbers in convective and diffusive flow within the atmospheric distillation column, and it shows at Figure 7 that corrosion increases significantly under critical and supercritical regimes.

The Sherwood number is a dimensionless parameter used to characterize mass transfer between a fluid and an interface. Figure 4 illustrates an increase in the Sherwood number, indicating that a significant quantity of corrosive crude oil particles is deposited on the cylinder walls.

The Peclet number is a dimensionless parameter representing the ratio of forced convection transport to diffusion transport. Figure 5 demonstrates a substantial increase in the Peclet number, indicating a low mass diffusivity compared to the convective transport of corrosive particles in contact with the cylinder walls.

The Schmidt number is a dimensionless quantity representing the ratio of momentum diffusivity ν (or kinematic viscosity) to mass diffusivity D. It is utilized to characterize fluids where viscosity and material transfer occur concurrently. In this study, the Schmidt number has a value of 1000. This implies that dynamic (or kinematic) viscosity is significantly more influential than mass diffusivity within the cylinder.

This numerical study of corrosion measurement is carried out by using existing correlations on the quantification of corrosion. This correlation is made on a cylinder geometry where a fluid with physicochemical characteristics flows. This measurement is thus only valid for perfectly cylindrical columns, and the fluid in question has a known flow speed and is dissipated by convection and advection substances (corrosion factors) during its upward movement. This constitutes the limitations of this study.

Figure 6 and Figure 7 show that critical turbulence regimes are favourable to corrosion proliferation. However, these regimes are characterized by large Reynolds numbers, but also high flow velocity.

Ahmed [3] has used the same model to quantify the corrosion rate in Arabic golf water on a modified aluminium alloy. They found that the Sherwood number increased linearly with the Reynolds number, suggesting that the rate of corrosion increases with the increased Reynolds number. In fact, the Sherwood number describes the maximum rate at which the ions are transported from the bulk solution to the metal surface.

We have correlated their results to ours in Figure 8 . Indeed, in a solution containing electrolytes, corrosion activity is much greater. The red and green curves respectively represent the evolution of the Sherwood number and corrosion in electrolytes, while the blue curve represents the evolution of the Sherwood number as a function of the Reynolds number in the Arabic golf water.

The correlation between the variation of the Sherwood number in the Arabic golf water represented by the curve in blue and the variation of the Sherwood number in an aqueous electrolyte represented by the curve in red and the corrosion C is perfect and is equal to 1.

That means that these three parameters increase proportionally with Re. Indeed, the evolution of the Sherwood number in Arabic golf water is proportional to the variation of the Sherwood number in an aqueous electrolyte. The proportional coefficient is 0.045^0.33, and we have seen previously that the proportionality between the evolution of the Sherwood number and the corrosion rate is 0.5. This indicates that these three equations translate the same phenomenon, namely the transport of ions from the solutions to the metal in question. Hence, the corrosion phenomenon was observed.

For example, with the increasing concentration of Cl⁻ and the extension of immersion time, the electrochemical noise resistance and charge transfer resistance of P110 steel decrease gradually, and the protective property of the corrosion-product film decreases, which is capable of forming steady pitting corrosion [17] .

Literature can be used to confirm the results of this study on the effect of turbulence and fluid flow velocity on corrosion progression, as evidenced in this study.

Liu [18] studied the corrosion behaviour of X70 in the water produced in oil fields and found that velocity affects the formation of corrosion product film, with sudden velocity changes leading to transient pressure instability in the pipeline [19] . Martinez [19] investigated the corrosion kinetics of turbulent X52 pipeline steel and observed that turbulence has a considerable impact on the electrochemical processes on the steel surface. The flow rate significantly influences the corrosion behaviour of steel [4] . Hydrodynamic mass transfers are addressed by presenting corrosion rates in terms of dimensionless parameters. The number of variables involved in mass transfer problems is extensive. The Reynolds and Schmidt numbers delineate the dependence between velocity and limit current under laminar and turbulent conditions. Utilizing dimensional and dimensionless variable analyses in mass transfer measurements is highly useful for obtaining information in induced flow corrosion control [3] .

The corrosion rate can be predicted without testing if certain parameters are known. For example, for flow in a smooth tube, the corrosion rate is determined by a correlation linking the Sherwood, Peclet and Reynolds numbers [3] . The Sherwood number quantifies the corrosion evolution. The corrosion rate evolved by half compared to the Sherwood number as a function of the Reynolds number [3] . Flows under critical and supercritical regimes are characterized by high fluid velocity. It is evident that uniform velocity around the tubes occurs within a range of 89 mm - 178 mm before velocity increases in tubes near the shell inlet, becoming non-uniform at the point of 356 mm, where corrosion occurs [7] . Severe corrosion primarily occurs near the elbow wall outlet, involving factors such as elbow material, bend angle, velocity, dimensions, and crude oil origin [8] .

Fluid velocity predominantly influences the corrosion rate. Severe corrosion is often found in areas with highly turbulent flow, while areas with minimal turbulence experience little corrosion. For example, variations in corrosion rates were observed in the hottest furnace tubes during the treatment of a particular crude oil. At a temperature of 270˚C, the corrosion rate remained low with a flow rate of 40 tons/day but became considerable with a flow rate of 70 tons/day or higher [6] . Unfortunately, the absolute critical value (which varies with temperature) below which the corrosion rate remains low is not yet known [6] .

Product volatility, operating pressure, and steam injections have considerable consequences on fluid velocity [6] . In turbulent flow regimes, a significant increase in metal mass loss is observed with increasing fluid velocity. The rate of erosion-corrosion is influenced by three main factors: the concentration of corrosive substances, the characteristic Reynolds number of the flow, and the duration of the experience [20] .

It is well established that the membrane caused by corrosion on metals can inhibit the proliferation of corrosion, but it must be preconditioned. During stripping corrosion, the protective membrane formed by Fe₃O₄ is destroyed by high flow velocity, leading to the loss of its protective effect [5] . The studies of [9] [10] indicate that for Reynolds numbers less than 2000, the shear forces of the fluid are low, which cannot destroy the protective layer on the metal. However, when the Reynolds number exceeds 2000, the shear forces accelerate the transport of corrosion factor reactants in the convective flow, carrying the protective layer on the metal surface, which can even scrape the metal surface, thus accelerating the corrosiveness of the fluid. High Reynolds values in natural gases accelerate widespread corrosion but can also lead to localized corrosion [11] [12] . As described, the formation of holes on the protective layer induced by corrosion on metals occurs when the Reynolds number exceeds 14,000, indicating that the protective layer is destroyed. In this case, the number of holes formed increases considerably compared to cases where low Reynolds numbers are operated. When the protective layer is removed, bare metal is exposed to act as an anode during electrochemical reactions [21] [22] . Small anodic surfaces accelerate the reaction rate of anode dissolution, leading to localized galvanic corrosion and pitting corrosion [5] .

The Reynolds number increases with a certain pH value, increasing the corrosion rate. This affects increasing the flow speed, and therefore, turbulence enhances the transport of species to and from the steel surface [23] .