TECHNICAL AND ECONOMIC OPTIMIZATION OF RADIOMETRIC CONTROL SYSTEMS USING ELECTRON ACCELERATORS

P. S. Moroz

(Presented by the scientific seminar of the Department of Economics)

Studies conducted on the radiometric method of recording the bremsstrahlung of electron accelerators (betatrons) have demonstrated the potential capabilities of the method and indicated ways of implementing it [1, 2].

Problems concerning the selection of individual parameters of a flaw detector were considered and solved, and the metrological characteristics of the developed flaw detectors were investigated. The researchers sought first of all to create operable control systems and adopted sensitivity, productivity, and cost of inspection as efficiency criteria. In this, the bremsstrahlung source was initially selected, and then the recording system was developed.

The accumulated theoretical and practical material has shown that, first, various types of electron accelerators can be created—such as betatrons, microtrons, and linear accelerators—over a broad and overlapping range of bremsstrahlung-beam parameters. At the same time, their cost depends significantly on the beam parameters. Second, there exist various systems for analyzing information about defects, for example, analog and discrete systems. Some of them make it possible to process information with higher quality, i.e., to detect smaller defects, but, as a rule, these systems are more complex and expensive.

Taking into account the contradictions among such criteria as sensitivity, productivity, and cost of inspection, which are most often applied at the level of purely engineering studies, at the stage of designing and developing industrial control systems the question arises: what technical capabilities—for example, in sensitivity and productivity—should a control system possess in order for the effect to be maximal?

An analysis of the functions of nondestructive-testing methods has shown that the problem of designing new optimal control systems and introducing them is connected with the concept of the integral quality of a product. It is indicated in [3] that the choice of an optimal control system is connected with optimal standards for defect detection and can be made through a joint analysis of the target return and the costs of inspection.

Taking into account that modeling and studying the target return is an independent problem and that the development of radiometric control systems is at the laboratory stage, it is proposed to re-

the following problem is posed for solution: to develop a methodology for the technical-and-economic optimization of a radiometric control system according to the criterion of minimum reduced costs, and for calculating the principal parameters at specified sensitivity, productivity, thickness of the material being inspected, and inspection geometry.

The problem is solved by the method of mathematical modeling. The most difficult stage is the mathematical description of the interrelations among the system parameters, whose existence is confirmed by numerous experiments and theoretical concepts.

For the investigation, an analog differential measurement scheme was selected, implemented in experimental installations [1]. Such a schematic arrangement consists of two channels, each of which has a gamma transducer (scintillator, PMT), a subtraction or ratio circuit, an integrator, an amplifier, and a recording instrument. Signals from the gamma transducers are fed to the subtraction or ratio circuit, and from the output the processed signal goes to an inertial integrator element with constant parameters.

As a result of analyzing the passage of the signal from a defect and the noise caused by statistical fluctuations of intensity, the following condition was obtained for detecting the signal from a defect against the noise background:

\[ \begin{aligned} M = \max \Biggl\{& \Biggl[ \Biggl[ \frac{\mu_{\mathrm{eff}} v_g}{S_k n_1} (1-e^{-\tau_1}) \sum_{j=1}^{n_1} j e^{-(n_1-j)(\delta_1-\tau_2)} \Biggr] e^{-n_2(\hat{\tau}_1+\hat{\delta}_2)} \\ &\quad + \frac{\mu_{\mathrm{eff}} v_g(1-e^{-\hat{\tau}_1}) \left[1-e^{-n_2(\hat{\tau}_1+\hat{\delta}_2)}\right]} {S_k\left[1-e^{-(\delta_1+\delta_2)}\right]} \Biggr] e^{-n_3(\hat{\tau}_1+\hat{\delta}_2)} \\ &\quad + \frac{\mu_{\mathrm{eff}} v_g}{S_k} (1-e^{-\tau_1}) \sum_{j=0}^{n-1} e^{-j(\hat{\tau}_1+\hat{\delta}_2)} \\ &\quad + \frac{\mu_{\mathrm{eff}} v}{S_k n_3} (e^{-\hat{\tau}_1}-1) \sum_{j=n_3}^{1} j e^{-(n_3-j)(\delta_1+\hat{\delta}_2)} \Biggr\} \Bigg/ \\ &\sqrt{ \delta_a^2+ \left(\delta_{\mathrm{n}}+\frac{2}{\bar N}\right) \left\{ \frac{ \left(1-e^{-n_\delta(\delta_1+\delta_2)}\right) \left(1-e^{-\delta_1}\right) }{ 1-e^{-(\delta_1+\delta_2)} } \right\}^2 } \ge Z, \end{aligned} \tag{1} \]

where \(\mu_{\mathrm{eff}}\) is the effective absorption coefficient, \(\mathrm{cm}^{-1}\); \(V_g\) is the volume of the detected defect, \(\mathrm{cm}^3\); \(S_k\) is the collimator area, \(\mathrm{cm}^2\); \(n_1, n_2, n_3\) are, respectively, the number of pulses during the time of entry, presence of the defect in the collimator field, and exit, and are a function of the inspection speed; \(\delta_1, \delta_2\) are, respectively, the ratio of the pulse duration to the charge time constant and of the pause to the discharge time constant; \(\delta_a\) is the instrumental component of the error; \(\delta_{\mathrm{n}}\) is the fluctuation component of the error, caused by nonidentity of the channels; \(n_\delta\) is the average number of consecutive pulses in the random pulse process entering the integrator input that exceed the root-mean-square deviation; \(\bar N\) is the average number of gamma quanta falling into the collimator per pulse, which is a function of the beam parameters, the collimator dimensions, and the absorber parameters; \(E_m\) is the maximum energy in the spectrum of the bremsstrahlung beam, MeV; \(P_x\) is the exposure dose rate, R/min; \(d\) is the absorber thickness, cm; \(F\) is the focal distance, m; \(Z\) is the threshold determining the probability of defect detection.

Expression (1), which is a simplified mathematical model of a single-channel flaw detector, is an implicit and nondifferentiable function of such parameters as the defect volume, collimator dimensions, and integrator parameters. Therefore, the study was carried out by a numerical method, which showed that, for a specified probability of detecting defects and other constant conditions, there exist optimal values of the integrator charging time constant and the collimator length at which the detectable defect is minimal. In this case, the discharge time constant is chosen from the requirements on resolving power.

Comparison of the experimental results with calculations by expression (1) gave satisfactory results, which demonstrates the validity of applying this model in subsequent studies.

Because of the low productivity of a single-channel flaw detector, the random location of a defect in the material, and the possibility that it may fall only partly within the collimator field, principles are being developed for constructing a multichannel radiation receiver. They are as follows: the number of scintillation counters, the dimensions of the collimators, and their arrangement must be such that, first, the inspection zone is covered; second, the minimally detectable defect, when randomly located in the volume of the material of the inspected zone, falls completely, at least, within the collimator field of one channel; and third, the radiation detector must be covered by the bremsstrahlung beam.

Mathematically, these conditions are described by the following equations:

\[ b_1 = \frac{d_k}{R} + a; \]

\[ d_k = 1.5\sqrt{b^2 + b_1^2} + 2h; \tag{2} \]

\[ (Rd_k)^2 = F^2 \tg^2 110\, E_m - \frac{B^2}{4}, \]

where \(d_k\) is the diameter of the crystals together with the side shielding, cm; \(h\) is the thickness of the crystal side shielding, cm; \(B\) is the width of the inspection zone, cm; \(R\) is the number of rows of scintillation counters; \(b, b_1\) are the length and width of the collimator, cm; \(a\) is the side of the cubic defect, cm.

Expression (2) made it possible to determine the number of scintillation counters

\[ n_c = \frac{2R}{d_k}\left(B + \frac{d_k - b_1}{2}\right) \tag{3} \]

and the capital costs for the radiation receiver

\[ K_{\mathrm{pr}} = n_c\left(Ts_{\mathrm{f}} + Ts_{\mathrm{kr}}\right) + C_z + C_p,\ \text{rubles}, \tag{4} \]

where \(Ts_{\mathrm{f}}\) is the list price of the FEU, rubles/unit; \(Ts_{\mathrm{kr}}\) is the list price of the crystal, rubles/unit; \(C_z\) is the cost of shielding material, rubles; \(C\) is the constant component of the receiver cost, rubles.

The resulting aggregated cost characteristics of the radiation source (betatron) [3] and of the receiver (4) made it possible to write the reduced costs as a function of a number of variables

\[ Z_{\mathrm{pr}} = \frac{Z_{\mathrm{pl}}}{W} + \frac{n_c Ts_{\mathrm{f}}}{T_1 W} + \frac{(K_i + K_{\mathrm{tp}})(P_a + E_n)}{T_1 W}, \quad \frac{\text{rubles}}{\text{m}^2}, \tag{5} \]

where \(W\) is the inspection productivity, m\(^2\)/h; \(Z_{\mathrm{pl}}\) is the hourly wage of the operators, rubles/h; \(T_1\) is the warranty service life of the FEU, h; \(K_i\) is the cost—

capacity of the source (betatron), rubles; \(P_a\) is the depreciation rate; \(E_n\) is the normative efficiency coefficient; \(T_1\) is the annual number of hours of use of the inspection system, h.

The set consisting of the objective function in the form of reduced costs (5), and the system of equalities and inequalities, constitutes a techno-economic model reflecting the principal technical and economic relationships among the parameters of the bremsstrahlung source, the parameters of the multichannel scintillation receiver, and production requirements.

The mathematical formulation of the problem of comprehensive optimization of a radiometric inspection system can be written in the following form: it is necessary to minimize the nonlinear objective function—the reduced costs

\[ Z_{\mathrm{pr}} = f(K_i, K_{\mathrm{pr}}, W(v, B)) \tag{6} \]

in the presence of the dependencies

\[ K_i = f(E_m, P_x, \text{type of accelerator}), \]

\[ K_{\mathrm{pr}} = f(n_c), \tag{7} \]

\[ n_c = f(R, d_k, B), \]

\[ M = f(E_m, P_x, \delta_1, F, b, b_1, v_g, v, d), \]

in the presence of constraints in the form of equalities

\[ f(F, R, d_k, B, E_m) = 0, \]

\[ f(b_1, d_k, R, a) = 0, \tag{8} \]

\[ f(d_k, b, b_1) = 0 \]

and inequalities

\[ M \geq Z, \]

when such initial conditions are specified as the minimum detectable defect, the thickness of the absorber, and the speed and width of the inspection zone.

Technical and economic constraints are imposed on the parameters of the bremsstrahlung beam, the focal distance, and the number of rows of scintillation counters.

The formulated problem is a complex, nonlinear, multifactor extremal problem, which was solved by a numerical method; in searching for the extremum of the objective function, the method of coordinate-wise descent and blind search over discretely varying parameters was applied.

The developed algorithm makes it possible, with the aid of a digital computer, to calculate the optimal parameters and operating modes of radiometric flaw detectors. By varying the initial conditions—the absorber thickness, sensitivity (defect volume), inspection speed, and bremsstrahlung energy—the behavior of the minimum reduced costs was investigated. As an example, Table 1 presents values of the principal parameters of optimal radiometric inspection systems.

Analysis of the computational material showed that the complexity of the radiometric inspection system is most significantly affected by the volume of the detected defect and the thickness of the absorber.

Analysis of the model and of the computational results obtained in the study of a single-channel flaw detector, when the collimator width is equal to the defect width, showed that reducing the collimator length until it is equal to the defect length leads to an increase in the relative value-

Table 1

\(E_m = 15\ \text{MeV},\ \varphi = 50\ \text{cm/min}\)

of the signal from the defect, but it reduces the time the defect remains in the collimator field and leads to an increase in the statistical component of the noise because of the decrease in the number of gamma quanta entering the collimator per pulse. A further reduction in the collimator length does not lead to an increase in the relative value of the signal from the defect. The statistical component of the noise can be reduced by decreasing the charge time constant and increasing the number of gamma quanta entering the collimator field per pulse, which is proportional to the exposure-dose rate and the collimator area. Increasing the charge time constant above the optimum value leads to a reduction of the noise down to the instrumental error, while the amplitude of the signal at the integrator output tends to zero. Conversely, as the charge time constant is decreased, the amplitude of the signal at the integrator output tends to the relative value of the signal from the defect, while the noise tends to the sum of the instrumental and statistical errors.

In multichannel radiometric flaw detectors, increasing the thickness of the absorber when detecting one and the same defect volume, or reducing the detectable defect at one and the same absorber thickness, leads to an increase in the number of defects in the radiation receiver and in the exposure-dose rate, i.e., to an increase in the cost of both the receiver and the radiation source.

Thus, the developed method of comprehensive optimization of radiometric flaw detectors indicates a way to create optimal models while taking into account the specific conditions of their application. The development of analogous models for other types of sources and methods of information processing, and their study, would make it possible to determine the ranges of effective application of various measurement methods.

The absence of cost characteristics for microtrons and linear accelerators does not permit a comparative analysis of their effectiveness.

References

1. V. I. Gorbunov, A. V. Pokrovskii. Defektoskopiya. 1965, 5, 38.
2. V. I. Gorbunov, A. V. Pokrovskii, A. K. Temnik. Defektoskopiya, 1961, 1.
3. V. I. Gorbunov, P. S. Moroz, A. V. Pokrovskii. Defektoskopiya. 1971, 3, 117.