On the Question of the Physical Nature of Ash Wear in Boiler Installations

I. K. Lebedev

The nature of ash wear, like abrasive wear in general, has at present been studied poorly; therefore in the modern literature one can sometimes encounter vague and indefinite interpretations of this phenomenon which do not reveal its physical essence. Thus, for example, research associate V. A. Lokshin of the All-Union Thermal Engineering Institute, on the question of the nature of ash wear of the heating surfaces of boiler units, writes: “Wear of tubes by flying ash occurs due to forces arising when particles strike the metal surface. The worn surface has a polished appearance, is uneven, but has no scratches” [5]. Research associate Syrkin of the Central Boiler-Turbine Institute, with regard to ash wear of tubular surfaces, which include the heating surfaces of boiler units, writes: “First of all we shall note the impact mechanism of destruction of tube walls in the process of ash wear. Destruction occurs mainly as a result of impacts by solid ash particles, which is proved both by observations of the worn surface under a microscope (alternating projections and depressions without elongated scratches) and by special laboratory experiments. Consequently, the regularities of the process of sliding wear or friction wear are not applicable to the present case” [7].

Corresponding Member of the Academy of Sciences of the USSR Prof. V. D. Kuznetsov, with regard to the nature of abrasive wear—of which, of course, ash wear is also a kind—writes: “The mechanism of wear in an abrasive medium, despite its apparent complexity, seems to me in principle quite simple. The process of abrasive wear reduces to the sum of a large number of elementary scratching processes. There must exist a deep connection between the phenomena of simple scratching and abrasive wear” [4, p. 234].

From consideration of the formulations cited above, two opposing points of view on the nature of ash wear become clear. If from the second part of Lokshin’s formulation one can only infer the inapplicability of the regularities of wear associated with friction to the ash wear of boiler installations, then Syrkin clearly denies this. Prof. V. D. Kuznetsov directly connects the process of abrasive wear with the process of elementary scratching that occurs when abrasive particles rub against the surface being worn. We note in passing that the formulations of Lokshin and Syrkin are very indefinite and unclear. They indicate the nature of the origin of forces between the abrasive particles and the worn surface and do not at all explain the mechanism of wear, denying only the regularities of “sliding” wear.

On the basis of numerous observations of ash wear during the operation of boiler installations, we arrive at the conclusion that the definition of the nature of abrasive wear given by Prof. V. D. Kuznetsov is also correct for cases of ash wear. Ash wear is the sum of a large number of elementary scratches produced by ab-

abrasive (ash) particles on the surface of the body being worn, irrespective of the nature and causes of the forces acting on the ash particles.

Derivation of equations expressing ash wear

Considering the phenomena of abrasive wear of plastic bodies and the phenomena of metal cutting, Prof. Kuznetsov comes to the conclusion that, both in cutting and in abrasive wear, the energy applied both to the cutter and to the abrasive particle is expended chiefly on the formation of plastic deformations in the layer of metal being removed. Thus, he assigns both cutting and wear to one and the same class of phenomena associated with plastic deformations of the layer being removed [4]. On this basis, it appears possible to apply the regularities established for cutting to the study of abrasive wear of metals. From these positions let us consider the wear of a cylindrical body washed by a transverse gas stream carrying abrasive ash particles.

In the discussion we shall adopt the following simplifying assumptions:

1. All ash particles moving with the gas stream, before striking the worn surface, have a velocity and direction of motion identical with the velocity and direction of the gas stream before the worn cylindrical body.

2. When the gas stream flows around the cylindrical body, the ash particles do not deviate from their initial direction and, consequently, by virtue of their energy, fall upon the surface of the body being flowed around.

3. We shall regard the ash particles as absolutely hard, not subject to destruction or deformation during impacts against the surface of the body being worn.

4. Each ash particle has several sharp corners or projections, with one or several of which at once it will produce scratches when moving along the surface of the body being worn.

Fig. 1

Let an ash particle fall on the surface of a cylindrical body at point \(A\), with velocity \(v_1\). Let the angle between the direction of motion of the ash particle \(BA\) and the tangent \(T\) to the surface of the cylinder be denoted by \(\alpha\) (Fig. 1). After a time \(\tau\), which we shall call the impact time, the ash particle rebounds from the surface of the cylinder with velocity \(v_2\). Resolve the velocities before and after impact into the normal and tangential components to the surface at the point of impact. The normal and tangential components will be denoted respectively by \(v_{1n}\), \(v_{2n}\), \(v_{1t}\), and \(v_{2t}\).

On the basis of Newton’s hypothesis,

\[ \frac{v_{2n}}{v_{1n}} = k, \tag{1} \]

whence

\[ v_{2n} = k v_{1n}, \tag{2} \]

where \(k\) is the coefficient of restitution, which depends on the elastic properties of the colliding bodies and does not depend on the velocity and dimensions of the colliding bodies.

On the basis of the theorem on the quantity of motion, the normal component of the impulse of the force can be expressed by the following equation

\[ S_n=m(v_{2n}-v_{1n})=m(1+k)v_1\sin\alpha . \tag{3} \]

The tangential component of the impulse can be expressed either by projecting the impulse of the force onto the tangent, taking into account the instantaneous coefficient of friction \(f\), expressing \(v_{2t}\) analogously to how this was done in equation (2) for \(v_{2n}\), or, as is often done, by assuming that the tangential component of the impulse of the force is related to the normal component in the same way as the ordinary friction force is related to the normal force, with the value of the coefficient of friction for finite, and not for instantaneous, normal forces being taken as the coefficient of friction [6].

Determining the magnitude of the tangential component of the impulse by the second method, we obtain

\[ S_t=-k_1S_n=-km(1+k)v_1\sin\alpha , \tag{4} \]

where \(k_1\) is the coefficient of friction between the colliding bodies.

The minus sign before the right-hand side of equation (4) determines the direction of the impulse.

On the other hand, on the basis of the impulse theorem,

\[ S_t=\int_0^\tau P\,dt, \tag{5} \]

where \(P\) is the magnitude of the tangential force,
\(t\) is the time of action of the tangential force.

If the mean value of the force is adopted, then after integration we shall have

\[ S_t=P\tau, \tag{6} \]

where \(\tau\) is the time of action of the force \(P\), or the impact time.

From equations (4) and (6) we find

\[ P=k_1\,\frac{m(1+k)}{\tau}\,v_1\sin\alpha . \tag{7} \]

Let us divide the impact time \(\tau\), as is customary [6, p. 129], into two intervals: \(\tau_1\)—the interval from the beginning of contact of the colliding bodies to the maximum approach of their centers due to elastic deformations, and \(\tau_2\)—the interval from the maximum approach of the centers to the moment when the surfaces separate from one another. During the period \(\tau_1\) the velocity of motion of the abrasive particle (with the worn body immobile) changes from \(v_1\) to zero; in doing so we shall regard the motion as uniformly retarded. During the period \(\tau_2\) the velocity will increase from 0 to \(v_2\), and the motion will be uniformly accelerated. In accordance with this, the tangential and normal components \(v_t\) and \(v_n\) will change. But during the impact period, in addition to elastic deformations, plastic deformations will also occur, and in particular, in the tangential direction there will be plastic compression of a layer of metal of a certain thickness, similar to the plastic compression of the layer being removed in metal cutting under the action of the force acting on the cutter in the direction of motion of the cutter and called, in cutting theory, the cutting force. Similarly, as in cutting, the length of the removed layer will be

is equal to the magnitude of the cutter travel; the magnitude of the layer cut off when an abrasive ash particle strikes the worn surface may be expressed as

\[ l=\frac{v_{1t}+0}{2}\tau_1+\frac{0+v_{2t}}{2}\tau_2 =\frac{v_{1t}}{2}\tau_1+\frac{v_{2t}}{2}\tau_2 . \tag{8} \]

Let us express \(v_{2t}\) in terms of \(v_{1t}\) in the following form

\[ v_{2t}=fv_{1t}, \tag{9} \]

where \(f\) is the coefficient of instantaneous friction. Then

\[ l=\frac{v_{1t}}{2}\tau_1+f\frac{v_{1t}}{2}\tau_2 =\frac{v_1}{2}\cos\alpha(\tau_1+f\tau_2). \tag{10} \]

Let us express the accelerations of the uniformly decelerated and uniformly accelerated motions of the abrasive particle during impact in the following form

\[ \frac{dv_{1n}}{dt}=-C_1 \tag{11} \]

\[ \frac{dv_{2n}}{dt}=C_2 . \tag{12} \]

Separating variables and integrating equation (11) over the limits from \(v_{1n}\) to 0 and from 0 to \(\tau_1\), and equation (12) over the limits from 0 to \(v_{2n}\) and from \(\tau_1\) to \(\tau\), we obtain

\[ v_{1n}=C_1\tau_1 . \tag{13} \]

\[ v_{2n}=C_2(\tau-\tau_1) \tag{14} \]

From equations (1), (13), and (14),

\[ \frac{v_{2n}}{v_{1n}}=k=\frac{\tau-\tau_1}{\tau_1}\cdot\frac{C_2}{C_1}. \]

Let us denote

\[ \frac{C_2}{C_1}=\delta, \]

then

\[ k=\frac{\tau-\tau_1}{\tau_1}\delta . \tag{15} \]

On the other hand,

\[ \tau=\tau_1+\tau_2 . \tag{16} \]

Solving equations (15) and (16) as a system, we find

\[ \tau_1=\frac{\delta}{\delta+k}\tau, \tag{17} \]

\[ \tau_2=\frac{k}{\delta+k}\tau . \tag{18} \]

The coefficient \(\delta\) characterizes the elastic properties of the worn surface. For steel it will probably be close to unity; therefore, to simplify the subsequent equations, we shall take

\[ \delta=1, \]

in this case

\[ \tau_1=\frac{1}{1+k}\,\tau, \tag{19} \]

\[ \tau_2=\frac{k}{1+k}\,\tau. \tag{20} \]

Let us now express the tangential component of the impulse through the velocities \(v_{1t}\) and \(v_{2t}\). On the basis of the theorem on momentum and equation (9),

\[ S_t=m(v_{2t}-v_{1t})=-m(1-f)v_1\cos\alpha. \tag{21} \]

Equating the right-hand sides of equations (4) and (21), we obtain

\[ f=1-k(1+k)\tan\alpha. \tag{22} \]

Substituting \(\tau_1\), \(\tau_2\), and \(f\) from equations (19), (20), and (22) into equation (10), we then obtain

\[ l=\frac{v_1}{2}\cos\alpha\,\tau(1-k_1 k\tan\alpha), \tag{23} \]

whence

\[ \tau=\frac{2l}{v_1\cos\alpha(1-k_1 k\tan\alpha)}. \tag{24} \]

Substituting the value of \(\tau\) obtained from equation (24) into equation (7), after simple transformations we have

\[ P=k_1\frac{m(1+k)}{2l}\,v_1^2(\sin\alpha\cos\alpha-k_1 k\sin^2\alpha). \tag{25} \]

Equation (25) expresses the magnitude of the forces acting on the ash particle tangentially to the worn surface at the point of impact. If the ash particle with pointed projections, by which it makes a scratch on the surface of the worn body, is regarded as a cutter, then the tangential component of the forces acting on the ash particle may be considered as cutting forces.

To determine the magnitude of the cutting forces as a function of the chip dimensions and other quantities characterizing the cutting conditions, various authors give many equations derived mainly on the basis of experimental data. Thus, for example, Chelyustkin [3] gives the following equation

\[ P=k_2ba^\xi, \tag{26} \]

where
\(P\) — cutting force;
\(b\) — width of cut;
\(a\) — depth of cut;
\(k_2\) and \(\xi\) — constants for the given material.

In view of the fact that the values of \(\xi\) for different materials being machined differ little from one another, Chelyustkin considers it possible for all materials to take the average value \(\xi=0.75\) and to express the equation in the following form

\[ P=k_2ba^{0.75}. \tag{27} \]

Professor V. D. Kuznetsov, considering the process of cutting metals as a process of plastic compression of the chip, derives for free cutting the following relationship [3, p. 157]:

\[ P=\sigma_0 f\cdot \zeta^{m_1}. \tag{28} \]

where \(P\) is the cutting force;
\(\sigma_0\) is the conditional yield strength in compression;
\(\zeta\) is the chip-shrinkage coefficient;
\(m_1\) is the exponent of the polytrope of plastic compression;
\(f_0\) is the cross-sectional area of the cut.

Equating the right-hand sides of equations (25) and (28),

\[ \sigma_0 f_0 \zeta^{m_1} = k_1 \frac{m(1+k)}{2l}\,v_1^2(\sin\alpha\cos\alpha-k_1 k\sin^2\alpha), \]

whence

\[ V = k_1 \frac{m(1+k)}{2\sigma_0 \zeta^{m_1}}\, v_1^2(\sin\alpha\cos\alpha-k_1 k\sin^2\alpha), \tag{27} \]

where \(V=f_0l\) is the volume of metal removed from the worn surface by one abrasive ash particle of mass \(m\).

If, on the surface of a cylinder washed by a gas stream carrying ash particles, we take the surface \(f_1\) (Fig. 2), then the number of impacts of ash particles on this surface per unit time will be

Fig. 2

\[ N=\frac{\mu}{mg}v_1s = \frac{\mu}{mg}v_1f_1\cos\beta = \frac{\mu}{mg}v_1f_1\sin\alpha, \tag{28} \]

where \(\mu\) is the ash concentration of the gas stream, \(\mathrm{kg/m^3}\);
\(mg\) is the average weight of one ash particle;
\(s\) is the surface normal to the direction of motion of the gas stream.

Then the wear of the surface \(f_1\) per unit time can be expressed as

\[ V=VN = k_1\frac{(1+k)}{2\sigma_0 \zeta^{m_1}g}\, v_1^3\mu f_1(\sin^2\alpha\cos\alpha-k_1 k\sin^3\alpha) \]

Express the wear in the following form

\[ V_t=a_tf_1, \tag{30} \]

where \(a_t\) is the depth of wear, or the thickness of the cut.

From equations (29) and (30)

\[ a_t = k_1 \frac{(1+k)}{2\sigma_0 \zeta^{m_1}} v_1^3 \mu \left(\sin^2\alpha \cos\alpha - k_1 k \sin^3\alpha\right). \tag{31} \]

The wear of the surface over a time interval \(t\) will be expressed as

\[ a = a_t t = k_1 \frac{(1+k)}{2\sigma_0 \zeta^{m_1} g} v_1^3 \mu t \left(\sin^2\alpha \cos\alpha - k_1 k \sin^3\alpha\right). \tag{32} \]

The derivation of the equations expressing wear was based, as was evident, on the equation of Prof. V. D. Kuznetsov, which expresses the cutting forces as a function of the geometric dimensions of the cut and the mechanical properties of the material being worn. In the derivation, it would have been possible to use any other one of the numerous equations expressing the cutting forces. We derived the wear equations taking Chelostkin’s equations as a basis. But the equations obtained in this way differ from those presented by us only in certain coefficients. The qualitative aspect of the equations remains the same.

In deriving the equations we took Prof. Kuznetsov’s equation as the initial equation also because it is derived from consideration of only the principal factor determining the magnitude of the cutting forces, namely, the plastic compression of the layer being cut. But, as studies in the field of cutting show, the influence of additional factors in cutting is insignificant in comparison with this principal factor.

In addition, Prof. Kuznetsov’s equation most simply expresses the dependence of the cutting forces on the geometric dimensions of the cut and the physical properties of the material being machined, which is also its advantage in determining theoretical relationships for abrasive wear, the coefficients of which, for applicability to the calculation of the quantitative aspect of wear, must be determined experimentally.

Some analysis of the derived equations and comparison of them with experimental data

From the equations of ash wear it is evident that the magnitude of wear depends on the value of the angle at which the abrasive particles collide with the worn surface. Let us determine the value of the angle at which the wear will be maximal. To do this, we take the first derivative of equation (27) with respect to \(\alpha\) and set it equal to zero:

\[ \frac{dV}{d\alpha} = k_1 \frac{m(1+k)}{2\sigma_0 \zeta^{m_1}} v_1^2 \left[ \cos^2 \alpha_{\mathrm{in}} - \sin^2 \alpha_{\mathrm{in}} - 2k_1 k \sin \alpha_{\mathrm{in}} \cos \right] = \]

\[ = k_1 \frac{m(1+k)}{2\sigma_0 \zeta^{m_1}} v_1^2 \left[ \cos 2\alpha_{\mathrm{in}} - k_1 k \sin 2\alpha_{\mathrm{in}} \right] \]

hence

\[ \tan 2\alpha_{\mathrm{in}} = \frac{1}{k_1 k}. \tag{33} \]

Equation (33) shows that the angle at which the wear will have its maximum value depends on two quantities: on the values of \(k_1\) and \(k\), in other words, on the nature of the colliding bodies and the quality of their surfaces.

To determine the angle of maximum wear, in Fig. 3 we present a nomogram constructed according to equation (33), from which it is evident that,

for a wide range of values of \(k_1\) and \(k\), \(\alpha_{\text{imp}}\) lies within the limits from \(30\) to \(40^\circ\) and cannot be greater than \(45^\circ\). This conclusion is confirmed by experimental data published in the literature. N. G. Zalogin, a staff member of VTI, writes [1]: “Specially conducted experiments showed that freely flying solid particles destroy metal most intensively if

Fig. 3

they strike its surface at an angle close to \(40^\circ\).” Zalogin then gives the results of these experiments in the form of the curve shown by us in Fig. 4 as a dashed line, while individual experimental points are plotted as circles. In the same figure, the solid line shows the relative

— theoretical curve for \(k_1 = 0.4\) and \(k = 0.8\)

– – experimental data of N. G. Zalogin

Fig. 4

wear calculated by us for different \(\alpha\) at \(k_1 = 0.4\) and \(k = 0.8\) according to equation (27). As can be seen from comparison of these curves, within the range of \(\alpha\) from \(20^\circ\) to \(60^\circ\) the experimental and theoretical curves agree very well.

In this same work, N. G. Zalogin gives data on the wear of a tube cut from the first row of a water economizer. For this tube, as Zalogin writes, not only was the metal wear around the circumference measured, but, by special experiments, the distribution of the number of impacts of ash particles around the circumference of the tube was also measured. After this, curves were constructed for the intensity of wear and for the distribution of impacts over the surface. By dividing the ordinates of the first curve by the ordinates of the second, a curve was obtained for the intensity of wear as a function of the angle of incidence of the particles on the surface of the tube. This curve is shown by us in Fig. 5 as a dashed line. In the same figure a theoretical curve is plotted (solid line) according to equation (27).

— theoretical curve for \(K_1 = 0.4\) and \(K = 0.8\)
– – experimental data of N. G. Zalogin

Vertical axis: Wear per ash particle
Horizontal axis: Angle of position of the tube surface relative to the direction of motion of the gas flow

Fig. 5

Comparison of these curves also shows their sufficiently good agreement.

For a quantitative determination of ash wear, at present there is only one empirical equation, expressing the thinning of the tube wall at the most dangerous point of an in-line staggered tube bank with transverse washing by a gas flow carrying abrasive particles. For this bank, the tubes of the second row in the direction of gas flow are the most dangerous with respect to wear, since, according to operating observations and experimental data of VTI, these tubes have the maximum wear.

This experimental equation has the following form [5]

\[ I = 3\varphi AKM(\beta_{\mu})(\beta_{\eta}v_{1})^{3}\tau, \tag{34} \]

where

\[ \begin{aligned} I \;&\text{— thinning of the tube wall at the dangerous point of the bank;}\\ 3 \;&\text{— ratio of the maximum thinning to the average over the perimeter}\\ &\quad \text{of a tube of the 2nd row of a staggered bank;}\\ \varphi \;&\text{— coefficient reflecting the geometrical forms of the bank;}\\ A \;&\text{— coefficient characterizing the abrasive properties of the ash}\\ &\quad \text{at an average size of ash particles of 30–40 microns;}\\ K \;&\text{— experimental coefficient taking into account the influence of the size}\\ &\quad \text{of ash particles;}\\ M \;&\text{— coefficient of abrasibility of the tube metal, equal for boiler}\\ &\quad \text{carbon steels to 1.0.} \end{aligned} \]

\(\beta_\mu\) and \(\beta_v\) are coefficients of nonuniformity of the fields of ash concentration and gas velocity;

\(v_1\) is the mean velocity of the incident flow before the tube bundle;

\(\mu\) is the mean concentration of ash in the gas flow;

\(\tau\) is the time in hours.

Experimental data are available for determining \(\varphi\), \(A\), and \(K\).

If the equation we have obtained theoretically for expressing wear is compared with the VTI equation just cited, it is not difficult to see that there is agreement in the influence of the main factors, which are \(v_1\), \(\mu\), and \(\tau\).

Thus, comparison of experimental data on ash wear with the theoretical equations derived above, proceeding from the premise that ash wear is the sum of elementary scratches, confirms this interpretation of the nature of ash wear.

Appearance and Shape of the Surface of Worn Parts

One of the reasons that served as grounds for rejecting the scratching mechanism in explaining the nature of ash wear was the appearance of the worn surface; as was seen above, Lokshin and Syrkin also refer to this. Without entering into a detailed description of the shape of the chip removed by an ash particle upon impact with the surface being worn, let us estimate the order of magnitude that determines the length of the layer cut off. Earlier we found that the length of the removed layer is determined by equation (23)

\[ l=\frac{v_1}{2}\cos\alpha\,\tau(1-k_1 k \tg\alpha). \]

If we take \(k_1 = 0.4\), \(k = 0.8\), and calculate the length of the cut-off layer at the point of maximum wear, that is, at an angle \(\alpha = 36^\circ\), we obtain

\[ l=\frac{v_1}{2}\,0.809\,(1-0.4\cdot 0.8\cdot 0.726)\tau =0.31\,v_1\tau. \tag{35} \]

From Hertz’s theory of the impact of spheres [2, p. 47],

\[ \tau_{\max}=2.9432\,\frac{\xi_{\max}}{v_0}, \tag{36} \]

where

\(\xi_{\max}\) is the maximum approach of the centers of the colliding bodies as a result of their deformation during impact;

\(v_0 = v_1\sin\alpha\) is the relative velocity of the colliding bodies at the beginning of impact.

Substituting the value of \(\tau_{\max}\) into equation (35), we find

\[ l=\frac{0.31\cdot 2.9432}{\sin\alpha}\,\xi_{\max} =1.57\,\xi_{\max}. \tag{37} \]

It would have been possible to calculate \(\xi_{\max}\), but even without these calculations it is perfectly clear that, in the collision of an abrasive particle of very small dimensions with an immobile cylinder, this quantity will be vanishingly small; consequently, the length of the chip will also be an extremely small quantity.

If we consider the trajectory of the sharp projection of an abrasive particle removing a chip from the surface of the body being worn during impact,

one can be convinced that this trajectory will not be located along a straight line parallel to the tangent at the point of impact, but will describe a certain curve of complex form, in consequence of which the longitudinal section of the chip being removed will also have a complex form, resembling the shape of an irregular lentil, while the scratch left on the surface of the body being worn will have the form of a somewhat elongated crater. If, in addition, one takes into account the continuous superposition of these craters one upon another in the process of wear, then it proves difficult to imagine the possibility of regarding in any way the “longitudinal scratches” on the worn surface in the form in which we have usually become accustomed to understand them and as Lokshin and Syrkin write about them.

As for the shape of worn parts, then at the angle of maximum intensity of wear on the surface, flats are formed, as is shown in Fig. 6; moreover, the size of these flats increases with wear, thereby causing an increase in the specific wear, if this wear is referred to the entire surface of the part. With respect to a cylinder, as a result of this process we obtain the shape of the worn part shown in Fig. 6.

Fig. 6

Conclusion

As a result of carrying out analytical investigations and comparing them with experimental data, we have established:

1. Ash wear is the sum of a large number of elementary scratches produced by abrasive ash particles on the surface of the parts being worn.

2. The process of wear of metal by abrasive particles is analogous to the process of cutting metals with a cutting tool.

3. The angle of incidence of particles on a surface at which maximum wear occurs depends on the elastic properties of the abrasive particle and of the body being worn, and on the degree of roughness of their surfaces, which determines the coefficient of friction between them. In all cases this angle cannot be greater than 45°.

4. For cylindrical surfaces (tube bundles of steam boilers), wear is expressed proportionally to the cube of the velocity of the incident flow and to the first power of the ash concentration and the time of action of the flow.

5. Worn parts should have a matte surface with alternating protrusions and depressions, which corresponds to the actual appearance of worn surfaces.

6. The method we have adopted for investigating wear, supplemented by experimental determination of the coefficients, opens broad possibilities for obtaining calculation formulas for various cases of ash wear.

References

1. Zalogin N. G. Combating abrasive wear of power-station equipment. “Electric Stations” No. 2, 1950.

2. Kilchevskii N. A. Theory of the impact of solid bodies. Gostekhizdat. Moscow–Leningrad, 1949.

3. Kuznetsov V. D. Physics of the Solid Body. Vol. III, Krasnoe Znamya Publishing House. Tomsk, 1944.

4. Kuznetsov V. D. Physics of the Solid Body. Vol. IV, Poligrafizdat Publishing House. Tomsk, 1947.

5. Lokshin V. A. Reducing the intensity of ash wear of heating surfaces of boiler units. “Izv. VTI” No. 7, 1947.

6. Loitsyanskii L. G. and Lur’e A. I. Theoretical Mechanics. Part Two. Dynamics. Gostekhizdat. Moscow–Leningrad, 1933.

7. Syrkin S. N. Ash wear of tubular heating surfaces. Collection of materials on the exchange of experience in power systems. Glavuralenergo, issue 1, Metallurgizdat, Moscow–Sverdlovsk, 1945.

Noted Misprints