Analysis of the Methodological Error of an IR Gas Analyzer

ANALYSIS OF THE METHODOLOGICAL ERROR OF AN IR GAS ANALYZER

M. S. Aleinikov, E. P. Koltsov, V. I. Lukovnikov,
K. L. Pesterev, V. S. Titov

(Presented by the scientific-technical seminar of the Department of Instruments and Devices of Automation Systems)

The methods of IR gas analyzers [1], based on determining concentration by logarithmizing the quotient from division (or difference) of signals proportional to the powers of IR fluxes, one of which passes only through the working cuvette with the medium under study, and the second through the cuvette with the medium under study and a reference cuvette with the limiting concentration of the measured gas, can be implemented with the aid of a gas analyzer whose block diagram is shown in Fig. 1.

Fig. 1. Block diagram of the gas analyzer.

With the aid of transmitting optical system 2, a collimated IR flux emitted by radiator 1 is formed; it passes through working cuvette 4, is modulated by modulator 5, and is divided by splitter 6 into two parts. One of the separated fluxes passes through a cuvette with vacuum (or with a gas that does not absorb the IR flux in the IR range), and the second through the reference cuvette. The fluxes focused by condensers 9 and 10 are received by photodetectors 11 and 12, whose output signals are fed to amplifiers 13 and 14, and then to synchronous detectors 15 and 16 synchronized with modulator 5. Output unit 17 carries out logarithmizing of the quotient from division (or difference) of the output signals of synchronous detectors 15 and 16.

According to Fig. 1, the expression for the static characteristic of the device for the case when the IR-radiation flux is collimated and equal-energy over the cross section, and a logarithmic divider is used as output unit 17, is easily obtained in the form

\[ U_1=K_{01}\ln\frac{K_{11}\cdot T_1}{K_{22}\cdot T_2}, \tag{1} \]

where \(U_1\) is the output voltage of the gas analyzer;
\(K_{01}\) is the transfer coefficient of the logarithmic divider;
\(K_{11}\) and \(K_{21}\) are the transfer coefficients of the first and second channels;
\(T_1\) and \(T_2\) are the equivalent integral transmittances of the optical devices corresponding to the first and second channels of the IR gas analyzer.

The quantities entering into expression (1) are determined from the relations:

\[ \begin{aligned} K_{11}&=\Phi_{\lambda\max 1}\cdot\eta_1\cdot\eta_2\cdot\eta_3\cdot\eta_4\cdot\eta_5\cdot\eta_{61}\cdot\eta_7\cdot\eta_9\cdot\eta_{11}\cdot S_{\lambda\max 11}\cdot K_{13}\cdot K_{15};\\ K_{22}&=\Phi_{\lambda\max 1}\cdot\eta_1\cdot\eta_2\cdot\eta_3\cdot\eta_4\cdot\eta_5\cdot\eta_{62}\cdot\eta_8\cdot\eta_{10}\cdot\eta_{12}\cdot S_{\lambda\max 12}\cdot K_{14}\cdot K_{16};\\ T_1&=\int_{\lambda_1}^{\lambda_2}\varphi(\lambda)\cdot T_2(\lambda)\cdot T_3(\lambda)\cdot T_4(\lambda)\cdot T_5(\lambda)\cdot T_{61}(\lambda)\cdot T_7(\lambda)\cdot T_9(\lambda)\cdot\sigma_{11}(\lambda)\cdot d\lambda;\\ T_2&=\int_{\lambda_1}^{\lambda_2}\varphi(\lambda)\cdot T_2(\lambda)\cdot T_3(\lambda)\cdot T_4(\lambda)\cdot T_5(\lambda)\cdot T_{62}(\lambda)\cdot T_8(\lambda)\cdot T_{10}(\lambda)\cdot\sigma_{12}(\lambda)d\lambda, \end{aligned} \tag{2} \]

where \(\eta_i\) is the loss coefficient corresponding to the \(i\)-th optical unit of the gas analyzer and determined by inaccurate matching of its entrance pupil with the exit pupil of the preceding unit;
\(K_i\) is the transfer coefficient of the \(i\)-th electronic unit;
\(S_{\lambda\max i}\) is the maximum monochromatic sensitivity of the IR-radiation receiver;
\(\Phi_{\lambda\max 1}\) is the maximum monochromatic flux of the emitter;
\(\varphi(\lambda)\), \(\sigma_i\lambda\), \(T_i(\lambda)\) are the relative spectral characteristics of the emitter, the receiver, and the transmittance of the \(i\)-th optical unit, respectively.

For the second case, when a logarithmic summator is used at the output of the gas analyzer, we shall have

\[ U_2=K_{02}\ln(K_{12}\cdot T_1-K_{22}\cdot T_2), \tag{3} \]

where \(K_{02}\) is the transfer coefficient of the logarithmic summator.

A comparison of methods for analyzing the accuracy of instruments [2] shows that the probabilistic method of analysis gives the estimate of the total error of an instrument closest to the actual one, in comparison with the root-mean-square method and the maximum–minimum method.

However, the simplicity of the maximum–minimum method and the optimistic nature of the results obtained (the calculated error is always overestimated in comparison with the actual one) lead to wide use of this method in engineering practice.

From relations (1) and (3), by the maximum–minimum method, we obtain calculation formulas for the limiting relative errors

\[ \begin{aligned} \delta U_1&=|\delta K_{01}|+\left|\ln^{-1}\left|K_{11}\cdot T_1(K_{21}\cdot T_2)^{-1}\right|\right|\cdot\left(|\delta K_{11}|+|\delta K_{21}|+\right.\\ &\qquad\left.+|\delta T_1+\delta T_2|\right)+(\delta U_{01});\\ \delta U_2&=|\delta K_{02}|+\left|(K_{12}\cdot T_1-K_{22}\cdot T_2)^{-1}\ln^{-1}(K_{12}\cdot T_1-K_{22}T_2)\right|\cdot\\ &\qquad\cdot\left[(|K_{12}\cdot T_1\delta K_{12}|+|K_{22}\cdot T_2\cdot\delta K_{22}|)+|K_{12}\cdot T_2\cdot\delta T_1-K_{22}\cdot T_2\cdot\delta T_2|\right]+(\delta U_{02}), \end{aligned} \tag{4} \]

where \(\delta K_i\) are the relative errors of the transfer coefficients;

\(\delta U_{0i}\) are the relative errors determined by the internal noise of the instrument, imbalance, and drift of the operating points of the electronic units;

\(\delta T_i\) are the relative errors of the integral transmission of the optical units.

The calculation of \(\delta K_i\) and \(\delta U_{0i}\) presents no fundamental difficulties. Thus, for example, \(\delta K_i\) can be obtained by summing the relative values of the errors of the component parameters, according to expressions (2).

To obtain a computational expression for \(\delta T_i\), let us analyze the quantities \(T_1\) and \(T_2\), which, in accordance with expressions (2), can be represented in the form

\[ T_1=\int_{\lambda_1}^{\lambda_2} T_{01}(\lambda)\cdot T_3'(\lambda)\cdot T_7'(\lambda)\,d\lambda; \]

\[ T_2=\int_{\lambda_1}^{\lambda_2} T_{01}(\lambda)\cdot T_3'(\lambda)\,d\lambda, \tag{5} \]

where \(T_3'(\lambda)\) and \(T_7'(\lambda)\) are the relative spectral transmissions of the gas in the working and reference cuvettes, without taking into account the spectral transmission of the cuvette walls;

\(T_{0i}(\lambda)\) is an equivalent relative spectral characteristic determined by the characteristics of the source, receiver of IR radiation, and materials of the optical elements, and not dependent on the composition of the gas mixture being analyzed.

According to Bouguer’s law [3], neglecting, as small, aerosol absorption and scattering, as well as Rayleigh scattering and scattering due to turbulence, the spectral transmission for a homogeneous medium of the working and reference cuvette can be obtained as

\[ T_3'(\lambda)=\exp\left[-\sum_{i=1}^{n}\beta_i(\lambda)\cdot W_i^*\right]\exp\left[\beta_x(\lambda)\cdot W_x^*\right]; \]

\[ T_7'(\beta)=\exp\left[-\beta_x(\lambda)\cdot W_{x\mathrm{ref}}^*\right], \tag{6} \]

where \(\beta_x(\lambda)\) and \(\beta_i(\lambda)\) are the relative spectral absorption coefficients per unit equivalent mass \(W^*\) of the investigated gas and \(W_i^*\) of the extraneous \(i\)-th gas in the analyzed medium;

\(W_{x\mathrm{ref}}^*\) is the equivalent mass of the investigated gas in the reference cuvette.

Taking into account (5) and (6), from (4) we obtain:

\[ \delta T_1=T_1^{-1}\left[ T_{01}^{-1}\frac{\partial T_1}{\partial T_{01}}\cdot\delta T_{01} +(W_i^*)^{-1}\frac{\partial T_1}{\partial W_i^*} +(W_{x\mathrm{ref}}^*)^{-1}\frac{\partial T_1}{\partial W_{x\mathrm{ref}}^*}\delta W_{x\mathrm{ref}}^* +(W_x^*)^{-1}\frac{\partial T_1}{\partial W_x^*}\delta W_x^* \right]; \tag{7} \]

\[ \delta T_2=T_2^{-1}\left[ T_{02}^{-1}\frac{\partial T_2}{\partial T_{02}}\cdot\delta T_{02} +(W_i^*)^{-1}\frac{\partial T_2}{\partial W_i^*}\delta W_i^* +(W_x^*)^{-1}\frac{\partial T_2}{\partial W_x^*}\delta W_x^* \right]. \]

Expressions (2, 4, 7) can be used to calculate the total error of the described IR gas analyzer for both versions of the design of its output unit.

Let us compare the variants of gas-analyzer design by the magnitude of their methodological errors; the computational expression is obtained from (5) and (7) for \(K_{11}=K_{12}\), \(\delta K_i=\delta U_{0i}=0\), \(W^*_i\), \(W^*_x\) — const, and unchanged functions \(T_{01}(\lambda)\).

\[ \begin{aligned} \delta U_{1\mathrm{M}}&=\ln^{-1}(T_1\cdot T_2^{-1}) \left[ T_1^{-1}\frac{\partial T_1}{\partial W_i^*}\Delta W_i^* - T_2^{-1}\frac{\partial T_2}{\partial W_i^*}\Delta W_i^* \right];\\ \delta U_{2\mathrm{M}}&=\ln^{-1}(T_2-T_1)(T_2-T_1)^{-1} \frac{\partial T_1}{\partial W_i^*}\Delta W_i^* - (T_2-T_1)^{-1}\frac{\partial T_2}{\partial W_i^*}\Delta W_i^* . \end{aligned} \tag{8} \]

Since, in accordance with formula (5), \(T_1\) and \(T_2\) are non-evaluated integrals, for convenience of calculation it is expedient to replace them approximately by the sum

\[ T_1=\sum_{m=1}^{q} T_{01}(\lambda)'\exp\left[-\sum_{i=1}^{n}\beta_i(\lambda_{0m})\cdot W_i^*\right] \exp\left[-\beta_x(\lambda_{0m})W_x^*\right]\times \]

\[ \times \exp\left[-\beta_x(\lambda_{0m})\cdot W_{\text{eq}}\right]; \]

\[ T_2=\sum_{m=2}^{q} T_{02}(\lambda)\cdot \exp\left[-\sum_{i=1}^{n}\beta_i(\lambda_{0m})\cdot W_i^*\right] \exp\left[-\beta_x(\lambda_{0m})\cdot W_x^*\right], \]

where \(q\) is the number of equal intervals \(\Delta\lambda\) into which the interval \(\lambda_1\div\lambda_2\) is divided;
\(\lambda_{0m}\) is the midpoint of the \(m\)-th interval \(\Delta\lambda\).

Using expression (8), the methodological errors were calculated. Here the equivalent spectral transmittances \(T_1\) and \(T_2\) were determined from expressions (9).

The calculation was carried out for the analyzed gas mixture presented in Table 1, with carbon monoxide CO taken as the gas being determined.

Table 1

The calculation of \(W_i^*\) and the determination of \(\beta_i(\lambda)\) were carried out in accordance with the recommendations of [3].

The relative spectral intensity \(\varphi(\lambda)\) of the emitter was taken equal to the spectral characteristic of an absolutely black body at a temperature of 800°K, and the relative spectral transmittances of the op-

... optical blocks \(T_i(\lambda)\), for the purpose of simplifying calculations of methodological errors, were taken as unity. The relative spectral sensitivity \(c'(\lambda)\) of the detector corresponded to the characteristic of an SF4-1 photoresistor.

Fig. 2. Dependence of the methodological error on the concentration of carbon monoxide.

Fig. 3. Dependence of the methodological error on the concentration of water vapor.

Analysis of the calculation results and of the dependences presented in Figs. 2 and 3 showed that the value \(\delta U_{2\mathrm{m}}\approx 0\), while \(\delta U_{1\mathrm{m}}\) does not exceed \(0.5\%\) when the relative humidity changes from 50 to 98% for a 100-fold increase in the concentration of the gas under study (CO), beginning from \(2\times 10^{-8}\ \mathrm{g/cm^3}\); moreover, with an increase in the concentration of water vapor, \(\delta U_{1\mathrm{m}}\) increases, while with an increase in the concentration of carbon monoxide CO, it decreases. The absorption bands of IR radiation by water vapor have the strongest influence on the error in measuring the CO concentration.

LITERATURE

1. M. S. Aleinikov et al. Method for determining gas concentration and a device implementing it. In the present collection.

2. V. I. Pamiura. Comparison of methods for error analysis. In the collection: Improving the Accuracy and Automation of Electrical and Magnetic Measuring Devices. Kiev, “Naukova Dumka,” 1968, 9.

3. V. E. Zuev. Propagation of Visible and Infrared Waves in the Atmosphere. Moscow, “Sovetskoe radio,” 1970.

4. N. I. Moskalenko, B. M. Golubitskii. Physics of the Atmosphere and Ocean. “Nauka,” 1968, 3, 4, 7, 9.