Resistance bias estimation of a liquid column in a cylindrical conductivity cell with lateral liquid supply

The article shows a physical model of a cylindrical two-electrode conductivity cell with inlet and outlet holes for filling located perpendicular to the cell axis. Based on the finite element method (FEM), the non-uniformity of the current density distribution inside the cell was determined. For a range of geometrical parameters of the cell, the resistance biases of the liquid column with respect to the idealized model—a cell with a uniform current density distribution (without holes) are calculated. A mathematical expression is given that describes calculating the electrolytic conductivity value using the geometrical parameters of a conductivity cell, taking into account the field distortion caused by holes for filling. It has also been found that at the ratio of the cell diameter to the hole diameter D/d ≥ 5, the entire field distortion inside the cell is provided by a liquid column in the holes with a length of only (h ≤ d) mm. Theoretical estimates and mathematical models covered in this article were used to create the primary differential conductivity cell. Structurally, such a cell consists of two tubes of the same diameter but of different lengths, at the edges of which platinized electrodes are placed.


Introduction
The principle of realization of the unit of electrolytic conductivity (EC) k (S•m −1 ) is to measure the electrical resistance R hom (Ω) of a liquid column of clearly defined geometry, provided that there is a uniform current density distribution J (A•m −2 ) [1]: where L (m) is the length of the liquid column, A (m 2 ) is the cross-sectional area, and D (m) is its diameter.The device (sensor) that forms this liquid column is called a conductivity cell.The ratio of the geometric dimensions of the cell-the length L (m) between the electrodes to the cross-sectional area A (m 2 )-is called the cell constant K cell (m −1 ).If the geometric parameters of the cells that define the constant can be accurately measured and checked from time to time, then such cells perform the functions of primary cells.They are used as part of the national measurement standards for the EC unit.For typical sizes of primary cells: length 30-100 mm, diameter 10-20 mm, the unevenness of the inner diameter should be less than 2 μm.This corresponds to technological errors of 0.005%-0.01%for length and 0.01%-0.02%for diameter.Such problems relate to technological problems of precision machining of the inner surfaces of the tube.
Several designs of such cells are known, which are used as primary ones.The most popular among national metrology institutes (NMIs) are Jones-type cells with a removable central extension tube [1][2][3][4][5][6][7] and cells with a moving electrode [6][7][8][9][10].Regardless of the cell design, the method for measuring EC remains the same.As a rule, this is a differential method whose main goal is to suppress the effect of electrochemical impedance at the metal/ solution boundaries [11][12][13][14][15][16][17][18].The method is implemented by measuring the difference in resistance between two liquid columns of the same diameter but different lengths.Thus, the basis of a cell of any design [1][2][3][4][5][6][7][8][9][10] is always the electrical resistance of a virtual liquid column with a known geometry, which ultimately determines the cell constant.For an idealized model (figure 1), the constant can be easily calculated by its geometric parameters L (m) and D (m), but the question arises of how to fill such a cell with liquid.
This article discusses simple physical models of a cylindrical conductivity cell that has holes in the side surface for filling.The purpose of the study is to determine the effect of the holes on the non-uniformity of the current density distribution J (A•m −2 ) inside the cell and to establish the value of the resistance bias of the liquid column with respect to an idealized model with a uniform field inside.Using data on the non-uniformity of the current density distribution J (A•m −2 ) and, accordingly, the resistance bias, it is possible to ensure the calculation of the EC value for such cells based on their geometric parameters, which is not possible for secondary Jonestype cells [2,5,19,20].
In order to theoretically calculate the resistance of a cell containing holes in its structure, mathematical models were used and the finite element method (FEM) was applied [21].

Physical and mathematical models 2.1. Physical model of a cell
The physical model of the cell, which is the subject of the study, is shown in figure 2. It is a cylindrical glass tube with an inner diameter of D (m) and a length of L (m), at the edges of which electrodes are placed.At some distance × (m) from the electrodes, holes with a diameter d (m) were made for filling the cell.Dimensions × (m) and d (m) are identical for both holes in the cell, the locations of which are centrally symmetrical to each other.The presence of side holes leads to an increase in the effective cross-sectional area A (m 2 ), and, in accordance with equation (1), this leads to a decrease in the resistance of the liquid column.
Thus, the resistance R w (Ω) of a filled cell that has side holes will be less than the resistance R wo (Ω) of an idealized model (figure 1) with a homogenous field inside, calculated by expression (1) by the relative value (bias) δ R (%), which is expressed as follows:  ( ) Since the geometry of the liquid column is fixed, this bias can be considered a systematic error.

Mathematical model of a cell
In order to calculate the resistance R w (Ω) and, accordingly, the bias (2), a mathematical model was used.It is based on a field theory in which the distribution of the scalar electric potential corresponds to the threedimensional Laplace's equation [16,17] in rectangular Cartesian coordinates: Calculation of the resistance of the liquid column R w (Ω) was carried out according to equation (4), where the measuring current I cell (A) was set as a constant and the electric potential U cell (V) was calculated by integrating the z-component of the electric field strength E Z (V•m −1 ) over the liquid column length L (m): Equations (3) and (4) are standard built-in functions of many software products designed for modeling various physical phenomena by the finite element method (FEM).In this work, in order to estimate the resistance of the filled cell and obtain quantitative characteristics for a range of geometrical parameters of the cell, COMSOL Multiphysics ® software was used.When modeling, the following parameters were set: EC of the solution k = 0.1 S•m −1 , current through a cell I cell = 1 mA, relative permittivity of the solution ε r = 77 [22], the boundaries of the model are electrically isolated, and the result of solving the model is the resistance between the electrodes.
The accuracy of the simulation result is greatly affected by the size of the mesh elements of the 3D model.This is especially critical for small models with high surface curvature.The following parameters for the size of tetrahedral mesh elements were used in the simulation: maximum element size 2.18•10 -3 m, minimum element size 2.18•10 −5 m, maximum element growth rate 1.3, curvature factor 0.2, resolution of narrow regions 1.These settings correspond to the predefined setting 'extremely fine'.A typical mesh of a liquid column 3D model is shown in figure 3.

Influence of a parameter h (mm)
In order to obtain quantitative characteristics of the resistance bias δ R (%) (2) for a range of cell geometric dimensions and parameters of holes for filling, the task was to choose a sufficient size of the parameter h (mm) of liquid column 3D models for the completeness of the bias assessment.For this purpose, a separate assessment of the effect of the size h (mm) of 3D models on cell resistance was carried out.The research results show that at the ratio of the cell diameter to the hole diameter D/d 2.5, all the effect due to the presence of the holes is provided by the length of the liquid column in the holes, h 1.5d (figure 4

Dependencies for a range of geometric dimensions
For a range of geometric dimensions of the cell, dependencies of the resistance bias of the liquid column δ R (%) (2) were obtained as a function of the cell length L (mm) (figure 5(a)) and diameter D (mm) (figure 5(b)) at hole parameters d = 2 mm and x = 2 mm.An analysis of the plots shows that for cells with an inner diameter D (mm) from 10 to 25 mm and a length L(mm) from 25 to 100 mm, the absolute value of the resistance bias |δ R | does not exceed 0.1%.

Influence of hole diameter d (mm)
The diameter of the holes for filling the cell has a significant effect on the resistance of the cell; this is especially noticeable when the ratios D/d and L/D decrease.The dependencies of the resistance bias δ R (%) for cells of different lengths with a diameter D = 10 mm are shown in figure 5(c).
It can be seen from the plot that a more rapid decrease in the cell resistance is observed with a decrease in the L/D ratio, and the bias δ R (%) is relatively small when the hole diameter is d 2 mm (i.e., D/d 5).

Influence of the position of the holes along the length of the cell
The position of the holes along the length of the cell tube has almost no effect on the value of the bias δ R (%) since it is about of thousandths of a percent.The simulation results show that the greatest effect on cell resistance reduction occurs when the holes are right next to the electrodes and the smallest when the holes are opposite each other (figure 5(d)).The size of the resistance bias δ R (%) when the holes are close to the electrode is especially important.The design, provided x d/2 makes it possible to more or less easily remove air bubbles near the electrode.

Accuracy of results
In order to evaluate the accuracy of the results of modeling by the finite element method, the resistance value R s (Ω) obtained by solving the 3D models of cylindrical liquid columns was compared to the resistance R c (Ω) calculated from the geometrical dimensions of the idealized model using equation (1).
The error of the simulation results δ s (%) was determined by the following: The results for small models are especially informative since they have the highest surface curvature.The results of calculating the error δ s (%) (5) for some models are given in table 1.In all cases of solving the models that were carried out, the error δ s (%) (5) did not exceed 0.0002 %.

How to use these data
The presence of holes for filling the cell significantly distorts the current density distribution J (A•m −2 ) (see figure 6), increasing the effective cross-sectional area A (m 2 ) and reducing the resistance of the cell, but nevertheless, this effect is constant.It can be corrected by applying the resistance bias value δ R (%) as a correction.
The research results can be used to improve the efficiency of the differential method for measuring EC.The operation algorithm of the differential method is as follows: First, the resistance R sh (Ω) of a short liquid column   is measured, and then the long one R l (Ω).EC is expressed by the equation [8-10]: where ΔL (m) is the length of a 'virtual' liquid column, which is the difference between the long and short ones.This basic equation is used to determine EC in almost all national measurement standards with a two-electrode conductivity cell.
The resistances R sh (Ω) and R l (Ω) correspond to the resistance R hom (Ω) of the idealized model in figure 1 and equation (1).The presence of holes reduces these resistance values.The resulting biases can be corrected by applying corrections to the resistance measurement results according to the expression: where K cell,l (m −1 ) and K cell,sh (m −1 ) are the cell constant values of a long and a short conductivity cell, R l (Ω) and R sh (Ω) are resistance measurement results of a long and a short cell, and δ R,l (%) and δ R,sh (%) are resistance biases (2) of a long and a short cell, respectively.The relative expanded uncertainty for accurate EC measurements ranges from 0.3% to 1%, and for the measurement standard level, from 0.05% to 0.3% (k = 2; hereinafter, k indicates a coverage factor), which is comparable to the resistance bias values δ R (%) in figures 4 and 5. Therefore, a correction of this bias is mandatory.The geometry of the liquid columns in such a differential cell is unchanged, while in a Jones-type cell with a removable central tube, radial displacements of sections can occur [23].The study of the unaccounted error that can occur in a Jones-type cell with a removable central extension tube due to radial displacements of the sections prompted the authors to design the differential primary conductivity cell depicted in figure 7. It is based on two quartz tubes with a precise inner surface with a nominal diameter of D = 9 mm and lengths of L l = 100 mm and L sh = 50 mm, at the edges of which platinized electrodes are placed.Holes for filling that have a diameter of d = 2 mm are located at a distance of x = 1.7 mm from the electrodes.The resistance biases δ R (%) (2) for the sensor cells are δ R,l = -0.030%and δ R,sh = -0.060%,respectively.The relative expanded uncertainty of the cell constants of such a sensor is U Kcell,l = 0,04% (k=2) for a long cell and U Kcell,sh = 0,05% (k=2) for a short one.Such metrological characteristics meet the requirements for primary conductivity cells.

Conclusions
The presence of holes for filling the cell significantly distorts the current density distribution J (A•m −2 ) inside the cell, increasing the effective cross-sectional area A (m 2 ) and reducing the resistance of the cell.This is especially noticeable with an increase in the diameter of the holes (a decrease in the D/d ratio) or a decrease in the L/D ratio.Nevertheless, since the form of the liquid column does not change, this effect is constant, and therefore it can be corrected.By estimating the bias δ R (%) for a cell with specific geometric parameters L (m), D (m), d (m), and x (m), it is possible to ensure the calculation of the constant of such a cell and its EC measurement result, which is impossible for secondary Jones-type cells.
The magnitude of the resistance bias δ R (%) is of the same order of magnitude as the technological errors in processing the inner surface of the cell tubes.Bias correction makes it possible to reduce the error in calculating the cell constant by 2-3 times.Based on their metrological characteristics, cells with short and long tubes can be used as primary cells in the national measurement standards of the EC unit.

Figure 1 .
Figure 1.A physical model of an idealized conductivity cell with a uniform current density distribution.

Figure 2 .
Figure 2. A physical model of a cell with inlet and outlet holes for filling (the current density lines are distorted due to the presence of holes).
(a)), and when D/d 5, it is only h d (figure 4(b)).A subsequent increase in the size h (mm) does not affect the change in resistance but only increases the computation time.

Figure 3 .
Figure 3.A typical mesh of a liquid column 3D model near the fill/drain point.

Figure 4 .
Figure 4. Dependencies of the resistance bias δ R (%) on the parameter h (mm).

Figure 5 .
Figure 5. Dependencies of the resistance bias δ R (%) on various cell parameters.

Figure 6 .
Figure 6.Current density distribution J (A•m −2 ) in the center slice of the liquid column model near the fill/drain point (D = 10 mm, d = 2 mm, x = 2 mm).