ACTA IMEKO 
ISSN: 2221‐870X 
June 2015, Volume 4, Number 2, 18‐22 

 

ACTA IMEKO | www.imeko.org  June 2015 | Volume 4 | Number 2 | 18 

Geometric part of uncertainties in the calculation constant of 
the primary four electrode conductivity cell 

Alexander Mikhal
1
, Zygmunt Warsza

2
 

1
 Institute of electrodynamics National Academy of Science, Av. Peremogy 56, 03680 Kiev, Ukraine  

2
 Industrial Research Institute of Automation and Measurements PIAP, Al. Jerozolimskie 202, 02‐486 Warszawa Poland 

 

 

Section: RESEARCH PAPER 

Keywords: conductivity; primary cell; geometric errors; standard deviation 

Citation: Alexander Mikhal, Zygmunt WARSZA, Geometric part of uncertainties in the calculation constant of the primary four electrode conductivity cell, 
Acta IMEKO, Acta IMEKO, vol. 4, no. 2, article 4, June 2015, identifier: IMEKO‐ACTA‐04 (2015)‐02‐04 

Editor: Paolo Carbone, University of Perugia, Italy 

Received April 30, 2014; In final form July 13, 2014; Published June  2015 

Copyright: © 2015 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited 

Funding: This work was supported by the Presidium of the National Academy of Sciences of Ukraine. 

Corresponding author: Alexander Mikhal, e‐mail: a_mikhal@ukr.net 

 

1. INTRODUCTION 

In the recent years, the leading economically developed 
countries have established national standards of electrolytic 
conductivity (ЕС), the basis of which uses the “absolute 
(direct)” method of reproduction of a physical quantity [1]. 
World practice knows a lot of ways to implement this method; 
however, the principle of its operation is almost the same [2-4]. 
The “absolute (direct)” method is based on the measurement of 
the liquid column resistance and on the calculation of the EC 
by the known length of the column cross-sectional area. 

The basic components of the EC standard of Ukraine are a 
four-electrode cell with the calculated constant, a special 
conductivity AC bridge, a thermostat for temperature control at 
25ºС and a precision digital temperature meter. The EC is 
determined in accordance with the expression: 

25(1 )k gK t    (1) 
where g is the conductivity of the liquid column obtained as 

a measurement result of the conductivity bridge; 

K is the cell constant obtained as a calculation result based 
on the actual geometric dimensions of the cell; 
α is the temperature coefficient for conductivity of 

potassium chloride solution; 
Δt25 is the temperature deviation from 25ºС obtained as a 

measurement result using the digital thermometer. 
The uncertainty (here referred to as standard uncertainty 

type B only) of the national standard  will be influenced by 
many factors. Most significant errors are the ones in calculating 
the primary cell constant. Error components being determined 
by the geometric parameters of the liquid column have the 
greatest influence. Let us consider these error components on 
the example of the four-electrode conductivity cell which is 
used in the EC standard of Ukraine [4]-[6]. The basis of the 
conductivity cell is the sensor element, a sketch of which is 
shown in Figure 1. 

The sensor element is a tube with internal diameter D. The 
inside of the tube is filled with an electrolyte solution. Typically, 
this is a solution of potassium chloride. The tube is used to fix 
the geometry of the liquid conductor and consists of three 
parts. The central part of the tube 1 has length L, two side 

ABSTRACT 
The paper presents the construction of a primary four electrode conductivity cell with calculated constant for the Ukrainian primary 
standard of electrolytic conductivity (EC). The equations for calculating the cell constant and the error budget for calculating uncer‐
tainty are presented. The components of the budget are: errors due to the non‐uniformity of the force lines of the electric field; errors 
due to the accuracy of measurement standards and measuring  instruments for determining  length and diameter of the tube; and 
errors due to manufacturing techniques of tubes and their assemblage. The article considers in detail the errors due to the non‐ideal 
profile of the central part of the tube. Two methods to reduce the standard deviation are given: the method of linear interpolation for 
compensation of the concave form which occurs along the axis of the tube and the method of equivalent triangles to compensate for 
deviations from a circle that occur across the axis of the tube.



 

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portions 2 have equal lengths l. The ends of the central portion 
of tube 1 are coated with circular potential electrodes 4. Their 
width corresponds to the tube wall thickness. Two discs 3 are 
fixed at the edges of the tube. The inner surface of the discs is 
coated with a metallic film 5 which performs the function of 
current electrodes. Discs 3 have central holes 6 with diameter d 
which serve for filling with the liquid. The inner disc surface 
has the form of a cone with angle α. Such configuration is 
intended to facilitate the removal of air bubbles when filling the 
cell with the liquid. The tube and the discs are made of quartz 
glass which has good insulating properties, temporal stability 
and a minimum coefficient of thermal expansion. Platinum is 
used as the metal for the electrodes as it has a minimum 
polarization effect for most electrolytes. The cell in its four 
points a, b, c and d is connected to the bridge which ensures the 
measurement of conductivity of the liquid column.  

The cell constant is determined by calculating the ratio of 
the length of the tube 1 to the cross-section area. However, this 
definition is true for an idealized object of measurements with 
uniform distribution of current flow lines. Distortion of such 
lines will be due to: the presence of holes for solution filling; 
the form of current electrodes and the presence of potential 
electrodes; and the non-ideal profile of the inner quartz tube 1, 
Figure 1. Therefore, the calculation of the constant will show 
errors. 

2. BUDGET OF UNCERTAINTY CELLS 

The error budget for the constant calculation of the cell, 
shown in Figure 1, can be written as [7]: 

[ , , ( , )]K St Cal Geom Tec PEu F       (2) 
where: δSt  is an error due to the accuracy of measurement 
standards and measuring instruments that determine the length 
and diameter of the tube; 

δCal  is an error due to the deviation of the calculation model for 
the cell constant in real conditions relative to the idealized 
model; 
δGeom  is an error in assessment of geometrical dimensions. 

We do not focus our attention on the method of processing 
errors (function F), which is used in calculating the uncertainty. 
This is regulated by international guidelines [8]. 

Each error in Eq. (2) has in its turn a number of 
components. 

Minimizing of the error δSt is limited by the level of 
metrological assurance for measurements of tube length and 

diameter. It is defined by the metrological characteristics of the 
standards and instruments for length measurement. 

Error δCal has two origins and two components accordingly: 
an error due to the alternating current measurement and an 
error due to the discontinuity of an electric field in the cell 
because of its finite dimensions and design features.  

The geometric error δGeom also has two components: 
δTec is an error due to manufacturing technology for the tube 

sections and their assemblage; 
δPE is an error due to the presence of the potential 

electrodes. 
The latter component of the error depends on the finite 

thickness of the potential electrodes and on changing position 
of a singular point of the potential electrodes upon assemblage. 
This component is related to the calculation of the electric field 
inside the cell. It will be considered in other papers. In this 
paper, we examine the component of error δTec. This error is 
due to the deviation of the actual profile of the inner surface of 
tube 1 (Figure 1) from the ideal profile of the tube. The latter 
one is presented as a rectangle along the longitudinal section 
and as a circle in cross-section.  

It should be mentioned that the cost of tube production 
from a monolithic quartz crystal is extremely high. As a rule, 
tubes are manufactured from work pieces (preform) which 
undergo precision machining. If precision machining of the 
inner surface is too deep, the mechanical resistance of the tube 
will reduce significantly. Tubes of less than 1 mm in thickness 
will crack (fracture) under elastic forces (adhesive 
polymerization, temperature differences). Therefore, grinding 
of the work piece inner profile should be of minimum depth. 
On the other side, the work piece inner surface can have 
wedge-like cracks which are in parallel to the axis of the work 
piece. These cracks are due to manufacturing techniques of the 
work piece production and depend on the quality of nozzles 
through which the work piece is pulled. Therefore, due to the 
lack of deep machining of tubes, we can observe deviations 
from a circle in cross-section along the entire profile. The 
second reason of a non-ideal profile may be the precession of 
the grinding tool. During processing, quality control of the tube 
is practically impossible. After final grinding, the tube profile 
may differ from the ideal rectangle. 

3. ERROR DUE TO MANUFACTURING 

To determine the actual profile of the tube, its diameter and 
length should be measured according to the following 
algorithm. We performed р measurements of the tube length L 
in different directions. These directions were distributed 
uniformly around the circumference. Conventionally, uniformly 
along the length, the tube is divided into m sections. To define 
the diameter, n measurements are made in the cross-section of 
each part of the tube in different directions. As a result, we 
obtain n×m measurements of the tube diameter and р values of 
the tube length. The constant can be determined from the 
measurement results through averageing the values of diameter 
Dav and tube length Lav. 

2

4 av

av

L
K

D
  (3) 

In modern technologies for processing quartz glass it is 
much easier to manufacture a tube with stable length value than 
a tube with stable internal diameter. From the experimental data 
we observe distortions of the internal profile of two types. The 

 
Figure 1. Construction of the primary conductivity cell.  



 

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first one is a deviation of the tube profile from a rectangle along 
the longitudinal section due to precession of the grinding tool. 
The second one is a deviation of the tube profile from a circle 
in cross-section due to the presence of wedge-like cracks on the 
inner surface of work pieces. 

The geometric dimensions of the actually manufactured tube 
can be measured with much higher accuracy than the error of 
profile. Parameter measurements are performed with a device 
with an LSB of 0.1 μm. Random components of the device are 
negligible. The actual tube profile deviates from the ideal 
rectangle by more than 10 LSB of this device. Therefore, the 
following expression can be taken as a metrological model for 
the constant calculation: 

2 2
0 (1 ), 2K K D LK K                  (4) 

where: K0  is the true value of the constant which is determined 
by the actual profile of the tube inner surface; 
δK is a systematic relative error of the constant calculation; 
K, D and L are standard deviations of the mean cell constant, 
diameter and length of the tube, respectively. 

Therefore, an error due to manufacturing δTec has two 
components: a systematic δK and a random K. 

Hereinafter, under such deviation we understand the 
standard deviation of the mean (SDM) of the diameter. The 
average diameter and the SDM of the diameter in each i-th 
section of the tube are determined from the expressions: 

,
1 11

m nm

iji av
i ji

av

DD
D

m m n
  



 
 (5a) 

2

1

1
( )

( 1)

n

iD ij iav
j

D D
n n




 

  (5b) 

where: Di,av is the mean diameter in the i-th section of the tube 
and Dij is the diameter in the j-th direction and in the i-th 
section of the tube. The SDM value from the m section (cut) 
can be expressed as: 

2

1

1

( 1)

m

D iD
im m

 




  (5с) 

3.1. Error in the longitudinal section 

The measuring results for the mean diameter of one of the 
sections of the tubes for n=8 and m=10 are shown in Figure 2. 

These data indicate that the average diameters in sections 1 
and 6 differ by almost 20 μm. In general, the profile of the tube 
internal section can be expressed through an arbitrary function 
D(x). The measuring results for average diameters along the 
length of the tube (Figure 2) show that this dependence has 

clearly a deterministic character. The discrete nature of the data 
allows us to use linear interpolation for the function D(x). The 
results are shown in the following formula: 

2 2
00 0

4 4

( ) ( )

iXL m

i

dx dx
K

D x ax b





 
  

   (6) 

where: a and b are linear interpolation coefficients; 
Xi  the length of the region between the i-th and i +1 section. 

The polynomial coefficients are expressed as: 

1i i

i

D D
a

X
 


 (6a) 

ib D  (6b) 

After simple transformations, the expression for calculating 
the constant with the proposed correction takes the form: 

1 , 1,

4 m i
i i av i av

X
K

D D  


   (7) 

We can use equation 3 and 5, but then we get an increase in 
the standard deviation. Let us show in one figure the graphs for 
the SDM of diameter measuring results with and without the 
deterministic component by using linear interpolation (Figure 
3). As it can be seen, the SDM of the diameter measurement 
results, taking into account the deterministic component, is 
close to almost one value and has a very small scatter of results 
compared with the case where the deterministic component is 
ignored and the average diameter value is calculated according 
to Eq. (3). As a result of this correction, the change of the SDM 
along the cross-section is reduced by 10 to 15 times. The SDM 
of the diameter with correction, calculated according to Eq. 
(5с), is less than 0.0004 mm. The same parameter without 
correction is 3 times larger. When calculating the random 
component of an error (Eq. 4), we obtain an error reduction by 
two. 

It should be noted that such correction method shifts the 
average value of the diameter. Thus, the constants calculated by 
Eqs. (3) and (7) for tube 1 in Figure 1 differ by 0.027 %. This is 
the rate by which systematic relative errors δK differ (4) when 
calculating the cell constant with and without correction. 

3.2. Error in the cross‐section 

Let us consider another tube for which the values of the 
mean radius in each of the sections are grouped along a virtually 
horizontal line. However, in each individual section the surface 
profile differs from a circle. Example of the function for the 
deviation from the mean Diav (figure represented by circle 
0.5D3,av = 4.569 mm) in one of the sections with m = 3 is shown 
in Figure 4. 

In all ten sections with m = 1 to 10 we observe triangular 
run outs along the lines 3 to 11 and 6 to 14 (Figure 4). Such 

 

Figure 2. Influence of linear interpolation on the level of SDM (σiD).  

10,15

10,155

10,16

10,165

10,17

10,175

1 2 3 4 5 6 7 8 9 10

D
i,

a
v
m
m

section number, m

Figure 3. Profile along the axis of the tube. 



 

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character of profile distortions allows using a method of the 
equivalent triangles. This method involves the assessment of 
the effective area Sief of the tube section and subsequent 
diameter corrections. The algorithm for calculation is to replace 
the diameter values in section 3 and 6 with the values of the 
mean diameter Dav, then to use standard formula for calculating 
the basic mean diameter Dbias and the basic area of the tube 
section Sbias: 

2( ) 4bias biasS D  (8) 

Next, we calculate the correction value in each section 
separately. It is represented as areas of triangles: 

2i i iS c h  (9) 

сi is the base of the assumed triangle in direction 3,11 or 
6,14 (Figure 4); hi is the height of the assumed triangle in the 
direction 3,11 or 6,14 (Figure 4). 

We take into account the influence of the deterministic 
component by forming the effective area of each section which 
is expressed as: 

3,6 3,6

2ief bias i bias i i
i i

S S S S ch
 

      (10) 

According to Eq. (11) we calculate the corrected Dicor which 
is put in Eqs. (2,3) instead of Diav: 

4icor iefD S   (11) 

The difference between SDM values for the measurement 
result of diameter Diav before correction and SDM values for 
the mean diameter Dicor after correction is shown in Figure 5: 

As can be seen in this figure, the SDM with the correction 
of the deterministic component is 2.5 to 3 times less than 
without correction. The use of an algorithm of effective areas 
shifts the mean diameter value. The constants calculated by 

Eqs. (5) and (11) differ by 0.015 %. Just as in the previous case, 
this value represents the difference of systematic errors (3, 4) 
for the cell constant calculation. 

4. CONCLUSIONS 

The above methods were used to calculate the corrections to 
the cell constant. For the primary ЕС standard of Ukraine, 
several cell designs were made. The primary cell is shown in 
Figure 6. Correctness, sufficiency and adequacy of the all 
selected models for correction constant cells is confirmed by 
international comparisons P22, P47, K36, which involved 
primary standard of Ukraine (laboratory UkrCSM). 

14 laboratories of the leading NMI of the world participated 
in international comparisons CCQM-K36: USA (NIST), 
Germany (PTB), Israel (INPL), Slovakia (SMU), Denmark 
(DFM), and others. To all participants samples of potassium 
chloride solutions with a nominal electrolytic conductivity 0.5 
S/m and 5 mS/m were sent. Each laboratory using equation (1) 
and own metrological features establishes the exact value EC 
klab and uncertainty u(klab) conductivity samples. 

The rules of Key Comparison specify that a Key 
Comparison Reference Value must be derived against which the 
participants’ results are compared, and a Degree of Equivalence 
of each laboratory must be inferred. In [6] the model of 
Reference Value kref  was presented. The Degree of 
Equivalence is given as: 

 

Figure 4. Real profile (blue) across the axis of the tube for section m=3 and 
0.5Di,av=4.569 mm (red).  

 

Figure 6. The cells for the primary standard.  

Figure 7. The results of international comparisons CCQM‐К36.а [6].  

0

0,0002

0,0004

0,0006

1 2 3 4 5 6 7 8 9 10D
e
v

ia
ti

o
n

, 
m
m

s ection number m

Diav

Dicor

 
Figure 5. SDM (σiD) without correction Diav and with correction Dicor.  



 

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 /  lab refD Sm m k k  (12) 
The results of the Degree of Equivalence with uncertainty 

on the international comparisons CCQM-K36 are shown in 
Figure 7. The proposed methods of correction constants cells 
(method of linear interpolation and method of effective areas) 
allowed us to solve two problems [7]. Firstly, the value obtained 
by the UkrCSM klab, practically coincides with the value kref [6]. 
Secondly, by minimizing the random component of the error 
(Eq. 4, 5c), a minimum value of uncertainty u(klab) was 
obtained.  

ACKNOWLEDGEMENT 

The authors express their gratitude to prof. M.N. Surdu, as 
supervisor of researches [4, 5] and V.G. Gavrilkin, as a scientist 
keeper of primary standards [5, 6], Head of the Department of 
metrological provision of physical and chemical quantitative 
measurement (lab. UkrCSM, Kiev). 

REFERENCES 

[1] Wu Y.C., A DC method for the absolute determination of con-
ductivities of the primary standard KCl solution from 0°C to 
50°C, J. Res. Nati. Inst. Stand. Tecnol., 1994, 99, №3, 241-246. 

[2] Maґriaґssy M., Pratt K.W., Spitzer P., Major applications of 
electrochemical techniques at national metrology institutes,  
Metrologia 46 (2009) 199–213 

[3] Shreiner R.H., Pratt K.W., Standard Reference Materials: Primary 
Standards and Standard Reference Materials for Electrolytic 
Conductivity, NIST Special Publication 260-142, 2004 Ed. 

[4] Brinkmann F. and etc., General paper: Primary methods for the 
measurement of electrolytic conductivity, Accred Qual Assur. – 
2003. – 8:346 – 353 DOI 10.1007/s00769-003-0645-5. 

[5] Gavrilkin V.G. and etc., State primary standard unit of electrical 
conductivity of liquids, Ukrainian Metrology Journal, 2006, №3, 
P. 47-51. (Ukr.) 

[6] Jensen H D 2006 Final Report of Key Comparison CCQM-K36. 
15 August 2006, available online at 

  http://kcdb.bipm.org/AppendixB/appbresults/ccqm-
k36/ccqm-k36_final_report.pdf 

[7] Miкhal A.A., Warsza Z.L. Influence of geometric Uncertainties 
on the Accuracy of calculated constant of the primary conductiv-
ity cell. 11th International Symposium on Measurement and 
Quality Control. IMEKO, Poland, Cracow-Kielce, 11-13 Sep-
tember 2013, p.8 

[8] Guide to the Expression of Uncertainty in Measurement, ISBN 
92-67-10188-9, 1st Ed., International Organization for Standard-
ization, Geneva, Switzerland, 1993.