ON THE THEORETICAL STUDY OF THE THERMAL METHOD OF NONDESTRUCTIVE TESTING

V. P. VAVILOV, V. I. GORBUNOV

(Presented by the scientific seminar of the Scientific Research Institute of EI)

Introduction

In recent years, methods for detecting hidden defects, inhomogeneities, and changes in physical parameters associated with the study of the temperature field of the surfaces of objects have been attracting increasing attention from specialists in nondestructive testing. A large number of works \[1–14\] have been devoted to questions of applying thermal methods of testing in various fields of science and technology; however, a systematized approach to the problems of thermal testing as a whole is still lacking. Such an approach, requiring the combined efforts of specialists in thermophysics, optoelectronics, and infrared technology, would make it possible to reveal the basic regularities in the detection of local surface gradients and would replace the intuitive approach that has existed until now \[2\] with a quantitative analysis of any specific case.

In the theoretical consideration of problems of thermal defectoscopy, in our view, two problems should be distinguished: the internal and the external. The internal problem includes the study of questions concerning the occurrence of local temperature gradients when a hidden defect is present in the material or product, as well as the derivation of the basic dependences of the gradient on the geometric characteristics, the depth of occurrence of the defect, and the ratio of the thermophysical properties of the main body and the defect. In this sense, the ideal would be the solution of a three-dimensional problem for simple geometric forms of the product and the defect. The external problem, in the final analysis, consists in finding the dependence of the magnitude of the signal at the detector output on the magnitude of the gradient in the presence of noise.

The internal problem. The solution of the heat-conduction equation describing the process of heat transfer in solids may be exact or approximate. Strictly speaking, an exact solution is possible only in analytic form, whereas an approximate solution may be obtained analytically, numerically, or by modeling.

To solve the internal problem it is necessary to solve the one-, two-, or three-dimensional nonstationary heat-conduction equation

\[ \frac{\partial t}{\partial \tau} = a \nabla^{2} t \tag{1} \]

in the region constituted by the given body, with the corresponding boundary conditions depending on the initial temperature distribution of the body and the medium and on the conditions of heat exchange between the body and the medium.

It should be noted at once that solving the problem exactly in this form...

is possible only in the presence of fairly simple boundary conditions which, with one degree of accuracy or another, will reflect the real testing conditions. Thus, the solution of the internal problem can be exact only in the mathematical sense; in practice, however, it will always be approximate.

From the considerations presented above one may derive a general algorithm for solving the internal problem:

1) pass from the real geometry of the object being tested to a certain simplified geometry in which the solution will be carried out; naturally, in such cases the body and the defect are to be approximated by simple geometric shapes: a plate, a cylinder, a sphere;

2) replace the real boundary conditions by conditions that are considered in the classical theory of heat conduction;

3) obtain the solution by using one of the known methods for solving the boundary-value problem of heat conduction.

Below we shall consider each of these points in detail.

1. Analytical heat-conduction theory deals with the solution for simple geometric shapes: a plate, a cylinder, a sphere. Apparently, it is expedient to choose these shapes also for the approximation of defects and to solve the heat-conduction equation in Cartesian, cylindrical, or spherical coordinate systems [15]. It should be noted that, since the dimensions of defects are, as a rule, much smaller than the dimensions of the object itself, and in order to detect the gradient it is necessary to inspect the object with small linear resolution, such an approximation gives a sufficiently good approximation and may be recommended even in numerical calculations in order to reduce the amount of computational work.

2. In heat-conduction theory four types of boundary conditions are considered [16].

With boundary conditions of the first kind, the temperature distribution on the surface of the body is specified for each instant of time:

\[ t_{\mathrm{п}} = f_1(x, y, z, \tau). \tag{2} \]

With boundary conditions of the second kind, the magnitude of the heat flux is specified for each point of the surface of the body and for any instant of time:

\[ g_{\mathrm{п}} = f_2(x, y, z, \tau). \tag{3} \]

Boundary conditions of the third kind specify the temperature of the surrounding medium and the law of heat exchange between the body and the surrounding medium. Usually the Newton–Richmann law is used for writing them:

\[ g_{\mathrm{п}} = \alpha (t_{\mathrm{п}} - t_{\mathrm{c}}), \tag{4} \]

where \(t_{\mathrm{п}}\) is the body temperature; \(t_{\mathrm{c}}\) is the temperature of the medium; \(\alpha\) is the heat-transfer coefficient.

With boundary conditions of the fourth kind, it is assumed that heat exchange of the body with the medium occurs only by heat conduction:

\[ \lambda_1 \left(\frac{\partial t_1}{\partial n}\right)_c = \lambda_2 \left(\frac{\partial t_2}{\partial n}\right)_c . \tag{5} \]

Condition (5) expresses the equality of heat fluxes on the contact surface of the body and the medium. Here \(\lambda\) is the thermal-conductivity coefficient;

\[ \left(\frac{\partial t}{\partial n}\right)_c \]

is the temperature gradient in the direction of the normal to the interface of the media.

Since in problems of thermal defectoscopy it is required to find the temperature distribution over the surface, the set of possible boundary conditions reduces to conditions of the second, third, and fourth kinds. Condition (3) will mean heating of the object in the active method of testing-

…; condition (4) will signify the process of heat exchange of the body with the medium during inspection; condition (5) will describe heat exchange between the body and the defect, since in most practical applications heat transfer inside the defect takes place only through thermal conduction, while convection and radiation may be neglected. In a number of cases, so-called mixed boundary conditions may occur. As the initial condition one may, practically always, take

\[ t(x, y, z)\big|_{\tau=0} = t_c = \text{const} \tag{6} \]

or

\[ t(x, y, z, \tau)\big|_{\tau=0} = \text{const}, \tag{7} \]

\[ t_c=\text{const}. \]

3. Let us consider methods for solving the heat-conduction equation. As analytical methods for solving equation (1), use is made of separation of variables, the method of instantaneous sources, and methods based on the application of Green’s functions, Dirac functions, etc. [16]. The operational method has become very widespread; its use for solving the problem of cooling of a spot weld with an internal defect is shown in [17]. The main difficulty of all analytical methods lies in the fact that they are suitable for solving simple (as a rule, one-dimensional) geometries, and therefore with their help it is difficult to estimate the subtle effects that arise when the form of the temperature field of the surface is disturbed owing to the presence of a small defect. This may explain the small number of theoretical works devoted to thermal inspection. Nevertheless, in many cases (extended defects of the delamination, lack-of-fusion, and crack types) analytical methods can give satisfactory results. The principal task here is the choice of the correct geometrical approximation, using either multilayer geometries of the heat—defect—body type [18], or differences in the thermophysical properties of the main body and the defect [19].

Since an analytical solution is practically possible only for one-dimensional problems, having obtained the dependence of the gradient on time and on one of the coordinates, we can say nothing about its dependence on the other coordinates; for example, we cannot obtain the temperature distribution over the surface in its real form. In such cases one has to assume that such a distribution has the form of a rectangular step, with one temperature level over the defect and another in adjacent regions. Such an assumption becomes incorrect when the extent of the defect in the direction perpendicular to the direction of heat propagation is small in comparison with the depth at which it lies, since here the effect of heat spreading through the layers surrounding the defect begins to have a strong influence. To take this into account it is necessary to solve at least a two-dimensional problem, which may be possible when using the grid method [15]. In [19] the solution is considered, by means of the grid method, of the problem of heating a metallic plate with an internal defect, finite in two dimensions.

Of considerable interest may also be the use of various models for studying the heat-conduction equation, based, for example, on hydrodynamic or electrical analogy. In [20] modeling of a weld is considered, but on the whole only the first steps have been taken in this direction.

External problem. Having calculated the process of formation of the gradient, one must establish a one-to-one correspondence between the magnitude of the gradient formed and the signal at the output of the device carrying out the inspection, whether it be a thermal imager, an infrared radiometer, or any other

another system for recording the thermal field of the object. In addition, in the external problem it is necessary to determine the required power of the device that creates the nonstationary regime and to relate it to the recording time. The effect of heat spreading along the shells of the defect causes partial or complete blurring of the gradient, which imposes certain requirements on the heater power and the inspection time [2, 17]. In this connection it should be taken into account that insufficient heating power may fail to produce the required gradient, while excessive heating power may lead to overheating of the product [5].

The detector signal for most inspected objects is calculated according to the general procedure for the energy calculation of infrared systems. As a result of such a calculation, the parameters of the optical path, the requirements for the infrared-radiation receiver, and the receiver-signal processing scheme are obtained. In general form, the magnitude of the signal at the receiver output is equal to [21]

\[ U = S \Gamma K \Delta F \int_{\lambda_1}^{\lambda_2} r(\lambda)\tau(\lambda)\varepsilon(\lambda) f(\lambda)\,d\lambda, \tag{8} \]

where \(\lambda_1 \div \lambda_2\) is the operating wavelength range, determined by the sensitivity range of the receiver or by the filter used;

\(r(\lambda)\) is the spectral density of the radiation of an absolutely black body having the temperature of the object being inspected;

\(\tau(\lambda)\) is the transmission coefficient of the medium;

\(f(\lambda)\) is the relative spectral sensitivity of the receiver;

\(\Gamma\) is a geometric factor associated with the optics and the geometry of inspection;

\(K\) is the transmission coefficient of the optics;

\(\Delta F\) is the resolution of the recording device on the surface of the object;

\(S\) is the integral conversion coefficient of the receiver;

\(\varepsilon(\lambda)\) is the emissivity coefficient of the object.

Formula (8) is derived for the Lambertian character of the radiation of the object, without taking scattered radiation into account. In thermal inspection, the useful signal is the signal caused by the change in \(r(\lambda)\) due to the presence of a gradient, and the principal noise signal is the change in the heat flux due to the surface variation of \(\varepsilon(\lambda)\). Since it may be assumed that, at room temperature, a change of \(\varepsilon(\lambda)\) by one percent corresponds to a gradient of one degree, it becomes evident that the capabilities of thermal inspection as a whole are limited by local variations in the emissivity of the object.

There are a number of ways to overcome this limitation:

1) applying, before the test, to the surface of the object an easily removable emulsion having a uniformly high emissivity coefficient;

2) recording thermal maps of the object in stationary and nonstationary regimes and subsequently subtracting them;

3) using systems sensitive only to temperature and independent of changes in \(\varepsilon(\lambda)\) [22];

4) receiving radiation in the short-wavelength range, where the dependence of energy on temperature is stronger than according to the Stefan–Boltzmann law [3];

5) “forcing” the radiation above the surface [3];

6) the method of two scanning radiometers proposed by D. R. Green [5].

From the standpoint of high inspection productivity, the last three methods are the most preferable.

Conclusion

Thus, the principal problems of nondestructive testing using infrared radiation are problems associated with overcoming the noise effect of the nonuniformity of the radiation coefficient and the problem of optimal heating. Here we do not touch upon questions related to the design of systems for recording the thermal image, since they have been considered in numerous Soviet and foreign publications \[21, 25\]. The principal parameters of thermal-imaging systems are given in \[26\]. The features of certain heating sources are described in \[5\]. On the whole, from the review of the literature on thermal methods of nondestructive testing it is clear that thermal methods of testing not only replace traditional ones, but in a number of cases are the only possible methods.

References

1. M. M. Miroshnikov. Optiko-mekh. prom., 1969, No. 12.
2. Green D. R., Mat. Ev., 1971, No. 11.
3. Methods of Nondestructive Testing. (Collection). Moscow, Mir, 1972.
4. A. K. Denel. Defectoscopy of Metals. Moscow, Metallurgiya, 1972.
5. Green D. R., Appl. Opt., vol. 7, 1968, No. 9.
6. Leftwich R. F., Ordway G. B., Appl. Opt., vol. 7, 1968, No. 9.
7. Overbo P. I. and others, Proc. IRE, 1959, No. 9.
8. Kutzscher E. and others, Mat. Ev., 1968, No. 7.
9. P. K. Oshchepkov. Zavodskaya laboratoriya, issue 6, 1959.
10. L. A. Vyatich, V. M. Skobelkin. Defektoskopiya, 1970, No. 5.
11. N. A. Bekeshko, P. K. Oshchepkov. Defektoskopiya, 1965, No. 5.
12. D. Vancetti. Elektronika, 1967, No. 7.
13. N. N. Gavrilov, B. A. Khokhlov. Trudy NIIIN, issue 6, 1972.
14. V. P. Sool, A. N. Bezus, G. S. Khulap. Trudy NIIIN, issue 6, 1972.
15. A. V. Lykov. Theory of Heat Conduction. Vysshaya shkola, 1967.
16. V. P. Isachenko, V. A. Osipov, A. S. Sukomel. Heat Transfer. Moscow, Energiya, 1969.
17. V. P. Vavilov, V. I. Gorbunov, V. B. Kuznetsov. Defektoskopiya, 1973, No. 1.
18. G. F. Muchnik, I. B. Rubashov. Methods of the Theory of Heat Exchange, Part I, Vysshaya shkola, 1970.
19. V. P. Vavilov. Diploma thesis. Research Institute of Electronic Introscopy at TPI, 1972.
20. Yu. A. Popov, N. A. Bekeshko. Trudy NIIIN, issue 4, 1970.
21. Yu. G. Yakushenkov. Fundamentals of the Theory and Calculation of Optoelectronic Instruments. Moscow, Sovetskoe radio, 1971.
22. Clair I. C. S., Mat. Ev., 1966, No. 8.
23. Green D. R., Nucl. Sci. and Eng., vol. 12, 1962, No. 2.
24. Astheimer R. W., Warmset E. H., IOSA, vol. 49, 1959, No. 2.
25. R. Hudson. Infrared Systems. Moscow, Mir, 1972.
26. N. A. Bekeshko. Defektoskopiya, 1972, No. 4.