On the Calculation of Conductometric Concentration Meters and the Estimation of Their Errors

A. T. Tarabanovskii

(Presented by Prof. Dr. I. N. Butakov)

In the course of performing contract work in the laboratory of thermochemical measurements, a conductometric concentration meter was developed for a caustic-soda regeneration solution*) with measurement limits of 0.5–6 percent by weight.

As is known [1], the principle of operation of a conductometric concentration meter is based on determining concentration by measuring the electrical conductivity of a solution flowing, in our case, between two cylindrical stainless-steel electrodes.

The measuring circuit of the concentration meter is an unbalanced alternating-current bridge. The supply voltage of the circuit is kept constant by means of a voltage stabilizer of the IP-6 type, used in GEUK-21 electric gas analyzers. An ordinary pyrometric millivoltmeter is connected into the measuring diagonal through a rectifier bridge assembled from DGTs-type semiconductor diodes.

A theoretical calculation of the instrument scale according to the theory of electrical conductivity in solutions [1] cannot be carried out at such high concentrations. Calibration of the instrument was performed using standard caustic-soda solutions.

During adjustment of the instrument under industrial conditions, it was found that the electrode-carbon electrodes recommended in the literature give a considerable instability of readings, expressed in periodically occurring large-amplitude oscillations of the pointer of the secondary instrument. In connection with this, we installed carefully polished stainless-steel electrodes, after which the indicated phenomena disappeared completely.

The sensor of the concentration meter is of the flow-through type; the analyzed caustic-soda solution, taken from the pipeline by means of a pulse line, flows through it continuously. Structurally, the sensor consists of two sleeves made of organic glass, into which the electrodes are screwed on threads. The distance between the axes of the electrodes is 45 cm. The electrode sleeves are connected by means of a glass tube. All transitions are made with rubber tubes. The developed flow-

*) Postgraduate student A. I. Gnatyuk and assistant T. A. Gnatyuk took part in the development, calibration, and adjustment of the concentration meter.

...of such a sensor is exceptionally simple and can be made by the forces of production laboratories.

The sensor makes it possible to use it for measuring concentrations in solutions of other substances that are used in the technological processes of chemical water treatment at thermal power stations:

1) in a solution of table salt for regeneration of cation-exchange filters, concentration 4–8%;

2) in a regeneration solution of sulfuric acid, concentration 0–3%;

3) in a solution of aluminum sulfate, concentration 0–8%. Since only a sodium hydroxide concentration meter was experimentally investigated, it is of interest to estimate by calculation the possibility of using the sensor to measure the concentration of other solutions. The cell constant of the sodium hydroxide concentration-meter sensor, calculated from the geometrical dimensions of the electrodes and the distance between them, is equal to [1].

$$ K=\frac{l}{s}=\frac{45}{0.9}=50\ \text{cm}^{-1}, \tag{1} $$

where \(l\) is the distance between the electrodes (\(l=45\ \text{cm}\)); \(s\) is the active area of the electrodes (\(s=0.9\ \text{cm}^2\)).

Experimentally, the dependence of the resistance of the sodium hydroxide solution in the sensor on concentration was established, as presented in Table 1.

Table 1

Dependence of the resistance of the sodium hydroxide sensor on concentration

From the experimental data, the cell constant of the sensor is calculated by the formula [1]

$$ K=R\cdot\chi,\ \text{cm}^{-1}. \tag{2} $$

Then, using the data of Table 1, we obtain

$$ K=810\cdot 4.65\cdot 10^{-2}=37.7\ \text{cm}^{-1}; $$

$$ K=500\cdot 8.87\cdot 10^{-2}=44.5\ \text{cm}^{-1}; $$

$$ K=275\cdot 16.28\cdot 10^{-2}=44.7\ \text{cm}^{-1}; $$

$$ K=200\cdot 22.42\cdot 10^{-2}=45.0\ \text{cm}^{-1}. $$

The mean value of the sensor cell constant is

$$ K_{\text{avg}}=\frac{37.7+44.5+44.7+45.0}{4}=42.7\ \text{cm}^{-1}. $$

The discrepancy between the theoretical and experimental value of the cell constant is

$$ \frac{(K-K_{\text{avg}})\cdot 100}{K_{\text{avg}}} = \frac{50.0-42.7}{42.7}\cdot 100 = \frac{7.3\cdot 100}{42.7} = 17\%. $$

Such a discrepancy is the reason why theoretical calculation of the concentration-meter scale is impossible. In what follows, for the calculation the experimental value of the cell constant is adopted: \(K=42.7\ \text{cm}^{-1}\).

We calculate the resistances of the sensors for other solutions under the condition that the geometrical dimensions of the electrode block of the sensors remain unchanged. The calculation is carried out by the formula

Table 2

Calculated dependence of the resistance of the aluminum sulfate concentration sensor on concentration

\[ R=\frac{K}{\chi}, \tag{2′} \]

where \(\chi\) is the specific electrical conductivity of the solutions, taken from tabular data [3]. Table 2 gives the calculation for the aluminum sulfate concentration sensor.

As is seen from Table 2, the values of the sensor resistances are so large that creating a measuring bridge circuit with practically acceptable values of the arm resistances is impossible. Then we reduce the distance between the electrodes in the ratio

\[ r=\frac{R^{1\%}_{\mathrm{NaOH}}}{R^{1\%}_{\mathrm{Al}_2(\mathrm{SO}_4)_3}} =\frac{800}{12200}=0{,}0663, \]

that is, the distance between the electrodes for the aluminum sulfate concentratometer should be

\[ l=r\cdot 45=0{,}0663\cdot 45=2{,}98\ \mathrm{cm}. \]

The cell constant of the sensor is then equal to

\[ K=\frac{l}{s}=\frac{2{,}98}{0{,}9}=3{,}31\ \mathrm{cm}^{-1}. \]

The dependence of the sensor resistance \(R\) on concentration \(c\) for this value of the cell constant is calculated in Table 3.

Comparison of the data of Tables 3 and 1 shows that the values of the sensor resistances as a function of concentration and the total change in resistance over the whole scale are of the same order as for the sodium hydroxide concentratometer sensor.

This makes it possible to use the measuring circuit of the sodium hydroxide concentratometer for measuring the concentration of aluminum sulfate with minimal modifications. It is only necessary to refine the value of the resistance of the shunt and of the resistance thermometer of the temperature-compensation circuit because of the different value of the temperature coefficient

Table 3

Calculated dependence of the resistance of the aluminum sulfate concentration sensor on concentration at the value of the cell constant

\[ K=3{,}37\ \mathrm{cm}^{-1}. \]

sulfate aluminum in comparison with caustic soda. The design parameters of sensors of concentration meters for sulfuric acid and common salt, calculated in an analogous manner, are given in Table 4.

Table 4

Design data for concentration sensors for regeneration solutions

The calculated data of Table 4 make it possible to draw the following conclusions.

1. The simple operation of changing the distance between the electrodes makes it possible to use the developed sensor design for measuring the concentration of various substances:

a) caustic soda within the range 1—6 wt. %;

b) coagulant within the range 1—8 wt. %;

c) sulfuric acid within the range 1—5 wt. %;

d) common salt within the range 1—10 wt. %.

1. The design dimensions of the electrode cells for all substances remain unchanged.

2. The parameters of the bridge measuring circuit of all concentration meters are close to those for the tested caustic-soda concentration meter. The presence in the measuring circuit of the latter of a variable resistance of 1000 ohm fully permits balancing the circuit of any concentration meter, provided that the initial division of the scales begins with a concentration of 1 wt. %.

3. The temperature-compensation circuit must be subjected to a separate calculation because of the differences in the temperature coefficients of the solutions.

A direct experimental verification of the calculation procedure presented has not yet been carried out. However, we intend to do this; at present the manufacture of a sulfuric-acid sensor, calculated according to the method we recommend, is being completed.

In designing conductometric concentration meters it is highly desirable to estimate, by calculation, the concentration errors.

To our knowledge, only one work [2] is devoted to this question; the calculation method in it is incomplete and contains direct errors, as our calculations have shown.

Having tabulated data on the dependence of the specific electrical conductivity of solutions [3] and using Lagrange’s interpolation formula [4], one can derive, for particular solutions, equations for the change in the specific electrical conductivity of solutions as a function of concentration.

The numerical values of the coefficients of these equations for particular solutions are computed with the required accuracy by the Lagrange formula:

\[ \varphi(x)=f(x_0)\cdot \frac{(x-x_1)\cdots(x-x_n)} {(x_0-x_1)\cdots(x_0-x_n)} +f(x_1)\times \]

\[ \times \frac{(x-x_0)(x-x_2)\cdots(x-x_n)} {(x_1-x_0)(x_1-x_2)\cdots(x_1-x_n)} +\cdots+f(x_n)\times \]

\[ \times \frac{(x-x_0)(x-x_1)\cdots(x-x_{n-1})} {(x_n-x_0)(x_n-x_1)\cdots(x_n-x_{n-1})}. \tag{3} \]

For the solutions of interest to us, the following equations have been obtained for the change in the specific electrical conductivity of the solutions as a function of concentration:

1) for an aluminum sulfate solution

\[ \chi\cdot 10^2=0.23280+0.08550\cdot c+0.03423\cdot c^2-0.00261\cdot c^3; \tag{4} \]

2) for a sodium hydroxide solution

\[ \chi\cdot 10^2=0.06592+4.77130\cdot c-0.18981\cdot c^2+0.00259\cdot c^3; \tag{5} \]

3) for a sulfuric acid solution

\[ \chi\cdot 10^2=-0.21240+4.38934\cdot c-0.21244\cdot c^2+0.03541\cdot c^3; \tag{6} \]

4) for a sodium chloride solution

\[ \chi\cdot 10^2=-0.5104+1.34569\cdot c+0.00774\cdot c^2-0.00207\cdot c^3, \tag{7} \]

where \(\chi\) is the specific electrical conductivity, \(\Omega^{-1}\cdot\mathrm{cm}^{-1}\);
\(c\) is the weight concentration of substances in the solution, %.

The first derivatives of these equations with respect to concentration \(c\) give the change in electrical conductivity with the change in the concentration of the substance in the solution:

1) for an aluminum sulfate solution

\[ \frac{d\chi}{dc}\cdot 10^2=0.08550+0.06846\cdot c-0.00783\cdot c^2; \tag{4′} \]

2) for a sodium hydroxide solution

\[ \frac{d\chi}{dc}\cdot 10^2=4.77130-0.37962\cdot c+0.00777\cdot c^2; \tag{5′} \]

3) for a sulfuric acid solution

\[ \frac{d\chi}{dc}\cdot 10^2=4.38934-0.42488\cdot c+0.10623\cdot c^2; \tag{6′} \]

4) for a sodium chloride solution

\[ \frac{d\chi}{dc}\cdot 10^2=1.34569+0.01548\cdot c-0.00621\cdot c^2. \tag{7′} \]

The error calculation is carried out as follows. From formulas (4′)—(7′), for each of the solutions, the changes in specific electrical conductivity with the change in concentration,

\[ \frac{d\chi}{dc}\cdot 10^2, \]

are calculated.

Then the relative change in electrical conductivity is found from the formula

\[ \delta_{\chi}=\frac{\dfrac{d\chi}{dc}}{\chi}\cdot 100,\% \tag{8} \]

where \(\chi\) is the specific electrical conductivity in \(\Omega^{-1}\cdot\text{cm}^{-1}\), corresponding to the given value of the weight concentration \(c\).

According to [2], the sensitivity of conductometric methods using technical-class instruments does not exceed \(S=\pm 3\%\). Then the absolute error of concentration measurement can be calculated by the formula

\[ \Delta c=\pm \frac{S}{\delta_{\chi}},\ \text{wt. percent,} \tag{9} \]

and the relative error by the formula

\[ \delta_c=\pm \frac{\Delta c}{c}\cdot 100,\%. \tag{10} \]

Calculations of the errors for different concentrations of the aluminum sulfate concentration meter are given in Table 5.

Table 5

Calculation of errors of the aluminum sulfate concentration meter,
scale 0–8 wt. percent, calculation equation

\[ \frac{d\chi}{dc}\cdot 10^2=0{,}08550+0{,}06846\cdot c-0{,}00783\cdot c^2. \]

It is seen from the table that the scale of the aluminum sulfate concentration meter must be limited to the measurement range of 2–8 weight percent, since at a concentration of 10 weight percent we are already on the descending part of the dependence of specific electrical conductivity on concentration. In this case a very large measurement error is obtained. When measuring concentrations below 2 weight percent, the measurement error is also large.

Within the range of 2–8 weight percent, the relative error will be no more than \(\pm 4{,}4\%\). If the relative error due to incomplete temperature compensation does not exceed \(\pm 0{,}6\%\), then the total error in measuring the concentration of aluminum sulfate will be no more than \(\pm 5\%\). Calculation of errors for concentration meters of other solutions is given in Table 6.

The data in the table show that the measurement errors will be greatest for the caustic soda concentratometer \((\delta_c = \pm 4,05\%)\), and smallest for the sulfuric acid concentratometer \((\delta_c = \pm 3,03\%)\).

Table 6

Calculation of errors of concentratometers for regeneration solutions

If it is assumed that the relative error due to incomplete temperature compensation will be of the same order as for the caustic soda concentratometer we investigated \((\pm 1\%)\), then the calculated error of the concentratometers for all the solutions considered will be on the order of \(\pm 5\%\) of the measured concentration value, provided that the composition of the water used to dilute the solutions is sufficiently constant.

Table 7

Experimental determinations of the errors of the caustic soda concentratometer

Theoretical and experimental studies by S. N. Elenin [5] for automatic coagulant feeders showed that when salt co-

the dilution-water content was rather high and the coagulant doses comparatively small, the readings of the coagulant concentratometer were random in character and did not reflect the true course of the coagulation process.

For a caustic-soda concentratometer under production conditions, the measurement errors were determined experimentally. To this end, for a given concentratometer reading, a sample of the solution was taken and its concentration was determined by chemical analysis; this concentration was taken as the true value.

The data from one of the experiments are given in Table 7. Comparison of the data in Table 7 with the calculated values of Table 6 for the caustic-soda concentratometer makes it possible to conclude that the experimental and calculated error values are of the same order.

REFERENCES

1. A. A. Mostofin. Electrical salinometers for monitoring the quality of steam and boiler water. Gosenergoizdat, 1951.

2. Edited by V. M. Glushkov and L. A. Kulsky. Intensification and automation of processes for regulating water quality. Institute of Technical Information, Kiev, 1962.

3. Chemist’s Handbook, Vol. 3. Goskhimizdat, 1952.

4. I. S. Berezin and N. P. Zhidkov. Methods of Computation, Vol. II. Fizmatgiz, 1960.

5. Academy of Municipal Economy named after K. D. Pamfilov. Mechanization and automation in municipal services. Collection of scientific papers, issue XV, ONTI, Moscow–Leningrad, 1962.