ANALYSIS OF SEVERAL METHODS FOR STARTING ELECTRIC CONTACT WIRE HEATING INSTALLATIONS

V. M. RIKKONEN, Yu. S. MELNIKOV

(Presented to the scientific seminar of the Department of Automation and Telemechanics)

At present, the method of electric contact heating of parts is becoming increasingly widespread in industry; this is especially true of the heating of moving wire. This method of heating has a number of advantages over previously used methods, the main ones being the creation of a continuous technological process with higher energy and economic indicators, the possibility of automation, and the combination of several technological operations.

Examples of such heating are: wire patenting and hot drawing, used at metalworking plants, as well as annealing of wire after cold drawing, used at cable plants.

Despite the importance of the theoretical study of electric contact heating installations, insufficient attention is paid to this question, and installations are built on the basis of experience and experiment.

For the proper design and operation of such installations, it is necessary first of all to reveal the regularities of electric heating of a moving wire, and then to find algorithms for controlling them.

A special case for installations is the process of starting them or accelerating them, which is a transient process from one state to another steady state. The present article is devoted to this question—to consider several possible starting methods.

The starting process of an electric contact heating installation is important because, on the one hand, in existing installations normal wire quality is not ensured over fairly long lengths (up to 150 m; either underheating or overheating occurs), while on the other hand, if the starting process is ensured, then the steady-state regime will also be of good quality.

In studying the process of heating a moving wire in the steady-state regime [1], an expression was obtained that relates the main parameters of the installation:

\[ U = c \cdot \gamma \cdot \Theta_k \sqrt{ \frac{v \cdot L \cdot K_1} {\ln\left(1+\dfrac{\Theta_k}{\tau_0+\tau_n}\right)} }. \tag{1} \]

where \(U\) is the voltage on the heated section of the wire;
\(c\) is the specific heat capacity of the wire metal;
\(\gamma\) is the specific weight of the wire;
\(\Theta_k\) is the final overheating temperature of the wire in the section;
\(v\) is the speed of motion of the wire;
\(L\) is the length of the heated section;

\[ K_1=\frac{1}{c\cdot\gamma\cdot\eta_n(\tau_0+\tau_n)}; \]

\(\tau_0 = 235^\circ\mathrm{C}\) — for copper wire;
\(\tau_n = 20^\circ\mathrm{C}\) — the initial value of the wire temperature;
\(\eta_n\) is the specific electrical conductivity of the wire at \(\tau=\tau_n\).

Expression (1) represents the operating algorithm of the automatic control system for heating a moving wire. It follows from (1) that, in order to obtain a constant overheating temperature of the wire, it is necessary to maintain a constant relationship between the speed of motion of the wire, the length of the section between the contacts, and the voltage.

Consequently, in the general case three control principles may be adopted for designing installations: the first is varying the voltage on the heated section of the wire as a function of the speed of its motion at an unchanged section length; the second is varying the length of the heating section as a function of the ratio of the voltage and the speed of motion of the wire; the third is varying the speed of motion as a function of the voltage on the heating section at an unchanged section length. The third method of control, which resembles the first but differs from it in that the controlled quantity is the speed rather than the voltage, may be disregarded because of the difficulty of practical implementation and its doubtful expediency.

Since, when the installation is started, it is accelerated, which means nothing other than a change in the speed of motion of the wire, it is appropriate to apply the first method of control.

According to the first variant, the voltage on the heated section of the wire must vary as a function of the speed of its motion according to the law determined by expression (1), regardless of the diameter of the wire. If it is taken into account that the voltage in the supply network usually varies, then it is necessary to have a stabilized and controlled voltage source.

An automatic control system constructed according to the second variant must consist of a calculating device with a control algorithm obtained from expression (1)

\[ L=\frac{\ln\left(1+\dfrac{\Theta_k}{\tau_0+\tau_n}\right)} {K_1\cdot c^2\cdot\gamma^2\cdot\Theta_k^2}\cdot\frac{U^2}{v} =K\cdot\frac{U^2}{v}. \tag{2} \]

and of a tracking system for changing the length of the section as a function of the output quantity of the calculating device.

The advantage of the second variant of regulation is that no stringent requirements are imposed on the installation power source; therefore, as such a source for supplying a group of devices for electric contact heating of wire, one powerful unregulated voltage source may be used. In this case the technical-and-economic indicators of the installations will be higher than those of installations according to the first variant.

Experimental tests of an installation for electric contact heating of moving wire, including the starting period, constructed according to the first

for the variant of the automatic-control algorithm showed that the quality of the wire meets the requirements of the GOST standard.

Tests of the control system built according to the second variant give the same results.

However, both of the automatic-control methods considered are incapable of ensuring normal heating of the so-called “filling” section of the wire, i.e., the segment of wire which, at the moment the installation is started and voltage is switched on, is located between the current-supplying contacts. These ends reach a considerable length, and therefore they must also be heated to the required quality.

If the stationary section of wire is preheated to the specified temperature before the installation is accelerated, and the installation is then started, the “filling” end will overheat; if no preliminary heating is done, then some part (the front part) of this section will prove to be underheated.

According to data in the literature, in existing electric-contact heating installations normal heating of the “filling” end is not obtained.

In addition, insufficient heating of the initial section of wire when electric-contact heating installations are started is explained by electrical and thermal transient processes in the system consisting of the power source and the heated section of wire, and the length of the underheated wire increases still more.

The problem of normal heating of wire when starting installations can be solved in various ways, in particular as proposed by engineers Yu. Kashin, I. Gelfand, and G. Kulachenkov.

The essence of the proposal is as follows (Fig. 1): before the heated wire is accelerated, a temperature distribution along the length of the wire between the current-supplying contacts is provided that is the same as

Fig. 1. Section of wire heating in an electric-contact installation: 1 — “hot” contact; 2 — “cold” contact (at the moment of movement); 3 — “cold” contact (working position); 4 — heated wire.

at steady-state speed. This is achieved by moving the “cold” contact (2) at a definite speed from the “hot” contact (1) to its working position (3). In this case the voltage is varied in such a way that the current flowing through the wire remains constant and corresponds to the speed of movement of the “cold” contact.

At the moment when the “cold” contact occupies its working position, the temperature distribution along the length between the contacts will be the same as in the steady-state process. It is not difficult to see that here the principle of relative motion is applied: it is not the wire that moves, but the roller. What will happen subsequently to the wire remains unclear.

The authors of the proposal do not establish according to what law the voltage between the contacts should vary, how the installation is to be accelerated, and how the wire speed will change in the process. This article analyzes the proposed starting method in order to reveal its shortcomings and to obtain recommendations for the proper design of such devices. First of all, we shall determine the required law of variation of the voltage between the contacts during the movement of the “cold” contact.

The law of variation of the excess temperature of a wire element through which a current of density $\sigma$ is passed has the form [1]:

\[ \Theta = (\tau_0+\tau_{\mathrm{н}})\left(e^{K_1\cdot\sigma^2\cdot t}-1\right)+\Theta_{\mathrm{н}}\cdot e^{K_1\cdot\sigma^2\cdot t}, \tag{3} \]

where $\Theta_{\mathrm{н}}$ is the initial excess temperature of the wire element above the ambient medium (it may be assumed that $\Theta_{\mathrm{н}}=0$).

The resistance of the wire element changes with time according to the law:

\[ R_{\Delta l}=R_{0\Delta l}(1+\alpha\Theta) =R_{0\Delta l}\left[1+\alpha(\tau_0+\tau_{\mathrm{н}})\cdot\left(e^{K_1\sigma^2\cdot t}-1\right)\right], \tag{4} \]

where $\alpha$ is the temperature coefficient of resistance of the material;
$R_{0\Delta l}$ is the resistance of the wire element at $\Theta=0$.

Transform expression (4), taking into account that

\[ \alpha=\frac{1}{\tau_0+\tau_{\mathrm{н}}}, \]

\[ R_{\Delta l}=R_{0\Delta l}\cdot e^{K_1\cdot\sigma^2\cdot t}. \tag{5} \]

Relations (3)—(5) are applicable both to the case of wire motion with stationary contacts and to the case of motion of the “cold” contact with the wire stationary.

At a constant speed of motion of the wire or of the contact, the heating time of an element is proportional to the distance $l$ of this element from the “cold” contact,

\[ t=\frac{l}{v}. \]

Then the total resistance of a section of wire of length $l$ is equal to

\[ R_l=\int_0^l \frac{\rho_0}{s}\cdot e^{K_1\cdot\sigma^2\cdot\frac{l}{v}}\cdot dl = \frac{\rho_0\cdot v}{s\cdot K_1\cdot\sigma^2} \cdot\left(e^{K_1\cdot\sigma^2\cdot\frac{l}{v}}-1\right). \tag{6} \]

The required law of variation of the voltage over the heating section during motion of the “cold” contact with speed $v$ is:

\[ U=\sigma\cdot s\cdot R = \frac{v\cdot\rho_0}{K_1\cdot\sigma} \cdot\left(e^{K_1\cdot\sigma^2\cdot\frac{l}{v}}-1\right). \tag{7} \]

The expression for the current density [1] is:

\[ \sigma=\sqrt{\frac{v}{L\cdot K_1}\cdot \ln\left(1+\frac{\Theta_{\mathrm{k}}}{\tau_0+\tau_{\mathrm{н}}}\right)}. \tag{8} \]

Substituting the value $\sigma=\mathrm{const}$ into (7), we obtain

\[ U= \frac{v\cdot\rho_0} {K_1\cdot \sqrt{\frac{v}{L\cdot K_1}\cdot \ln\left(1+\frac{\Theta_{\mathrm{k}}}{\tau_0+\tau_{\mathrm{н}}}\right)}} \cdot \left[ e^{\frac{l}{L}\cdot \ln\left(1+\frac{\Theta_{\mathrm{k}}}{\tau_0+\tau_{\mathrm{н}}}\right)} -1 \right]. \tag{9} \]

This expression represents the interrelationship between the voltage, the speed of motion of the “cold” contact, the distance between the contacts \(l\), and the temperature of the final overheating of the wire on the section.

In calculating the curves \(U = f(l)\) (Fig. 2), the initial data adopted are those for the case of recrystallization annealing of copper wire [1]:

Fig. 2. Voltage dependences on the heated section of wire as a function of the distance between contacts for different speeds of motion of the “cold” contact.

\[ \Theta_k = 580^\circ\mathrm{C}; \qquad L = 3.76\ \mathrm{m}; \qquad K_1 = 1.97 \cdot 10^{-5}; \]

\[ \rho_0 = \frac{1}{56}\ \Omega \cdot \mathrm{mm}^2/\mathrm{m}; \qquad (\tau_0 + \tau_n) = 255^\circ\mathrm{C}. \]

The speed was varied from \(12.2\ \mathrm{m/sec}\) to \(1.0\ \mathrm{m/sec}\). In these calculations we neglected heat transfer.

From Fig. 2 it is evident that the lower the speed, the straighter the dependence of voltage on length. If the movable contact is moved very slowly, it is possible to select a steady thermal regime that will depend mainly on the heat transfer of the wire when it is in the stationary state. When the roller assumes its working position, the installation can be started from zero speed. Subsequent control must be carried out according to the first control principle, i.e., by changing the voltage as a function of speed.

If, however, the roller moves sufficiently rapidly, then the voltage must be changed in accordance with expression (9). When the roller reaches the working position, (3) must be such that the wire at that moment, i.e., almost instantaneously, acquires the speed of the moving roller when the roller stops. Subsequent control must again be carried out by voltage as a function of speed.

It is necessary to verify whether constancy of the temperature of the final section of the wire, which is the principal indicator of the quality of the process, is ensured during acceleration of the electric-contact installation and variation of the voltage on the heated section according to expression (1).

Since the overheating temperature \(\Theta\) is related to the current density by dependence (3), one can determine the required law of variation of the dens—

current density on the heating section during acceleration of the wire, in order to ensure constancy of \(\Theta_k\).

Consequently, if during acceleration the distribution of heating temperature along the length of the section is taken as unchanged—which also means constancy of the total active resistance when the voltage is varied according to expression (1)—the current density must be maintained as a function of speed according to expression (8).

Figure 3 shows this dependence \(\delta = f(v)\). The determination of the overheating temperature of individual points of the wire at the moment of acceleration was carried out by the method of mathematical modeling on an analog computing machine, MNB-1.

Fig. 3. Dependence of the heating current density on the speed of wire motion.

Figure 4 shows a block diagram of the model of the heated section. The procedure for compiling such a block diagram of the model and the purpose of the individual elements of the circuit are given in [2]. For simplification, the law of variation of the wire speed was taken to be linear.

Fig. 4. Block diagram of the mathematical model of the heated section of wire.

On the model it was obtained that the final heating temperature of individual points of the wire during its acceleration, while observing the required law—

the end \(U = f(v)\) remains constant and equal to the specified value (\(\sim 580^\circ\mathrm{C}\)), i.e., all points of the “feeding” end of the wire are heated to the same temperature.

The adopted control algorithm is valid for obtaining high-quality heating of the wire both during the startup period of the installation and in normal operation in the event of a deviation of the speed from the normal value.

Conclusions

1. To ensure high-quality heating of the wire during the startup period, including the “feeding” end, the proposal of the group of engineers (Kashina et al.) may be used, with the recommendations of items 2 and 3.

2. The speed of movement of the “cold” contact may be different, but constant during the startup period; for this it is necessary to maintain the relationship between the voltage and the position of the contact (the length of the section traversed), according to Fig. 2.

During the period when the contact is stopped, the wire must instantaneously acquire the speed of the contact.

1. The automatic control system for wire heating must be built on the principle of changing the voltage as a function of the speed of its movement.

References

1. V. M. Rikkonen, Yu. S. Melnikov. Calculation of the parameters of electric annealing of a moving wire, Elektrotekhnika, No. 9, 1964.

2. V. M. Rikkonen, Yu. S. Melnikov. Investigation of the process of heating a moving wire using mathematical modeling, Izvestiya TPI, vol. 141, 1966.