# Invited Paper: Liquid flow electrification phenomenon: Review

Thierry Paillat, Gérard Touchard, Paul Leblanc

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Thierry Paillat, Gérard Touchard, Paul Leblanc. Invited Paper: Liquid flow electrification phenomenon: Review. Journal of Electrostatics, 2025, pp.104068. 10.1016/j.elstat.2025.104068. hal-05057356

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# Liquid flow electrification phenomenon: Review

T. Paillat, G. Touchard, P. Leblanc

Institut PPRIME – Université de Poitiers – CNRS – ISAE ENSMA,
11, Boulevard Marie et Pierre Curie, 86073, Poitiers, Cedex 9, France

Corresponding author: Thierry Paillat
e-mail address: thierry.paillat@univ-poitiers.fr

**Abstract**— Flow electrification of liquids is the transfer of electrical charge due to liquid's flow. This phenomenon is known for several ten years and also called "streaming current" or "streaming potential". By the way, since the sixties, the Founding Editor of Journal of Electrostatics, Dr J. C. Gibbings, has made important contributions in this field [1-5]. Similarly, during the sixties and the seventies, the fundamental aspects of the phenomenon have been examined [6-10]. On the other hand, industrial applications, which were among the first investigations, were undertaken on electrostatic hazards in relation with the phenomenon, especially for safety reasons in petroleum industry [11]. Useful applications was rather rare and almost exclusively in the field of high-voltage generator development or velocity probe using the phenomenon [6,14].

Flow electrification phenomenon is particularly complex as it involves materials, physics, chemistry, and fluid mechanics. It originates from the electric double layer (EDL) present at each solid/liquid interface, which results in a polarization of the interface. This physics equilibrium is determined by the nature of the materials and their physico-chemical properties. It is disrupted by the liquid flow toward a new equilibrium, described by the flow electrification phenomenon. Studied within the research framework in electrochemistry, colloid chemistry, and electrical engineering, EDL remains poorly understood. It relies on models whose viability has yet to be confirmed, as well as on often indirect experimental approaches. Zeta-metry and the flow electrification measurements constitute the main methods of study.

The evaluation and control of electrostatic hazard in an industrial and applied context can be achieved either by acting on the genesis of charge generation at interfaces or by addressing the various parameters governing the phenomenon of electrification through flow. Thus, after a description of the EDL, the phenomenon of electrification through flow will be analyzed in detail for both laminar and turbulent regimes. Particular attention will be given to studying this phenomenon in the context of electrostatic hazard in power transformers. Then, a new theoretical analysis of the flow electrification phenomenon under turbulent flows is described. Finally, an overview of new experimental approaches for analyzing the EDL will be presented.

## APPENDIX

$D_0(m^2 s^{-1})$: ionic diffusivity

$D_t(m^2 s^{-1})$: eddy diffusivity

$I_0, I_1, I_2$: modify Bessel function on 0, 1 and 2 order

$i_w(Am^{-2})$: wall current density

$K(m s^{-1})$: constant which control de time dynamic of charge formation

$Q_l(m.s^{-3})$: liquid flow rate in laminar flow

$R(m)$: capillary, tube radii

$r(m)$: radial coordinate

$Re$: Reynolds number

$U(m.s^{-1})$: liquid flow velocity

$U_m(m.s^{-1})$: mean liquid flow velocity

$u^*$: friction velocity

$V_{rel}(m.s^{-1})$: relaxation velocity

$v^*(m.s^{-1})$: radial fluctuant velocity

$z(m)$: axial coordinate

$\delta_0(m)$: diffuse layer thickness

$\epsilon$ (F.m$^{-1}$): liquid permittivity

$\sigma$ (S.m$^{-1}$): liquid conductivity

$\rho(C.m^{-3})$: space charge density

$\rho_w(C.m^{-3})$: space charge density at the wall

$\rho_{wd}(C.m^{-3})$: space charge density at the wall at the equilibrium

$\rho_m$: specific mass of the liquid

$\tau$ (s): relaxation time

$\tau_w$: wall shearing stress

$\nu$ (m$^2$.s$^{-1}$): liquid kinematic viscosity

$\psi$ (V): electric potential

+ subscripted: dimensionless value

## I. OVERVIEW OF THE PHENOMENON

At the interface between a solid and a liquid, physicochemical processes lead to the polarization of the interface into two regions with opposite electrical charges: one in the solid and one in the liquid. This physics equilibrium, known as the electrical double layer (EDL), results from a balance between the chemical mechanisms that generate charges at the interface and the forces of diffusion and migration. The EDL is specific to each solid/liquid couple. For instance, just a few ppm of impurities or additives in the liquid can significantly affect the properties of the EDL.

The flow of liquid over the solid surface disrupts this equilibrium by partially transporting the electrical charges from the liquid's EDL, inducing the phenomenon of flow electrification. In order to restore this balance, new charges are generated at the interface. The dynamics and
quantity of charge produced at the interface increase as the EDL imbalance becomes more pronounced and are particularly affected by the nature of the flow (laminar or turbulent).

If the solid, such as a pipeline, is a dielectric or an isolated conductor, the charges from the EDL accumulate, leading to a local increase in potential. The liquid and the EDL charges swept by the floware typically transported to the tank. If this one is also a dielectric or a floating conductor, its potential may also increase. These local potentials can reach the dielectric breakdown of the medium, leading to electrostatic discharges of varying nature and intensity. For hydrocarbons with low ignition thresholds (such as fuels or liquefied gases requiring only a few millijoules or less), the emitted energy can, under stoichiometric conditions, trigger an accident and ignite the liquid. For less flammable materials, a more insidious mechanism can take place. The repeated occurrence of electrostatic discharges, particularly on the solid surface, gradually weakens the dielectric properties of both solid and liquid. In the presence of high-voltage electricity, such as in electrical components like transformers, electrical insulation may be lost, leading to a powerful and destructive discharge.

Flow electrification and the associated electrostatic hazard are major concerns for numerous industries, including land, aerospace, and space transportation, electrical engineering (where high-power components are cooled and insulated by dielectric liquids), as well as the plastics, food, and pharmaceutical industries.

Over the years, several models of the EDL have been developed. One of the most used and realistic is Stern's model [12]. This model (Fig. 1) predicts that in fact in the liquid two sub-layers exist one close to the solid (called the compact layer) and one farther in the liquid (called the diffuse layer) whose thickness $\delta_0$ (often called the Debye length [13]), is proportional to the root square of the liquid electrical resistivity:

$$ \delta_0 = \left( \frac{\varepsilon D_0}{\sigma} \right)^{\frac{1}{2}} \qquad (1) $$

$\varepsilon$ and $\sigma$ are respectively the permittivity and the electrical conductivity of the liquid and $D_0$ is the ions diffusion coefficient inside the diffuse layer. All investigations are generally made with the hypothesis that positive and negative ions have the same valence (Z=1) and the same diffusion coefficient$D_0$. Another important parameter is the relaxation time $\tau$ of the diffuse layer:

$$ \tau = \frac{\varepsilon}{\sigma} \qquad (2) $$

The compact layer, due to its proximity to the solid surface, cannot be affected by the flow. Therefore, flow electrification is the result of the convection of a portion of the charges within the diffuse layer caused by the liquid flowing past the surface (Fig. 2). This leads to three main areas of investigation: the structure of the diffuse layer, the

characteristics of the flow and its influence on charge behavior, and the geometry of the solid surface.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Stern Electrical double layer charge distribution.</figcaption></figure>

<caption>Fig.2: Flow electrification phenomenon description.</caption>

### A. Diffuse layer structure for a liquid at rest

When the liquid becomes in contact with the solid surface a physicochemical process arises, leading to an electrical current density at the interface ($i_w$). It is maximum at the beginning of the contact when no charge exists in the liquid and vanish when the diffuse layer becomes fully developed:

$$ i_w = K(\rho_{wD} - \rho_w) \qquad (3) $$

with $\rho_w$ the wall space charge density, $\rho_{wD}$ the wall space charge density for a fully developed diffuse layer and K a coefficient function of several parameters. Depending primarily to the coefficient K, the development of the diffuse layer has different behavior. If K is rather high, the charge accumulates near the interface then migrates in the liquid, leading to a kind of accumulation of charges near the interface. In contrary, if K is very low, which seems to be the case for many dielectric liquids, the charge has a profile similar to a fully developed diffuse layer but with a charge at the interface $\rho_w$ evolving slowly to $\rho_{wD}$. For a liquid at rest and for a fully developed diffuse layer we have the following system of equations:

$$ \left\{ \begin{array}{l} \nabla_+ \rho_+ + \sqrt{\rho_+^2 + 1} \nabla_+ \psi_+ = 0 \\ \Delta_+ \psi_+ = -\rho_+ \end{array} \right. \qquad (4) $$

The index + signifies dimensionless quantities, $\rho$ is the space charge density and $\psi$ the electrical potential. The first term of the first equation correspond to the diffusion phenomenon and the second one to the phenomenon of migration, the second equation is the Poisson equation. Numerical solution of space charge density evolution in the diffuse layer for different geometries: single electrode, two parallel walls and pipe of circular cross section can then be established. However, only in the case of a single electrode analytical solution exist in the whole diffuse layer whatever the value of the space charge density on the wall [14]. For other geometrical configuration (parallel walls or pipe of circular cross section), according to the value of the non-dimensional space charge density three zones must be distinguished: the zone of weak space
charge density which has an analytical solution, the zone of mean space charge density which must be computed numerically and the zone of strong space charge density which has an analytical solution. In fact, in bibliography, especially in the case of industrial applications, nearly always weak space charge density is considered. The error made with this approximation can be rather important and has been evaluated in the case of a pipe of circular cross section [15].

Whatever the geometry and the flow characteristic, $\rho_{WD}$ is one important parameter governing the intensity of the flow electrification phenomenon. Many investigations have been made to determine this parameter and now new kind of methods based on the measurement methods of space charge in material will be presented later.

## B. Physicochemical process at the interface solid/liquid

Several processes have been expected to be at the origin of the diffuse layer formation. Chronologically, the first is the adsorption.

### 1) The adsorption model (Fig. 3)

<figure><img src="image_3.png" /><figcaption>Fig. 3: Schematic diagram of electric charge generation in EDL according to the Adsorption Model.</figcaption></figure>

This scenario of charge generation is divided into three phases. Phase 1 corresponds to the situation before the solid/liquid contact: atoms in the solid and polar impurities or additives weekly dissociated in the liquid. In the phase 2, after the contact polarization appears in the solid and at the same time negative ions begin to be attracted by the solid. Finally, in the phase 3 negative ions are adsorbed on the solid.

This process has been the most widely used to explain the formation of the diffuse layer. But it seems not to be realistic, indeed, when one site is already occupied by one ion from the liquid then it does not more attract other ions. Thus, for a liquid flowing past a surface the adsorption phenomenon must decline and even vanish after some time and this has never been observed. This is the reason why another model called the corroding model has been developed [16].

### 2) The corroding model (Fig. 4)

In this model, phase 1 and phase 2 are the same as in the adsorption model. In the solid at the interface dipoles exist as in adsorption model, but even these dipoles can break into positive ions and negative ones.

Negative ions from the liquid come in contact with the positive ions of the interface, then a chemical reaction appears the positive ions of the solid combine with the negative ions of the liquid and these new molecules come into the liquid bulk. One might think that to confirm this model it would be useful to chemically detect traces of the solid ions within the liquid, but the low electrical charges densities measured (in the order $10^{-4}$ C m⁻³) corresponding around 1 over $10^{13}$ molecules cannot be detectable chemically. Anyway, this model seems more realistic because, unlike the adsorption model it has no time limit.

<figure><img src="image_4.png" /><figcaption>Fig. 4: Schematic diagram of electric charge generation in EDL according to the Corroding model.</figcaption></figure>

## II. FLOW ELECTRIFICATION BY LAMINAR FLOW

In a fundamental point of view, flow electrification by a laminar flow through a pipe has been widely investigated for a long time, especially because the flow behavior is well known and has often analytical expression [5,6]. It is also the case of industrial applications concerning high voltage transformer in which the flow of oil is generally laminar.

With the hypothesis of weak space charge density, another interesting advantage to analyze streaming current at the exit of pipes long enough for the diffuse layer to be fully developed, is the possibility to compute the space charge density on the wall for a fully developed diffuse layer from the streaming current measured, so, to get one important parameter of diffuse layer appearing at the contact liquid/solid.

### A. Evolution of coefficient K

The coefficient K entering in the expression of the wall current density (equ. 3) was supposed to be function of the physicochemical reaction on the wall for the two models (adsorption or corroding), even for corroding an expression was given [16], but the expression given takes into account only the chemical reactions but not a possible influence of the flow characteristics.

Then, as part of an investigation of electrostatic hazards in high power transformers, an analysis of oil flow electrification in a rectangular channel [17,18] revealed that the coefficient K is influenced not only by chemical reactions but also by wall shear stress or in other words by the flow velocity and the Reynolds number.

Thus, a new model call corroding model revisited has been established [19]. This model is based on experiments carried out on a stainless-steel capillary of 0.24mm of radius R and initially 4 m long then shortened to 3m, then 2m, 1m, 0.5m and 0.08m. The flow was laminar, and the
streaming current was measured at the exit of the capillary. For the capillary of 4m long, clearly, the space charge density transported Ql was constant for Reynolds number less than 200. This means that the diffuse layer was fully developed. It is then possible to compute the space charge density on the wall (ρWD) for a fully developed diffuse layer in the hypothesis of weak space charge density:

$$ \rho_{WD} = \frac{Ql}{8} \left(\frac{R}{\delta_0}\right)^2 \frac{I_0\left(\frac{R}{\delta_0}\right)}{I_2\left(\frac{R}{\delta_0}\right)} \quad (5) $$

I0 and I2 are the modified Bessel functions of 0 and 2 order.

Then, for a given length z of the pipe the wall current density is computed:

$$ i_w(z) = \frac{1}{2\pi R} \frac{d \int_0^R 2\pi \rho(r, z) U(r) r dr}{dz} \quad (6) $$

where U(r) and ρ(r, z) denotes, respectively, the local axial velocity and the local space charge density.

with:

$$ \int_{0}^{R} 2\pi \rho(r, z) U(r) r dr = 8\pi \rho_{w}(z) \delta_{0}^{2} U_{m} \frac{I_{2}\left(\frac{R}{\delta_{0}}\right)}{I_{0}\left(\frac{R}{\delta_{0}}\right)} \quad (7) $$

U_m being the mean flow velocity. Finally:

$$ \rho_w(z) = \rho_{wD} \left[ 1 - \exp \left( - \frac{K \left(\frac{R}{\delta_0}\right)^2 I_0\left(\frac{R}{\delta_0}\right)}{2R_e v I_2\left(\frac{R}{\delta_0}\right)} z \right) \right] \quad (8) $$

thus:

$$ K = \frac{-2\nu R_e I_2\left(\frac{R}{\delta_0}\right)}{\left(\frac{R}{\delta_0}\right)^2 I_0\left(\frac{R}{\delta_0}\right)} \ln \left[ 1 - \frac{Ql}{8\rho_{WD}} \frac{\left(\frac{R}{\delta_0}\right)^2 I_0\left(\frac{R}{\delta_0}\right)}{I_2\left(\frac{R}{\delta_0}\right)} \right] \quad (9) $$

v is the kinematic viscosity. These values of K are plotted in terms of the Reynolds number (Re = 2UmR/v) for different length z in figure 5. Obviously, K appears to be non-constant and cannot be predicted solely by the physicochemical processes on the wall. Instead, it also depends on the wall shearing stress, which is proportional to the Reynolds number. Nevertheless, it remains compatible with a corrosion process, as the corrosion can be enhanced by the shear stress, but not with the adsorption process.

On the other hand, the evolution in terms of capillary length is not clear and may be only inside experimental approximation.

<figure><img src="image_5.png" /><figcaption>Fig. 5: K evolution versus Reynolds number for different duct lengths.</figcaption></figure>

Finally, it seems that the evolution of K can be expressed as follows:

$$ K = aR_e + b \quad (10) $$

where a and b are coefficients that depend on the physicochemical reaction occurring at the liquid/capillary material interface.

In a different approach, P. Leblanc corroborates this result. He carried out flow electrification experiments using stainless steel capillaries of varying lengths and under different flow velocities. By analyzing the results for a solid-liquid contact time constant, he shows that space charge density transported is not constant, contrary to the predictions of classical theories. Instead, it increases linearly with flow velocity and wall shearing stress. [20]. Numerical simulation further reinforces these conclusions [21], without relying on the assumptions of low charge density.

## B. Charging tendency, applications to transformer monitoring

Charging tendency is an important process for fundamental aspect but especially for applications in particular concerning electrostatics hazards occurring in high power transformers. A. Sierota and J. Rungis reviewed the state of research in this field in 1994 [22], even if it concerns only the liquid and not the solid. In particular, he explicated the different apparatus used by research and industry to determine the charging tendency of the oil. Except flow in pipes system, three others systems, commonly used, are presented following.

### 1) The ministatic charge tester.

The method employed by this system was originally developed by D. A. Young at EXXON to measure the electric charge tendency in fuel (Fig. 6) [23].
<figure><img src="image_6.png" /><figcaption>Fig. 6: Schematic setup of the ministatic charge tester.</figcaption></figure>

It was later modified by T. V. Oommen and E. M. Petrie at Westinghouse [24]. Its principle is to characterize the flow electrification phenomenon during the circulation of oil through filter paper. Its advantage lies in its simplicity; however, the protocol recommended by Westinghouse is not stringent enough to ensure reliable results. Thus, a similar apparatus was developed with a new protocol [25], which was subsequently used to test the oil of all transformers in the nuclear power plants operated by EDF in France. This measurement principle has been adapted for other liquids and context with varying insulating properties by replacing the filter paper with a metallic or dielectric capillary [26,27].

## 2) *The spinning disk system.*

Another simple apparatus consists of a grounded disk rotating in the liquid contained in a metallic vessel grounded through an electrometer. This simple apparatus designed and used by J. Kedzia [28] for transformer oils, provided unreliable results. For this reason, it has been decided at the CIGRE Study Committee Ex-15 held in Paris in 1990 to modify the spinning disk system. The principle of this new spinning disk apparatus is presented fig. 7. The rotating disk is covered of pressboard of same quality than the one inside the transformer as well the oil is the same. At first vacuum is made in the chamber then it is filled with nitrogen heated gas. After drying, the pressboard is impregnated and the cell is filled of oil arriving directly from the transformer. The rotational speed recommended is 130 rpm over 8h, then 16h rest and 8h again of rotation. During rotation the leakage current of the chamber is measured. This method has been widely employed, notably to analyze the impact of the chemical composition of mineral oils, whether new or aged [29, 30], and more recently to study the charging tendency of ester-based dielectric liquids [31,32,33], or oil including nanoparticle [34]. Additionally, it has been applied to other dielectric liquids, such as toluene, hexane, and ethanol for a metallic rotating electrode [35,36].

<figure><img src="image_7.png" /><figcaption>Fig. 7: Schematic setup of the CIGRE spinning disk apparatus.</figcaption></figure>

## 3) *The Couette charger associate with the absolute charge sensor (ACS).*

This apparatus performed by the group of M. Zahn in M.I.T. [37] is schematically shown fig. 8. It is a very sophisticated system. The charge in oil is generated by the flow inside a Couette charger acting like a current generator in the circuit. Generally, the inner and the outer electrode of the Couette charger are made of metal bare, but could be covered with a pressboard sheet which is a situation more similar to the flow electrification occurring in a transformer.

<figure><img src="image_8.png" /><figcaption>Fig. 8: Schematic setup of the Couette charger associate with the absolute charge sensor (ACS)</figcaption></figure>

## 4) *Oscillating system.*

The principles for studying the phenomenon of flow electrification were developed by the research group at Opole University in Poland. They are based on oscillating systems with two distinct geometries. The first is similar to a couette charger, but with an oscillation replacing the electrode's rotation [38]. The second adopts a planar geometry [39,40], where the relative motion between the solid and the liquid is achieved through the oscillation of a plane driven by an eccentric mechanism controlled by a servomotor (Fig 9).
<figure><img src="image_9.png" /><figcaption>Fig. 9: Schematic setup of the plane oscillation set-up.</figcaption></figure>

The behavior of the currents is particularly complex. Their frequency analysis is not only correlated with the oscillation frequency of the system but also contains specific information related to the electrical double layer (EDL). Wotska [41] focused on simulating flow electrification phenomena in an oscillating Couette system. He specifically investigated the influence of excitation frequency on the organization of the liquid's electric charges following such a stimulus. In the future, this information could provide valuable insights into the dynamics of charge formation at the interface, as well as the mobility of charges within the EDL.

5) Apparatus based on flow inside pipes or channels.

The first is probably Boumans [6]. Then, Klinkenberg [11] which research was devoted to petroleum industry and perform industrial size experiment on pipes of 350 m long and tanks of 1000 m³ (fig. 10).

<figure><img src="image_10.png" /><figcaption>Fig. 10: Schematic Klinkenberg experiment setup.</figcaption></figure>

Experiments were made with gasoline in which they had a percentage of water, so, they determined the percentage of water which generates the maximum streaming current, thus the most hazardous conditions. An explosion of one tank due to flow electrification has been even published in the report [11].

In Poland, Brzostek, Kedzia and Szuta [42, 43] performed a device with a pipe to analyse the flow electrification phenomenon of transformer oil.

As well the group of Poitiers University has studied the phenomenon of flow electrification in high voltage transformers since 1991 and during around twenty years in a program of research supported by "Electricité De France" (EDF). For this purpose, two large flow loop facilities and one, smaller but with more measurement capability, have been performed.

The first large flow loop facility is presented figure 11 [44]. Initially, it was composed of three channels (2) of pressboard, 75 cm long and whose section dimensions are (2-0.2) cm, (4-0.4) cm and (6-0.6) cm. There was a constant ratio of ten between the two dimensions of each section. That is the reason why it was assumed that the flow of the oil inside the duct is almost the same as the one between two infinite plates. For one particular study, the upper channel was replaced by a cylindrical duct to analyze the effect of an impinging jet. Each of those three channels is placed inside a larger metallic one and the set is linked together with the two large vessels (1)a and (1)b. A system of valves directs the flow to one of the channels, and a set of three flow meters (4) allows for measuring flow rates of up to 15 m³/h.

Two kinds of experiments have been made. The first one makes the oil flow through one channel without recycling. The oil goes from (3)a to (3)b by the action of the pressure difference of N2 between the two vessels. Once (3)a is empty and (3)b is full, the experiment stops and the oil is sent back to (3)a after having been discharged; the experiment can start again. In that type of experiment, the electrical current generated by the flow in the channel chosen can be measured, with electrically neutral oil at the entrance. The second kind of experiment is done with a pump (7) switched on. The flow of oil is then continuous, thus it is possible to measure the influence of charged oil at the entrance of the duct, on the wall current.

<figure><img src="image_11.png" /><figcaption>1: stainless steel vessel, 2: testing vein, 3: pressurization vessels, 4: flow meters, 5: faraday cage, 6: insulator, 7: pump current 8: conductivity cell, 9: oil moisture measurement.</figcaption></figure>

<caption>Fig. 11: Diagram of the first large flow loop facility.</caption>
<figure><img src="image_12.png" /><figcaption>1: Testing vein, 2: Pressboard disc, 3: Electrodes, 4: PTFE holder, 5: Tube, 6: Adjustment device</figcaption></figure>

<caption>Fig. 12: Diagram of the impinging jet vein.</caption>

In each case the current due to the charge accumulated in the pressboard and as well the streaming current transported by the oil are measured and correlated.

The vein devoted to analyse impinging jet (fig. 12) has been also tested on this setup [18]. The aim of this research was to understand the reason why the pressboards of shell transformers situated in front of the oil flow entering in the bottom of the transformer, in the 45° zone, showed traces of electrical discharges. Thus, the experiments with this vein correspond to a kind of simulation. Indeed, experiments showed that the wall shearing stress is reinforced in that zone, which leads to a much greater generation of charges.

Whatever the facility used, in order to avoid any disturbance due to the contact of the oil or pressboard with air and moisture, the whole flow loop was previously emptied with a vacuum pump and then filled up with nitrogen gas before the oil was filling the loop.

The second large flow loop facility is presented in figure 13. The aim of the perfection of this facility was to determine if the leakage current measured on a transformer is related with the charge growing on some well insulated pressboard parts of the transformer which then could be at the origin of creeping electrical discharges [45]. In other word, if it could be an indicator of electrostatic hazard in the transformer better than ECT or tan δ.

This flow loop facility includes two ducts. The first one (4) is a long (1.5m) Teflon channel inside which can be inserted a pressboard channel simulating well insulated pressboard part of a transformer. At the middle of the pressboard channel, on its external face is fixed an electrode (a thin sheet of metal) which make possible to measure the charge accumulated in the pressboard due to the oil flow.

The second one (5) is metallic simulating the body of the transformer. It is filled of different kind of metallic

windings with their insulated paper assembled with press board plates. Oil is flowing inside channels provided between stacks of pressboard plates on which have been stuck several blocks of pressboard, similar to those encountered inside transformer channels to slowdown the flow.

<figure><img src="image_13.png" /><figcaption>1: Flow meter, 2: Stainless steel vessel, 3: Heat Exchanger, 4: Well insulated vein, 5: Leakage current vein, 6: Pump, 7: Conductivity cell, 8: Moisture control, 9: Ultra thermostat, 10: Pico-ammeter</figcaption></figure>

<caption>Fig. 13: Second flow loop facility.</caption>

This duct electrically insulated to the rest of the facility and inside a faraday cage is intended to simulate the leakage current measured on a transformer.

The experiments made shown that the capacitive current pass by a maximum and then vanish, meaning the charge accumulated in the pressboard channel is limited while the leakage current remain constant. Thus, it seems no correlation exists between them. Finally, it seems that the better way to predict Electrostatic hazard is to record directly the charge accumulation on a piece of well insulated pressboard, leakage current does not seem to be a pertinent indicator.

### 6) Small flow loop facility.

This facility, shown in figure 14, is also called "capacitive sensor for flow electrification in transformer" [46]. This facility has several advantages compared to the two previous one. First it is much less bulky and can be transported rather easily. More, it needs much less oil and pressboard to analyse. Finally, all the currents measured can be correlated, indeed, it is possible to observe and record the accumulation current, the upstream and downstream leakage currents and the streaming current.

As we can see on figure 15 the positive current due to the charges transported by the flow of the liquid is equal and opposite to the negative charge accumulated in the pressboard added to the negative charge leaking along the pressboard upstream and downstream.

In fact, the process responsible of the phenomenon of flow electrification can be considered like a current generator regarding the flow of the liquid or a kind of charge separator between the solid (the pressboard in that case) and the liquid (the oil).
<figure><img src="image_14.png" /><figcaption>1: Pump, 2: Heat regulation, 3: Flow meter, 4: Oil tank, 5: Manometer, 6: Sensor, 7: Accumulation current 8: Upstream leakage current 9: Downstream leakage current 10: Streaming current</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 15: Typical currents measurements on the capacitive sensor.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 16: Electric equivalent circuit of the electrification phenomenon at the dielectric solid/liquid interface.</figcaption></figure>

In the solid one part of the charge are captured and accumulated and one-part leaks along the interface. Finally, the process, throughout the pressboard, is well described and illustrated in figure 16. Many experiments have been made on this facility supported by EDF [46,47,48] and as well for fundamental research. It has been possible to test several oils combined with several kind of pressboards, even pressboards which was predefined, to see the influence of the different pressboard components. Also, different oils with and without additives have been tested.

The increasing use of High Voltage Direct Current (HVDC) equipment raises new questions regarding the impact of a DC electric field on electrostatic hazard [49,50]. In this context, the capacitive sensor has been modified to study the evolution of the accumulated charge trapped in the pressboard [51] for different DC voltage values. At low amplitudes of the DC electric field (central zone), charge generation appears to be controlled by phenomena related to the electric double layer (EDL) (Fig. 17). However, at higher values (outer zone), a break in the charge accumulation behaviour is observed; with the processes being dominated by mechanisms induced by the external DC field.

<figure><img src="image_17.png" /><figcaption>Fig. 17: Evolution of Accumulated Charge versus the external DC Electric Field.</figcaption></figure>

According to a principle similar to that of the "capacitive sensor", Liu [52] designed a device in which the pressboard vein is electrically isolated inside a large oil reservoir, thereby more faithfully reproducing the insulation conditions of transformers. He observed a reduction in the dielectric strength of the oil, whether the electric field is direct or alternating, in the presence of the flow electrification phenomenon. This reduction becomes even more pronounced as the oil flow velocity increases.

### III. FLOW ELECTRIFICATION BY TURBULENT FLOW

A turbulent regime considerably complicates the phenomenon of flow electrification.

Few researches have been made on the subject apart the important work made by Abedian and Sonin using the
eddy diffusivity to explicit the turbulent effect [53]. Smarzly made an analytical study [54] but in the hypothesis of weak space charge densities and with important approximations on the velocity profiles. Experimentally, flow electrification of liquid nitrogen has been investigated [55,56]. In our laboratory we have also carried out theoretical researches since 1978 and experiments on highly turbulent flows since 1982, which are mainly reported in [57,58]. Following we analyze the phenomenon.

Turbulence homogenizes the distribution of EDL charges within the liquid volume. In contrast with laminar flow, electrification in turbulent flows is not only a function of two parameters ($\rho_{w+}$ and $R_+$), but also of two others ($R_e$ and $V_R = 10 D_0/v$)[10]. $V_R$ is similar to an inverse Schmidt number made with the ratio of the diffusion coefficient of the ions divided by the kinematic viscosity of the liquid, highest is this parameter and more the diffuse layer will relax rapidly from a perturbation.

In the hypothesis of a fully developed diffuse layer: ($\rho_{w+} = \rho_{wD+}$), three models of diffuse layer behavior in turbulent flows have been perfected successively [58].

## A. First model

The first model is based on a competition between the relaxation velocity of the diffuse layer and the radial fluctuant velocity of the flow. The relaxation velocity of the diffuse layer is given by the following equation:

$$V_{rel} = \frac{\sigma\delta_0}{\varepsilon} \sqrt{\rho_+^2 + 1} \quad (11)$$

$\rho_+$ is the dimensionless space charge density, it is a local quantity, which, like $\sigma$, varies with the radial coordinate, thus, as well $V_{rel}$ is a function of the radial coordinate. The greater is the dimensionless space charge density then the greater is the relaxation velocity. The charges in the diffuse layer being greater near the wall, the relaxation velocity is greater near the wall than in the center.

In contrast, the root mean square value of the radial fluctuating velocity ($v'$) is null on the wall and maximum in the center (Laufer [59]).

We introduce two quantities:

$$- \text{The friction velocity: } u^* = \sqrt{\frac{\tau_w}{\rho_m}} \quad (12)$$

$\tau_w$ being the wall shear stress and $\rho_m$ the specific mass of the liquid

$$- \eta : \eta = \frac{(R-r)u^*}{v} \quad (13)$$

Since $\rho_+$, is computed (numerically or analytically) in the same way as $V_{rel}$, thus, in order to facilitate comparison between relaxation velocity and fluctuating velocity two analytic laws (depending on the value of $\eta$) have been perfected, following correctly the experiments made by Laufer:

$$\text{for } \eta \le \eta_3 = \frac{\text{sh}(\frac{0.9}{0.171} - 1.46) + \text{sh}(1.46)}{0.3911}$$

$$\frac{v'}{u^*} = 0.171[\text{Argsh}(0.391\eta - \text{sh}(1.46)) + 1.46] \quad (14)$$

for $\eta > \eta_3$

$$\frac{v'}{u^*} = 0.9 \quad (15)$$

With these tow analytical expressions of the root mean square value of the radial fluctuating velocity, $v'$ and $V_{rel}$, could be compared in the whole cross section of the pipe.

In the center of the pipe, where the space charge is weaker and the turbulence intensity greater, then $v' > V_{rel}$, this region is considered to be homogenized by the turbulence, while near the wall, where the turbulence intensity is weaker but the space charge higher ($v' < V_{rel}$), the diffuse layer is unperturbed (Fig. 18 and Fig. 19).

<figure><img src="image_18.png" /><figcaption>Fig. 18: $v'$ and $V_{rel}$.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 19: Unperturbed and homogenized zones.</figcaption></figure>

## B. Second model

In this model, the eddy diffusivity $D_t$ enters in the fundamental system of equations governing the evolution of the space charge density of the diffuse layer.

Indeed, for a diffuse layer at rest or in a laminar flow, the fundamental system (equ. 4) written in the case of a pipe of circular cross section is:

$$\left\{ \begin{array}{l} \frac{d\rho_+}{dr_+} + \sqrt{\rho_+^2 + 1} \frac{d\psi_+}{dr_+} = 0 \\ \Delta_+ \psi_+ = -\rho_+ \end{array} \right. \quad (16)$$

Thus, this system becomes for a turbulent flow:

$$\left\{ \begin{array}{l} (1 + D_{t+}) \frac{d\rho_+}{dr_+} + \sqrt{\rho_+^2 + 1} \frac{d\psi_+}{dr_+} = 0 \\ \Delta_+ \psi_+ = -\rho_+ \end{array} \right. \quad (17)$$

with $D_{t+} = (D_t/D_0)$.

Then, knowing the evolution of $D_t$, it is possible to compute numerically the evolution of the space charge in all the cross section.

An example of the evolution of the dimensionless space charge density of the diffuse layer is shown figure 20 in the case of the first model and in figure 21 for the second
one, while the four parameters $\rho_{w+}$, $R_+$, $R_e$ and $V_R$ are the same. The dimensionless radius of the perturbed zone is also written in figure 20 for the first model and the ratio of the total space charge density for the second model over the total space charge density for the first one (parameter Sp), (Fig.21). Clearly the total charge in the whole section is greater in the second model but, this is in contradiction with the fact that the diffuse layer is fully developed and that the charge evolution in the laminar sub-layer is the same for the two models. Thus, even if the second model appears better than the first one because it has no discontinuity, it is not acceptable like that. This is the reason why a third model has been perfected.

<figure><img src="image_20.png" /><figcaption>Fig. 20: Charge density evolution for model 1</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 21: Charge density evolution for model 2</figcaption></figure>

### C. Third model

In this third model two zones in a section of the pipe are defined, one, very close to the wall which is call the laminar sub layer in which the Diffusivity due to turbulence is equal to zero, and the rest of the pipe section until the axis where $D_t \neq 0$.

Then, the system of equations which govern the diffuse layer in the laminar sub layer would be the same that in a laminar flow or for a diffuse layer at rest (equ. 16). In another hand, the equations of the diffuse layer in the other

part of the pipe (the more central region) are the same than those given in the second model (equ. 17).

The boundary conditions being:

- The space charge density on the wall $\rho_{w+}$, (thus for the laminar sub layer) is the same than in the two other models and the same that in the case of a laminar flow or even of a liquid at rest in the pipe.

- At the junction of the two zones the sum of the space charge density in the central zone (where the diffusivity due to turbulence $D_{t+}$ is not equal to zero) must be equal to the value of the sum of the space charge that would exist in the case of a laminar flow.

Thus, for this model the two important conditions: same space charge density on the wall and same total charge quantity in the whole section are respected while keeping a charge evolution due to the turbulence intensity in the central region.

However, as for the first model, a discontinuity exists at the connection with the laminar sub-layer. Nevertheless, this small discontinuity only slightly modifies the charge transported by the flow.

<figure><img src="image_22.png" /><figcaption>Fig. 22: Charge density evolution for model 3 and $R_+=10$</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 23: Charge density evolution for model 3 and $R_+=1$</figcaption></figure>

We have plotted in figure 22 and figure 23 the space charge density profile given by this model for the same parameters value used for the first and second model (3
values of $R_e$, $V_R = 100$, $\rho_{w+} = 10$) and for two different dimensionless radii ($R_+ = 10$ and 1). The discontinuity is visible for $R_+=1$ and $R_e=3000$.

In figures (24,25,26) is shown the ratio ($Q_p$) of the space charge transported by a turbulent flow over the space charge transported by a laminar flow predicted by the three different models.

<figure><img src="image_24.png" /><figcaption>Fig.24: $Q_p$ in terms of Re (Model 1).</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig.25: $Q_p$ in terms of Re (Model 2).</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig.26: $Q_p$ in terms of Re (Model 3).</figcaption></figure>

It is clear, that for very large Reynolds number a big difference exists between model 2 and the two others but looking the data for Reynolds numbers in the range 3000 to 10 000, there is not a so big difference.

The particular interest in developing models for high Reynolds numbers lies not only in a fundamental perspective but also in their application to the handling of conventional fuels (hydrocarbons) [60,61], as well as cryogenic liquids, particularly hydrogen. Indeed, the low viscosity of liquid hydrogen allows reaching very high Reynolds number, that is to say, highly turbulent flows even in small diameter tubes.

Even, if in the past experiments have already been made with hydrogen liquid, Liquid Natural Gas (LNG) and nitrogen liquid [62,63], which at the time was truly innovative, but could not give sufficient reliable results to be able to confirm one model rather than another. However, now, taking into account the difficulties encountered, it seems possible to make new experiments more reliable, (especially avoiding the past quasi-inevitable appearance of a two-phase flow due to the boiling of liquid hydrogen in the capillary tube) thus to compare such experimental result to the theoretical predictions presented above. Then, one of these models being validated it would be very useful especially for the development of hydrogen as a transport fuel.

#### IV. ALTERNATIVE METROLOGY TO IMPROVE THE UNDERSTANDING OF THE EDL.

The main aim of the studies presented previously was to determine the intensity of the flow electrification phenomenon, generally in order to predict and limit this phenomenon, with the main objective of preventing electrostatic hazards in industry. Only the studies carried out in tubes of circular cross-section in the case of laminar flow and for tubes long enough for the diffuse layer to be fully developed at the tube outlet have made it possible to obtain the structural elements of the diffuse layer, namely $\rho_{WD}$ and $\delta_0$ [19]. We will now present methods for obtaining these elements directly.

##### A. Kerr effect measurements

Kerr effect measurements represent the first alternative technique for measuring the distribution of electric charges within the electrical double layer [64, 65]. The Kerr effect is based on the birefringence properties exhibited by some liquids when subjected to an electric field. These measurements typically rely on the use of monochromatic polarized lasers passing through a cell containing the liquid. The polarization of the light is modified proportionally to the square of the local electric field intensity, thanks to polarization analyzers and photodetectors. The grayscale levels in the resulting images provide information on the spatial distribution of the electric field, which can then be used to calculate a charge density map (Fig 27).

<figure><img src="image_27.png" /><figcaption>Fig. 27: Experimental cell for electric field measurement using Kerr effect.</figcaption></figure>
Synchronized approaches using alternating electric fields (AC) enable the exploration of charge dynamics and electrode polarization. Kerr effect measurements are particularly well-suited for polar liquids and electrolytic solutions, especially aqueous solutions, but they have limitations for low-polarization dielectrics. This metrology was initially extensively tested on nitrobenzene, which has a particularly high Kerr constant ($3.5 \times 10^{-12} \text{ m} \cdot \text{V}^{-2}$). However, other liquids, such as ultrapure water, mineral oils, esters, and HydroFluoroEthers (HFE), have also been tested [66]. The main limiting factors of this measurement technique lie in its lack of sensitivity, linked to optical metrology, which could be partially addressed by further advancements.

## B. Stimuli measurements

Over the past decade, stimulus-based methods, initially developed to measure the distribution of electric charges in solids, have been considered for their potential adaptation to analyzing the distribution of electric charges in liquids, particularly within the EDL. However, this adaptation presents numerous challenges, primarily due to the weak charge densities to be measured.

In the case of the EDL involving dielectric liquids (such as mineral oil or heptane), the maximum charge densities are low, on the order of $10^{-4} \text{ Cm}^{-3}$, but they are distributed within the liquid over depths reaching several hundred micrometers, or even up to a millimeter. In contrast, for more conductive solutions, such as pure water, the maximum charge densities are much higher, around $10^3 \text{ Cm}^{-3}$, but they are concentrated over very shallow depths, in the order of a few nanometers. Depending on the type of liquid, it is therefore necessary to adjust the sensitivity and resolution of the method employed. An additional challenge stems from the inherent nature of the method itself. These techniques rely on the application of an external stimulus, such as a pressure wave, a temperature step, or an electrical pulse, which disturbs the electrostatic equilibrium of the EDL and generates an electrical signal detected by a pair of electrodes. The stimulus must create a temporary imbalance in the electric charges without altering their organization, for instance by avoiding the generation of liquid flow for example.

### 1) Thermal stimuli.

The Thermal Step Method (TSM) is likely one of the first techniques to have been explored. This approach is based on the use of a temperature step propagating through a chamber containing the liquid, bounded by two electrodes (Fig. 28). As it travels from one electrode to the other, the stimulus disrupts the electric charges of the electrical double layer at both interfaces. However, only the signal from the excitation electrode is detectable due to the strong attenuation of the thermal step during its propagation.

The cell used features an axisymmetric stainless steel geometry with a liquid thickness of 3 mm [67,68]. It enables the generation of positive or negative temperature steps of up to 20 K, with heating rates reaching several

$K \cdot s^{-1}$. Various dielectric liquids have been studied: mineral oils, new or used, with or without additives (modifying their electrical conductivity), silicone oils, as well as cyclohexane. A temperature step of a few degrees is sufficient to produce a signal. This signal appears as a unipolar transient, characteristic of each liquid, and reflects the properties of the EDL in terms of signature, amplitude, and dynamics (Fig. 29). However, depending on the viscosity of the liquid used and the imposed level of excitation, thermoconvective phenomena may disrupt the organization of the EDL. The estimation of the Rayleigh number, combined with the analysis of signals from two measurements imposing identical but oppositely sloped temperature step, allows this impact to be assessed. Experimental measurements have been compared to numerical simulations based on a charge distribution within the EDL, established according to the Stern model (neglecting the compact layer) and data from the literature [69,70]. The simulations produce currents with a shape similar to the experimental signals but with amplitudes one to two orders of magnitude lower. This difference is attributed to the effects of the compact layer, which are not accounted for in the modeling.

<figure><img src="image_28.png" /><figcaption>1: Thermal stimulus chamber using a coolant liquid, 2: Liquid sample chamber, 3: Chamber at constant temperature</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Fig. 29: Transient thermal step currents measured on mineral oil for different thermal steps.</figcaption></figure>

This difficulty in precisely interpreting the experimental signal has led the authors to refrain from proposing
mathematical deconvolution methods to establish the charge distribution within the EDL. Finally, tests conducted with more conductive liquids (such as pure water) failed to detect any signal, which seems to indicate the resolution limits of the method in its current configuration.

## 2) Pressure stimuli.

PWP (Pressure Wave Propagation) measurements rely on the use of a pressure wave generated by a piezoelectric transducer [71,72]. The experimental cell has a planar geometry, with the inter-electrode distance, filled with the liquid under study, adjustable between a few tens of microns and 3 millimeters. The pressure wave, which can be modulated in frequency and amplitude, can reach a pressure of 5 MPa with a half-width duration of 40 ns. It propagates from the aluminum emitting electrode, through the liquid, to the brass measuring electrode (Fig. 30). The voltage signal (Fig. 31), measured in open circuit between the two electrodes, displays two trains of transient signals. These are induced by the propagation of the pressure wave near the two interfaces: emitter electrode/liquid and liquid/measuring electrode. The time separation between these two signal trains is closely correlated with the travel time of the pressure wave in the cell and thus depends on the speed of sound in the liquid

<figure><img src="image_30.png" /><figcaption>Fig. 30: PWP Measuring cell.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 31: PWP measurement for aluminum/glycerol and glycerol/brass interfaces.</figcaption></figure>

and the inter-electrode spacing. For most of the liquids studied, each signal train consists of two transients of opposite polarities. The amplitude, the signature, and the duration of these transients are correlated with the direction of propagation of the pressure wave at the

interface, as well as the nature of the materials and, therefore, the properties of the EDL. The shape and characteristics of the signals have been confirmed by numerical studies [73].

The sensitivity and resolution of the measurement are optimized by increasing the amplitude of the pressure wave and reducing the inter-electrode spacing, provided this spacing allows the EDL of the liquid to develop without mutual interaction between the two interfaces (2*5δ₀). According to numerical simulations, the measured signal becomes more intense as the charge densities increase. Thus, for electrolytic solutions, glycerol, ethanol solutions, or additive-containing oils, covering a conductivity range from 10⁻⁹ Sm⁻¹ to 10⁻² Sm⁻¹, the PWP signal is entirely satisfactory. Conversely, for dielectric liquids (such as new or used mineral oils) with a conductivity below 10⁻¹¹ Sm⁻¹, no significant signal was observed, highlighting the current sensitivity limitations of this method.

However, as soon as a DC electric field is applied between the two electrodes—whether using a dielectric liquid like mineral oil or glycerol—a time-dependent signal is measured. This signal reflects the creation, transfer, or movement of electric charges based on the orientation of the field, leading to a new equilibrium state in the charge distribution within the liquid [74]. For a dielectric liquid, this geometry resembles that of a charged capacitor, which could serve as a step in calibrating the method. Ultimately, this could allow the PWP technique to map the charge distribution of the EDL.

## 3) Electro-acoustic stimuli.

Although widely used for the study of impregnated solids (such as oil-impregnated paper [75,76] and other dielectric solids immersed in liquids [77,78]), the application of the PEA (Pulsed Electro-Acoustic) method to the analysis of EDL remains relatively recent [79]. This method relies on the use of an electrical pulse as a stimulus, reaching an amplitude of 2 kV with a half-width duration of 20 ns. By destabilizing the charges in the medium under study, the pulse generates a pressure wave, which is measured using a piezoelectric transducer. The experimental cell geometry is similar to that used in PWP measurements (Fig 32). The electrical response obtained (Fig 33) also resembles that observed in PWP measurements: two wave trains are induced at the interfaces, each composed of two transients of opposite polarities.

The initial studies conducted with this method are promising, as a consistent response has been observed for liquids such as ethanol and glycerol.

# V. CONCLUSION AND FUTURE PROSPECTS

The phenomenon of flow electrification, primarily studied in the context of electrostatic hazard, remains a current topic of interest. Recent research has particularly focused on the influence of wall friction on the development dynamics of the electric double layer and the impact of a direct current electric field on its organization.
<figure><img src="image_32.png" /><figcaption>1: Aluminium exitation electrode, 2 Liquid:, 3: Aluminium measurement electrode 4: Piezoelectric sensor.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Fig. 33: PEA measurement for aluminum/ethanol interfaces.</figcaption></figure>

The widespread adoption of new dielectric liquids, such as esters and nanofluids, in electrical power components, transportation, HVDC power grids, as well as the emergence of cryogenic liquids like hydrogen (H2) as an energy carrier, raises new challenges. These advancements bring electrostatic hazard back into focus in unprecedented industrial contexts. Similar to historical approaches, the study of these new issues relies mainly on empirical expertise. These approaches take into account the specific properties of materials, flow regimes, and system dimensions while remaining closely tied to industrial realities. However, the lack of fundamental data concerning the physics and the behavior of the EDL for a liquid at rest and in highly turbulent flows (in the case of hydrogen) still limits the ability to develop predictive and reliable numerical simulations. The collection of such data is a major challenge. It requires the implementation of alternative methods to characterize the distribution of electric charges within the EDL, whether at equilibrium or during its development and transport. This is essential for deepening our understanding of this complex phenomenon.

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