Cooling of Particles in the Spray Cone During Electric Metallization by Spraying
Abstract Generated abstract
This paper examines the temperature decrease of metal particles in the spray cone during electric arc spray metallization, a problem complicated by very small particle sizes, short flight times, and inconsistent experimental measurements. The authors formulate a theoretical cooling calculation for steel spherical particles, treating radiative heat loss as dominant and applying heat-transfer similarity criteria to particles of several diameters over flight times of about 0.002 to 0.003 seconds. The resulting temperature curves are compared with published experimental values and are used to show that discrepancies among earlier measurements can be partly explained by differences in particle diameter and flight duration under varying metallization conditions. The authors present the method as a practical first approximation, noting that latent heat of crystallization was not included.
Full Text
COOLING OF PARTICLES IN THE SPRAY CONE DURING ELECTRIC ARC SPRAY METALLIZATION
A. F. KRUGIN, G. P. BOIKOV
(Presented by Prof. A. N. Dobrovidov, Doctor of Technical Sciences,
and Prof. G. I. Fuks, Doctor of Technical Sciences)
The study of the temperature of particles in the spray cone during spray metallization is of great practical importance and is of theoretical interest.
Initial information on the temperature of particles at the moment of their impact on the surface being metallized was purely qualitative in character. Only later was the temperature of particles in the spray cone determined experimentally by the calorimetric method [1, 2], with the aid of an optical pyrometer [3 and others], by a thermocouple [4], and from the shape of the particles after their impact on a polished steel plate [5]. The difficulty of solving the problem is connected with the fact that, according to data from various investigators, the sizes of metal particles range from 0.001 to 0.015 mm [2, 6], and their flight time from the melting zone to the surface of the article being metallized varies from 0.002 [6] to 0.003 sec [7].
Experimental methods for studying the temperature of particles in the spray cone make it possible to determine only the average temperature of the metallic particles, without indicating their size. The measurement results obtained by various authors are contradictory, as is shown by the data in Table 1; therefore, researchers’ opinions on the mechanism of layer formation and on the nature of the adhesion forces of particles in metallized layers also differ.
Table 1
| Author and source | Method of temperature measurement | Material | Metallization method | Temperature of particles on impact |
|---|---|---|---|---|
| N. V. Kats [2] | Calorimetric | Steel | Electric | \(0.55—0.80\,t_{kr}^{1)}\) |
| A. F. Troitskii [1] | Calculated-calorimetric | Steel | Electric | \(0.85—0.95\,t_{pl}^{1)}\) |
| G. A. Ioannidi [3] | Optical | Steel | Electric | \(820—1167^\circ\mathrm{C}\) |
| L. V. Krasnichenko [5] | From particle shape | Steel | Electric | \(t_{pl}\) |
| T. Terston [4] | Thermocouple | Various materials | Gas | \(t_{pl}\) |
We have attempted to solve this question theoretically for any sizes of metal particles occurring in electric arc spray metallization. As initial data, we used the values available in the literature for particle sizes, particle flight times, and also the values of the thermophysical characteristics of the materials.
\(^{1)}\) \(t_{kr}\) — crystallization temperature; \(t_{pl}\) — melting temperature of the metal.
Physical picture of the process
Under the action of the heat of the voltaic arc in the metallizing gun, the steel electrodes are melted; their metal serves as the material for the required coating. At the moment of melting, the liquid mass of metal is atomized by a free jet of air into the finest particles, which, under the action of surface-tension forces, tend to assume the shape of a sphere. A liquid particle, having flown out of the melting zone of the metallizing gun, begins to cool sharply, mainly owing to radiation.
Formulation of the problem
A steel sphere of diameter \(2R=5\) microns at temperature \(t_0 \cong 1550^\circ\mathrm{C}\) at the initial instant of time is placed in air space and cools during the flight time \(\tau_{\text{fl}}=0.002—0.003\) sec.
For calculating the cooling of the steel sphere we adopt the following data [8]:
coefficient of thermal conductivity \(\lambda=25\ \text{kcal}/\text{m}\,\text{h}\,\text{deg}\),
coefficient of thermal diffusivity \(a=0.025\ \text{m}^2/\text{h}\),
radiation coefficient of the material \(C=4.2\ \text{kcal}/\text{m}^2\text{h}\,^\circ\mathrm{K}^4\).
We neglect the influence of convection on the cooling process of the sphere.
Basic notation
\[
\Theta_m=\frac{T_m}{T_c}
\]
— criterion of the dimensionless variable temperature;
\[
\Theta_0=\frac{T_0}{T_c}
\]
— criterion of the dimensionless initial temperature;
\[
K_i=\frac{q_c\cdot R}{\lambda\cdot T_c}
\]
— Kirpichev criterion;
\[
F_{01}=\frac{a\cdot\tau_1}{R^2}
\]
— Fourier criterion;
\[
Q_{ci}=\frac{q_{ci}}{q_c}
\]
— criterion of the dimensionless radiant flux at the surface of the body
\(R\) — radius of the metallic particle, m;
\(\tau\) — time from the beginning of cooling, sec;
\(\tau_1\) — calculation time interval, h;
\(T\) — absolute temperature of the body, \({}^\circ\mathrm{K}\);
\(t\) — temperature of the body on the centigrade scale, \({}^\circ\mathrm{C}\);
\(m\) — number indicating how many time intervals \(\tau_1\) have passed since the beginning of cooling;
\(\mu_n\) — characteristic numbers obtained from the equation \(\tan\mu=\mu\) [9].
In order to carry out the required calculation, we shall use the expression for the heating of a “thin” sphere [10, p. 89], which may be represented in criterial form
\[ \Theta_m=\Theta_0+3\cdot K_i\cdot F_{01}\cdot\sum_{i=1}^{i=m} Q_{ci}. \tag{1} \]
Since the Kirpichev, Fourier, and dimensionless radiant-flux criteria are characteristics of both heating and cooling, then, changing
changing the sign before carrying out these criteria to the opposite one, we reduce the heating problem to a cooling problem. Then expression (1) is rewritten as follows:
\[ \Theta_m=\Theta_o-3.K_i.F_{01}\cdot \sum_{i=1}^{i=m} Q_{ci}. \tag{2} \]
Replacing the similarity criteria by the corresponding complexes and complexes, for calculating the cooling of a sphere we obtain the final formula:
\[ t_m=t_o-3\cdot \frac{a\cdot \tau_1}{\lambda\cdot R}\cdot \sum_{i=1}^{i=m} q_{ci}, \tag{3} \]
where
\[ q_{ci}=C\cdot \left(\frac{T_{i-1}}{100}\right)^4 - \tag{4} \]
is the radiant flux from the surface of the sphere; \(\tau_1\) is the computational time interval, determined with the aid of the formulas [10, pp. 63 and 100]
\[ \tau_1=\frac{0.0155\cdot T_o\cdot \lambda\cdot R}{q_{ci}\cdot a} \tag{5} \]
and
\[ \varphi_m< \frac{5\cdot(q_{c1}-q_{c2})} {q_{c1}\cdot\left(3+2\sum_{n=1}^{\infty} e^{-\mu_n^2\cdot \frac{a\cdot \tau_1}{R^2}}\right)}, \tag{6} \]
here \(\varphi_m\) is the actual error of the calculation.
Using relations (3–6), we carry out the following operations:
- Determination of the computational time interval.
According to formula (5), \(\tau_1=15.2\cdot 10^{-8}\) h.
- Determination of the temperature of the sphere after the expiration of the first computational time interval \(\tau_1\).
According to formula (3),
\[ t_1=t_o-3\cdot \frac{a\cdot \tau_1}{\lambda\cdot R}\cdot q_{c1}=1465^\circ\mathrm{C}. \]
Then the radiant flux acting during the second computational time interval \(\tau_1=15.2\cdot 10^{-8}\) h will be
\[ q_{c2}=C\left(\frac{t_1+273}{100}\right)^4 =384\cdot 10^3\ \frac{\mathrm{kcal}}{\mathrm{m}^2\cdot \mathrm{h}}. \]
- Estimation of the approximation of the calculation.
Substituting the corresponding numerical values into inequality (6), we obtain \(\varphi_m<0.29\). Thus, the computational error for the selected computational time interval \(\tau_1=15.2\cdot 10^{-8}\) h is less than \(29\%\).
Wishing to carry out the calculation with a higher degree of accuracy, we assign a smaller value of the computational time interval, \(\tau_1=3.47\cdot 10^{-8}\) h. Then
\[ t_1=t_o-3\cdot \frac{a\cdot \tau_1}{\lambda\cdot R}\cdot q_{c1}=1530^\circ\mathrm{C}; \]
\[ q_{c2}=C\left(\frac{t_1+273}{100}\right)^4 =446\cdot 10^3\ \frac{\mathrm{kcal}}{\mathrm{m}^2\cdot \mathrm{h}}. \]
Substituting into the right-hand side of inequality (6), instead of the corresponding expressions, their values, we obtain \(\varphi_m<0.043\), i.e., the computational error for the selected computational time interval is less than \(4.3\%\). Considering
considering such an error admissible, for further calculations we finally take \(\tau_1 = 3.47 \cdot 10^{-8}\) h.
4. Determination of the temperature of the sphere after the expiration of a second, identical interval of time from the start of heating.
According to formula (3),
\[ t_2 = t_o - 3 \cdot \frac{a \cdot \tau_1}{\lambda \cdot R}(q_{c1}+q_{c2}) = 1512^\circ \mathrm{C}. \]
Then the radiant flux acting during the third interval of time will be
\[ q_{c3} = C\left(\frac{t_2+273}{100}\right)^4 = 425 \cdot 10^3 \frac{\mathrm{kcal}}{\mathrm{m}^2 \cdot \mathrm{h}} . \]
Knowing the value of \(q_{c3}\), we determine the temperature at the end of the third such interval of time \(\tau_1\)
\[ t_3 = t_o - 3 \cdot \frac{a \cdot \tau_1}{\lambda \cdot R}\cdot(q_{c1}+q_{c2}+q_{c3}) = 1494^\circ \mathrm{C}. \]
Similarly we find \(t_4, t_5, t_6\), etc. The results of the calculation are plotted on the graph \(t=f(\tau)\) (Fig. 1). The results of similar calculations for particles with diameters of 1 and 15 microns are also given here.
Fig. 1. Change in particle temperature as a function of time
Vertical axis: \(t^\circ \mathrm{C}\).
Horizontal axis: time in seconds, \(\tau\).
Curves: \(\varnothing 15\,\mu\), \(\varnothing 5\,\mu\), \(\varnothing 1\,\mu\).
In order to determine the temperature of the particle under investigation at the end of its flight, we find on the abscissa axis the values \(\tau = 0.002\) sec and \(\tau = 0.003\) sec and from these points erect perpendiculars to their intersections with the corresponding curves \(t = f(\tau)\).
The calculation carried out for the temperature of particles in the spray cone permits the following conclusion to be drawn.
The temperature of the particles at the moment of their impact on the surface being metallized agrees well with the experimental data of various authors (Table 1 and Fig. 1).
The theoretical studies carried out make it possible in part to explain the discrepancies in the experimental data that occur in various works. From analysis of the curves it becomes evident that the results of the experimental data are strongly influenced both by the particle sizes and by their flight time, which, in turn, depend on the metallization conditions.
The calculation carried out is not final, since the latent heat of crystallization of the metal, which to some extent slows the rate of cooling of the particles, was not taken into account in the investigation.
The proposed method of investigation may be recommended for practical purposes as a first approximation.
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