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

Transcription

1 ACTA IMEKO ISSN: 87X June 5, Volue 4, Nuber, 8 Geoetric part of uncertainties in the calculation constant of the priary four electrode conductivity cell Alexander Mikhal, Zygunt Warsza Institute of electrodynaics National Acadey of Science, Av. Pereogy 56, 368 Kiev, Ukraine Industrial Research Institute of Autoation and Measureents PIAP, Al. Jerozoliskie, 486 Warszawa Poland ABSTRACT The paper presents the construction of a priary four electrode conductivity cell with calculated constant for the Ukrainian priary standard of electrolytic conductivity (EC). The equations for calculating the cell constant and the error budget for calculating uncertainty are presented. The coponents of the budget are: errors due to the non unifority of the force lines of the electric field; errors due to the accuracy of easureent standards and easuring instruents for deterining length and diaeter of the tube; and errors due to anufacturing techniques of tubes and their asseblage. The article considers in detail the errors due to the non ideal profile of the central part of the tube. Two ethods to reduce the standard deviation are given: the ethod of linear interpolation for copensation of the concave for which occurs along the axis of the tube and the ethod of equivalent triangles to copensate for deviations fro a circle that occur across the axis of the tube. Section: RESEARCH PAPER Keywords: conductivity; priary cell; geoetric errors; standard deviation Citation: Alexander Mikhal, Zygunt WARSZA, Geoetric part of uncertainties in the calculation constant of the priary four electrode conductivity cell, Acta IMEKO, Acta IMEKO, vol. 4, no., article 4, June 5, identifier: IMEKO ACTA 4 (5) 4 Editor: Paolo Carbone, University of Perugia, Italy Received April 3, 4; In final for July 3, 4; Published June 5 Copyright: 5 IMEKO. This is an open access article distributed under the ters of the Creative Coons Attribution 3. License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original author and source are credited Funding: This work was supported by the Presidiu of the National Acadey of Sciences of Ukraine. Corresponding author: Alexander Mikhal, e ail: a_ikhal@ukr.net. INTROUCTION In the recent years, the leading econoically developed countries have established national standards of electrolytic conductivity (ЕС), the basis of which uses the absolute (direct) ethod of reproduction of a physical quantity []. World practice knows a lot of ways to ipleent this ethod; however, the principle of its operation is alost the sae [-4]. The absolute (direct) ethod is based on the easureent of the liquid colun resistance and on the calculation of the EC by the known length of the colun cross-sectional area. The basic coponents of the EC standard of Ukraine are a four-electrode cell with the calculated constant, a special conductivity AC bridge, a therostat for teperature control at 5ºС and a precision digital teperature eter. The EC is deterined in accordance with the expression: k gk( t5) () where g is the conductivity of the liquid colun obtained as a easureent result of the conductivity bridge; K is the cell constant obtained as a calculation result based on the actual geoetric diensions of the cell; α is the teperature coefficient for conductivity of potassiu chloride solution; Δt 5 is the teperature deviation fro 5ºС obtained as a easureent result using the digital theroeter. The uncertainty (here referred to as standard uncertainty type B only) of the national standard will be influenced by any factors. Most significant errors are the ones in calculating the priary cell constant. Error coponents being deterined by the geoetric paraeters of the liquid colun have the greatest influence. Let us consider these error coponents on the exaple 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 eleent, a sketch of which is shown in Figure. The sensor eleent is a tube with internal diaeter. The inside of the tube is filled with an electrolyte solution. Typically, this is a solution of potassiu chloride. The tube is used to fix the geoetry of the liquid conductor and consists of three parts. The central part of the tube has length L, two side ACTA IMEKO June 5 Volue 4 Nuber 8

2 Figure. Construction of the priary conductivity cell. portions have equal lengths l. The ends of the central portion of tube 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 etallic fil 5 which perfors the function of current electrodes. iscs 3 have central holes 6 with diaeter d which serve for filling with the liquid. The inner disc surface has the for of a cone with angle α. Such configuration is intended to facilitate the reoval of air bubbles when filling the cell with the liquid. The tube and the discs are ade of quartz glass which has good insulating properties, teporal stability and a iniu coefficient of theral expansion. Platinu is used as the etal for the electrodes as it has a iniu polarization effect for ost electrolytes. The cell in its four points a, b, c and d is connected to the bridge which ensures the easureent of conductivity of the liquid colun. The cell constant is deterined by calculating the ratio of the length of the tube to the cross-section area. However, this definition is true for an idealized object of easureents with unifor distribution of current flow lines. istortion of such lines will be due to: the presence of holes for solution filling; the for of current electrodes and the presence of potential electrodes; and the non-ideal profile of the inner quartz tube, Figure. Therefore, the calculation of the constant will show errors.. BUGET OF UNCERTAINTY CELLS The error budget for the constant calculation of the cell, shown in Figure, can be written as [7]: uk F[ St, Cal, Geo( Tec, PE )] () where: δ St is an error due to the accuracy of easureent standards and easuring instruents that deterine the length and diaeter of the tube; δ Cal is an error due to the deviation of the calculation odel for the cell constant in real conditions relative to the idealized odel; δ Geo is an error in assessent of geoetrical diensions. We do not focus our attention on the ethod of processing errors (function F), which is used in calculating the uncertainty. This is regulated by international guidelines [8]. Each error in Eq. () has in its turn a nuber of coponents. Miniizing of the error δ St is liited by the level of etrological assurance for easureents of tube length and diaeter. It is defined by the etrological characteristics of the standards and instruents for length easureent. Error δ Cal has two origins and two coponents accordingly: an error due to the alternating current easureent and an error due to the discontinuity of an electric field in the cell because of its finite diensions and design features. The geoetric error δ Geo also has two coponents: δ Tec is an error due to anufacturing technology for the tube sections and their asseblage; δ PE is an error due to the presence of the potential electrodes. The latter coponent 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 asseblage. This coponent is related to the calculation of the electric field inside the cell. It will be considered in other papers. In this paper, we exaine the coponent of error δ Tec. This error is due to the deviation of the actual profile of the inner surface of tube (Figure ) fro 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 entioned that the cost of tube production fro a onolithic quartz crystal is extreely high. As a rule, tubes are anufactured fro work pieces (prefor) which undergo precision achining. If precision achining of the inner surface is too deep, the echanical resistance of the tube will reduce significantly. Tubes of less than in thickness will crack (fracture) under elastic forces (adhesive polyerization, teperature differences). Therefore, grinding of the work piece inner profile should be of iniu 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 anufacturing 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 achining of tubes, we can observe deviations fro a circle in cross-section along the entire profile. The second reason of a non-ideal profile ay be the precession of the grinding tool. uring processing, quality control of the tube is practically ipossible. After final grinding, the tube profile ay differ fro the ideal rectangle. 3. ERROR UE TO MANUFACTURING To deterine the actual profile of the tube, its diaeter and length should be easured according to the following algorith. We perfored р easureents of the tube length L in different directions. These directions were distributed uniforly around the circuference. Conventionally, uniforly along the length, the tube is divided into sections. To define the diaeter, n easureents are ade in the cross-section of each part of the tube in different directions. As a result, we obtain n easureents of the tube diaeter and р values of the tube length. The constant can be deterined fro the easureent results through averageing the values of diaeter av and tube length L av. 4L K av (3) av In odern technologies for processing quartz glass it is uch easier to anufacture a tube with stable length value than a tube with stable internal diaeter. Fro the experiental data we observe distortions of the internal profile of two types. The ACTA IMEKO June 5 Volue 4 Nuber 9

3 first one is a deviation of the tube profile fro a rectangle along the longitudinal section due to precession of the grinding tool. The second one is a deviation of the tube profile fro a circle in cross-section due to the presence of wedge-like cracks on the inner surface of work pieces. The geoetric diensions of the actually anufactured tube can be easured with uch higher accuracy than the error of profile. Paraeter easureents are perfored with a device with an LSB of. μ. Rando coponents of the device are negligible. The actual tube profile deviates fro the ideal rectangle by ore than LSB of this device. Therefore, the following expression can be taken as a etrological odel for the constant calculation: K K ( ), (4) K K L where: K is the true value of the constant which is deterined by the actual profile of the tube inner surface; δ K is a systeatic relative error of the constant calculation; K, and L are standard deviations of the ean cell constant, diaeter and length of the tube, respectively. Therefore, an error due to anufacturing δ Tec has two coponents: a systeatic δ K and a rando K. Hereinafter, under such deviation we understand the standard deviation of the ean (SM) of the diaeter. The average diaeter and the SM of the diaeter in each i-th section of the tube are deterined fro the expressions: av i n iav, ij i j (5a) n n ( ) (5b) nn ( ) i ij iav j where: i,av is the ean diaeter in the i-th section of the tube and ij is the diaeter in the j-th direction and in the i-th section of the tube. The SM value fro the section (cut) can be expressed as: i (5с) ( ) i 3.. Error in the longitudinal section The easuring results for the ean diaeter of one of the sections of the tubes for n=8 and = are shown in Figure. These data indicate that the average diaeters in sections and 6 differ by alost μ. In general, the profile of the tube internal section can be expressed through an arbitrary function (x). The easuring results for average diaeters along the length of the tube (Figure ) show that this dependence has i,av,75,7,65,6,55, Figure 3. Profile along the axis of the tube. section nuber, Figure. Influence of linear interpolation on the level of SM (σ i ). clearly a deterinistic character. The discrete nature of the data allows us to use linear interpolation for the function (x). The results are shown in the following forula: L X i 4 dx 4 dx K x ( ) i ( ax b) (6) where: a and b are linear interpolation coefficients; X i the length of the region between the i-th and i + section. The polynoial coefficients are expressed as: i i a (6a) X i b i (6b) After siple transforations, the expression for calculating the constant with the proposed correction takes the for: X K (7) 4 i i iav, i, av 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 SM of diaeter easuring results with and without the deterinistic coponent by using linear interpolation (Figure 3). As it can be seen, the SM of the diaeter easureent results, taking into account the deterinistic coponent, is close to alost one value and has a very sall scatter of results copared with the case where the deterinistic coponent is ignored and the average diaeter value is calculated according to Eq. (3). As a result of this correction, the change of the SM along the cross-section is reduced by to 5 ties. The SM of the diaeter with correction, calculated according to Eq. (5с), is less than.4. The sae paraeter without correction is 3 ties larger. When calculating the rando coponent of an error (Eq. 4), we obtain an error reduction by two. It should be noted that such correction ethod shifts the average value of the diaeter. Thus, the constants calculated by Eqs. (3) and (7) for tube in Figure differ by.7 %. This is the rate by which systeatic relative errors δ K differ (4) when calculating the cell constant with and without correction. 3.. Error in the cross section Let us consider another tube for which the values of the ean radius in each of the sections are grouped along a virtually horizontal line. However, in each individual section the surface profile differs fro a circle. Exaple of the function for the deviation fro the ean iav (figure represented by circle.5 3,av = ) in one of the sections with = 3 is shown in Figure 4. In all ten sections with = to we observe triangular run outs along the lines 3 to and 6 to 4 (Figure 4). Such ACTA IMEKO June 5 Volue 4 Nuber

4 Figure 4. Real profile (blue) across the axis of the tube for section =3 and.5 i,av =4.569 (red). character of profile distortions allows using a ethod of the equivalent triangles. This ethod involves the assessent of the effective area S ief of the tube section and subsequent diaeter corrections. The algorith for calculation is to replace the diaeter values in section 3 and 6 with the values of the ean diaeter av, then to use standard forula for calculating the basic ean diaeter bias and the basic area of the tube section S bias : S bias (8) ( bias ) 4 Next, we calculate the correction value in each section separately. It is represented as areas of triangles: Si ch i i (9) с i is the base of the assued triangle in direction 3, or 6,4 (Figure 4); h i is the height of the assued triangle in the direction 3, or 6,4 (Figure 4). We take into account the influence of the deterinistic coponent by foring the effective area of each section which is expressed as: S S S S ch () ief bias i bias i i i3,6 i3,6 According to Eq. () we calculate the corrected icor which is put in Eqs. (,3) instead of iav : 4S () icor ief Figure 6. The cells for the priary standard. Eqs. (5) and () differ by.5 %. Just as in the previous case, this value represents the difference of systeatic errors (3, 4) for the cell constant calculation. 4. CONCLUSIONS The above ethods were used to calculate the corrections to the cell constant. For the priary ЕС standard of Ukraine, several cell designs were ade. The priary cell is shown in Figure 6. Correctness, sufficiency and adequacy of the all selected odels for correction constant cells is confired by international coparisons P, P47, K36, which involved priary standard of Ukraine (laboratory UkrCSM). 4 laboratories of the leading NMI of the world participated in international coparisons CCQM-K36: USA (NIST), Gerany (PTB), Israel (INPL), Slovakia (SMU), enark (FM), and others. To all participants saples of potassiu chloride solutions with a noinal electrolytic conductivity.5 S/ and 5 S/ were sent. Each laboratory using equation () and own etrological features establishes the exact value EC k lab and uncertainty u(k lab ) conductivity saples. The rules of Key Coparison specify that a Key Coparison Reference Value ust be derived against which the participants results are copared, and a egree of Equivalence of each laboratory ust be inferred. In [6] the odel of Reference Value k ref was presented. The egree of Equivalence is given as: The difference between SM values for the easureent result of diaeter iav before correction and SM values for the ean diaeter icor after correction is shown in Figure 5: As can be seen in this figure, the SM with the correction of the deterinistic coponent is.5 to 3 ties less than without correction. The use of an algorith of effective areas shifts the ean diaeter value. The constants calculated by eviation,,6 iav,4, icor section nuber Figure 5. SM (σ i ) without correction iav and with correction icor. Figure 7. The results of international coparisons CCQM К36.а [6]. ACTA IMEKO June 5 Volue 4 Nuber

5 S/ klab k ref () The results of the egree of Equivalence with uncertainty on the international coparisons CCQM-K36 are shown in Figure 7. The proposed ethods of correction constants cells (ethod of linear interpolation and ethod of effective areas) allowed us to solve two probles [7]. Firstly, the value obtained by the UkrCSM k lab, practically coincides with the value k ref [6]. Secondly, by iniizing the rando coponent of the error (Eq. 4, 5c), a iniu value of uncertainty u(k lab ) was obtained. ACKNOWLEGEMENT 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 priary standards [5, 6], Head of the epartent of etrological provision of physical and cheical quantitative easureent (lab. UkrCSM, Kiev). REFERENCES [] Wu Y.C., A C ethod for the absolute deterination of conductivities of the priary standard KCl solution fro C to 5 C, J. Res. Nati. Inst. Stand. Tecnol., 994, 99, 3, [] Maґriaґssy M., Pratt K.W., Spitzer P., Major applications of electrocheical techniques at national etrology institutes, Metrologia 46 (9) 99 3 [3] Shreiner R.H., Pratt K.W., Standard Reference Materials: Priary Standards and Standard Reference Materials for Electrolytic Conductivity, NIST Special Publication 6-4, 4 Ed. [4] Brinkann F. and etc., General paper: Priary ethods for the easureent of electrolytic conductivity, Accred Qual Assur. 3. 8: OI.7/s [5] Gavrilkin V.G. and etc., State priary standard unit of electrical conductivity of liquids, Ukrainian Metrology Journal, 6, 3, P (Ukr.) [6] Jensen H 6 Final Report of Key Coparison CCQM-K36. 5 August 6, available online at [7] Miкhal A.A., Warsza Z.L. Influence of geoetric Uncertainties on the Accuracy of calculated constant of the priary conductivity cell. th International Syposiu on Measureent and Quality Control. IMEKO, Poland, Cracow-Kielce, -3 Septeber 3, p.8 [8] Guide to the Expression of Uncertainty in Measureent, ISBN , st Ed., International Organization for Standardization, Geneva, Switzerland, 993. ACTA IMEKO June 5 Volue 4 Nuber