Motion Error Mechanism Analysis and Simulation Verification of Ocean Electric Field Sensor

The electric field measurement based on Unmanned Underwater Vehicle (UUV) can monitor and evaluate real-time electric field of underwater targets in motion, which has the ability to adapt to harsh marine environments. However, electric field sensors generate new errors during underwater motion, and there is still absence of studies on the error theory and simulation. To address this, the error generation mechanism of Ag/AgCl electrodes induced by the flow field was analyzed in this paper. A coupled mathematical model of flow-chemical-electric multi-physical field was established for the first time, and finite element simulation analysis was carried out. The results show that the change value of electrode surface potential under the influence of flow rate is basically in agreement with the empirical formulae. R-squared value of the two fits reaches up to 0.968, and RMSE value is less than 0.003, confirming the robustness of the model and simulations. The multi-physical field coupling model and simulation method proposed in this paper can provide theoretical and methodological support for the design of electric field sensors and motion error correction.


Introduction
The oceanic electric field sensor, employed for the measurement of underwater electric fields, assumes a pivotal role in diverse domains including seabed geological exploration, marine biological investigations, oceanic current monitoring, and national defense.This sensor exhibits considerable potential for application [1] .It functions by situating an electrode within the seawater medium to gauge the electric field.This electrode captures the potential difference present in the seawater, transmuting it into an electrical signal which is subsequently subjected to post-processing and recording procedures.In comparison with alternative electrode types, the Ag/AgCl electrode presents distinct advantages such as heightened stability, facile preparation, and an extensive potential range [2][3][4] .Consequently, this electrode finds widespread utilization within the realm of oceanic electric field measurement.Motion electric field measurement endows notable advantages including a versatile operational scope, real-time monitoring, and the motion evaluation of electric fields of underwater targets, thus enabling comprehensive spatial assessment.Nevertheless, the process of motion introduces new errors, posing significant challenges to the precision of electric field measurement.
There have been some researches on the error influence of electric field sensor in underwater measurement in motion.Russian scholar Maksimenko V has published a series of results [5][6][7][8][9] , and studied the measurement error of electric field sensor caused by water flow.He believed that the electrode potential change in the flow originated from the episodic flow deformation of the bilayer at the leading edge of the electrode, and also deduced the relationship between the electrode potential change and the flow velocity change.However, there are limitations in his theory: the role of electrochemical reactions is not considered, the coupling analysis between physical fields is lacking, and the magnitude of the influence of each factor is not further investigated by numerical simulation methods.Then Luo W et al. [10] of China University of Geosciences (Wuhan) also pointed out that the chloride ion concentration in seawater changes due to ocean currents and affects the stability of the electrodes, but there is a lack of theoretical basis and in-depth analyses of the influence mechanism and multi-field coupling relationships.Xie X et al. [11] of Naval University of Engineering believe that the change of electric double layer structure causes the motion noise of towed antenna, but it is only speculation and lacks theoretical analysis.
In summary, although research on motion-induced errors in electric field sensors has been conducted both domestically and internationally, several limitations persist.These include an unclear understanding of the underlying generation mechanisms, the absence of comprehensive systematic modeling studies, and challenges in adapting to real-world scenarios.Particularly significant is the lack of a multi-physics field coupled simulation method for quantitatively assessing the impact of factors contributing to errors in electric field sensors during motion.Therefore, there is a need to introduce a comprehensive mathematical model and simulation approach that incorporates multiple physical fields.By employing finite element analysis, this approach can effectively investigate the influential factors contributing to errors in electric field sensors during motion, thus furnishing a theoretical foundation for sensor design and the correction of errors under motion conditions.

Electric double layer effect of electrode surface
According to the theory of surface science, the Electric Double Layer (EDL) is formed on the surface of Ag/AgCl electrode in contact with seawater.Gouy-Chapman-Stern (GCS) model is used to describe EDL (as shown in figure 1).The GCS model posits that EDL comprises two parallel charge layers encompassing the object's surface.The first layer, denoted as the surface charge layer (in the case of the Ag/AgCl electrode, it carries a negative charge), is composed of ions adsorbed onto the object's surface due to intermolecular forces and similar interactions.This layer is typically compact and termed the Stern layer.The second layer, generally more diffuse, is constituted by ions that are adsorbed due to Coulombic forces and influenced by thermal motion, allowing them to move freely within the fluid.This layer is referred to as the Diffuse layer.As a result, the EDL generates an electric potential gradient in the normal direction of the electrode surface.The GCS model accounts for the diffusive properties of ions within the electrolyte solution.It employs the Poisson-Boltzmann equation (Equation 1) to describe the distribution of ion concentration and charge density within the electric layer.

Electrochemical reactions on the electrode surface
During electrode measurements in seawater, electrochemical reactions also occur at the electrode surface.The operational characteristics of the Ag/AgCl electrode in seawater fall under the category of the second type of reversible electrode.This type of electrode is composed of a metal inserted into a solution of a sparingly soluble salt that shares the same anion as the sparingly soluble salt.In electrode reactions, anions undergo dissolution and precipitation (forming sparingly soluble salts) at the solidliquid interface.The electrode reaction occurring with the Ag/AgCl electrode in seawater can be expressed as follows: AgCl e Ag Cl Ag Ag e The electrode reaction of Ag/AgCl will be formed on the electrode surface, resulting in the accumulation of negative charges on the electrode surface and the decrease of electrode potential.As the surface concentration accumulates, the positive reaction rate of electrode reaction (equation ( 3)) gradually decreases until a dynamic equilibrium of the reaction is reached.
According to the Nernst equation, which quantitatively describes the diffusion potential formed by ions between two systems, we can deduce the electrode potential from the ion concentration in the solution.The Nernst equation is: Substituting the solubility product of AgCl and the activity of Cl -into equation (5), the expression of electrode potential can be written as: It should be noted that concentration and activity are not synonymous concepts.Corresponding relationships need to be obtained through reference tables or approximative models before they can be employed for calculations.

Effects of seawater flow
The relative motion of the electric field sensor to the seawater in motion makes the electric field sensor subject to influences from the flow, which introduces new errors.This process consists of the following mass transfer modes: 1) Electromigration.The electric double layer will generate a potential difference in the normal direction of the electrode surface, which forces the charged ions near the electrode surface to move.In addition, the electrochemical reaction of the electrode and the applied electric field will cause the charged ions in the solution to move; 2) Convection.Due to the effect of the electric double layer and electrochemical reaction, there is a concentration gradient near the electrode surface, and convection promotes the transfer of matter in the direction of flow velocity; 3) Diffusion.There is a concentration gradient near the electrode surface, which causes the material to spontaneously move from a region of high concentration to a region of low concentration.This process mainly affects the transport of substances in the direction of concentration gradient.
The dilute mass transfer equation is used to describe the mass transfer process of charged ions on the electrode surface during seawater flow: where, is the rate of change of the concentration of the i-th species ion over time, i J  is the divergence of the ion current density of the i-th species ion, u is the velocity vector, and i c  is the divergence of the concentration of the i-th species ion.F is the Faraday constant, and V  is the gradient of the potential, indicating the degree of spatial variation of the potential.

Error contribution equation in motion
Substituting equation (2) into equation (1) gives: Equation ( 8) contains potential gradient ∇ = ()/, which is obtained from equation ( 9): Insert the equation (10) Equation ( 11) is combined with equation ( 7): Equation ( 12) encompasses the influence of electrochemical reaction terms, incorporating convection, diffusion, and migration as the three mass transfer mechanisms.It yields the variations in ion concentration under the collective influence of electric potential gradient within the double layer, electrochemical source terms, and the effects of dynamic operational conditions.Based on the Nernst equation depicted in Equation ( 6), electrode surface potential changes can be deduced from ion concentration, offering a quantitative depiction of sources of error in oceanic electric field sensors under dynamic operational conditions, along with their influencing factors.

Parameters and variables
The main parameters involved in this model are shown in table 1

Coupling relationship analysis
The coupling relationships among various physical fields in the simulation model are illustrated in figure 2. With chloride ion concentration as the central focus, the Electric Double Layer (EDL) influences its migration process through the potential difference generated by surface charge density.Furthermore, the reaction rate of chloride ions in electrochemical reactions, as well as the diffusion and convection effects driven by laminar flow, directly impact the variation in chloride ion concentration.Additionally, the spatial location of the EDL constitutes an active site for electrochemical reactions.The process of how laminar flow affects the ion distribution within the EDL is described by the convection-diffusion equation.Moreover, the reaction rate and equilibrium constant of electrochemical reactions are indirectly influenced by laminar flow.

Simulation results and analysis
Utilizing steady-state analysis for numerical solutions, the simulation model yields chlorine ion concentration distribution maps on the electrode surface under varying flow velocities, as depicted in figure 3. Adjacent to the electrode surface, there exists a concentration gradient of Cl -ions.Specifically, in proximity to the electrode surface, Cl -concentration reaches its maximum at approximately 1180 mol/m 3 , rapidly diminishing thereafter to around 620 mol/m 3 , representative of the bulk solution concentration.This phenomenon of diminished Cl -concentration predominantly occurs within the Stern layer of the double layer structure.To further validate the accuracy of the simulation model and its outcomes, a comparison was conducted between simulation results and theoretical values [12] , with the average electrode surface potential serving as the representative simulated output.The comparison of potential simulation results and theoretical values is presented in figure 5.As discernible, theoretically, the electrode surface potential exhibits a rising trend with increasing flow velocity.The curve is steeper at low flow velocities and gradually levels off at higher flow velocities.This overall trend aligns with the simulation outcomes.Both sets of data closely match at low flow velocities.When the flow velocity surpasses 2m/s, they exhibit a similar rate of ascent, with discrepancies of no more than 10 μV.Table 3 demonstrates the evaluation indicators of the fitting results between the simulation results and the theoretical values, and the fitted R-square is 0.968, while the RMSE value is less than 0.003, which indicates that the simulation results are highly fitted to the theoretical values with high confidence.
Table 3. Indicators of the degree of fit of the simulation result.

Conclusion
In this paper, by analysing the error generation mechanism of Ag/AgCl electrode under the induction of flow field, a coupled mathematical model of multi-physical fields of flow field-chemical fieldelectrical field was established for the first time, and finite element simulation analysis was carried out.The results show that under the simulation conditions, the chloride ion concentration on the surface of

Figure 1 .
Figure 1.Gouy-Chapman-Stern model structure schematic diagram.The GCS model posits that EDL comprises two parallel charge layers encompassing the object's surface.The first layer, denoted as the surface charge layer (in the case of the Ag/AgCl electrode, it carries a negative charge), is composed of ions adsorbed onto the object's surface due to intermolecular forces and similar interactions.This layer is typically compact and termed the Stern layer.The second layer, generally more diffuse, is constituted by ions that are adsorbed due to Coulombic forces and influenced by thermal motion, allowing them to move freely within the fluid.This layer is referred to as the Diffuse layer.As a result, the EDL generates an electric potential gradient in the normal direction of the electrode surface.The GCS model accounts for the diffusive properties of ions within the electrolyte solution.It employs the Poisson-Boltzmann equation (Equation1) to describe the distribution of ion concentration and charge density within the electric layer.

iR
is the source term, which indicates the generation or consumption of the i-th ion during the chemical reaction.i J is the ion current density of the i-th ion.ii Dc − is the diffusion term and i D − is the diffusion coefficient of the i-th ion., i m i i z u Fc V − is the migration term under the action of an electric field, i z is the valence of the i-th ion, , mi u is the ion mobility,

Figure 2 .
Figure 2. Schematic diagram of physical field coupling relationship of simulation model.

Figure 3 .
Figure 3. Distribution of chloride ion concentration on electrode surface under different flow rates.(a) 10 -6 (b)1 (c)3 (d)5, unit: m/s.Three distinct points on the electrode surface were selected as reference points, and the relationship between chloride ion concentration and potential variation at these points under varying flow velocities was graphed, as depicted in Figure4(a).The parameterized sweep was conducted with inlet velocity

Figure 4 (
b), based on Figure4(a), computes the potential variation at the three reference points on the electrode surface using the Nernst equation.The conclusion drawn is that, with increasing inlet velocity, the electrode potential at the same reference point exhibits an ascending trend.Over the in v range from 10 -6 m/s to 5m/s, the electrode potential experiences a change of 50~60μV.Notably, the change in electrode potential from in v = 10 -6 m/s to 1m/s is approximately 30μV, indicating that at low flow velocities, alterations in flow velocity have a relatively significant impact on the electrode surface potential.

Figure 4 .
Figure 4. (a) The variation of chloride ion concentration with velocity of fluid on the electrode surface;(b) The variation of potential with velocity of fluid on the electrode surface.To further validate the accuracy of the simulation model and its outcomes, a comparison was conducted between simulation results and theoretical values[12] , with the average electrode surface potential serving as the representative simulated output.The comparison of potential simulation results and theoretical values is presented in figure5.As discernible, theoretically, the electrode surface potential exhibits a rising trend with increasing flow velocity.The curve is steeper at low flow velocities and gradually levels off at higher flow velocities.This overall trend aligns with the simulation outcomes.Both sets of data closely match at low flow velocities.When the flow velocity surpasses 2m/s, they exhibit a similar rate of ascent, with discrepancies of no more than 10 μV.

Figure 5 .
Figure 5.Comparison between potential simulation results and theoretical value.

Table 1 .
: Main parameters of simulation model.The surface charge density is used in the electrostatic interface to model the potential distribution of the electric double layer, the variables shown in table 2 are defined as follows:

Table 2 .
Definition of simulation model variables.