Study of the Heating Process of a Moving Wire Using Mathematical Modeling

V. M. Rikkonen, Yu. S. Melnikov

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

At present, in the cable industry much attention is being paid to the creation of continuous technological processes. One of the questions requiring solution in this connection is the annealing of wire. Under the new technology, steel annealing is carried out by briefly passing an electric current of large magnitude through the wire. However, the theory of heating a moving wire has not yet been developed sufficiently fully. In the authors’ work [1] an analysis is given of the dynamics of wire heating in the steady-state regime, that is, at established values of the wire speed, the annealing current, and constant lengths of the annealing sections. Expressions were obtained for the superheating of the wire above the ambient medium, the law for regulating the annealing regime was determined, and the principal factors influencing the heating process were identified. On this basis, the main parameters of an installation for electric annealing of wire were determined: the lengths of the annealing sections and the annealing voltages at different wire speeds.

Since the annealed section of wire is an object of regulation, it is necessary to determine its dynamic properties, that is, the law of variation of the final temperature of the wire when voltage is applied to it.

The first attempt to obtain analytical expressions for the final temperature of the wire was made by K. D. Kolesnikov in [2]. However, in the analysis the author passes from the heat-balance equation for an elementary link of the wire to the heat-balance equation for the entire annealed section. In doing so, naturally, the temperature of the elementary link is replaced by the average temperature of the wire over the entire section. But the quality of wire annealing is determined by the final temperature at the outlet of the annealing section, while the temperature distribution along the length of the annealed section is nonlinear in character [1]; consequently, the average temperature of the wire in the section cannot be regarded as the principal characteristic of annealing quality.

The present work is devoted to determining the actual law of variation of the final temperature of the wire with time when voltage is applied to it, that is, to determining the quality curve of the transient process.

To establish this law, it is first of all necessary to know the character of the change in current density in the two arms of the annealed section of wire, which, as shown in [1], is the principal quantity determining the degree of annealing of the wire. For this it is necessary to determine the—

variation of the values of the resistance of both annealing sections, integrated over length, with time. An analytical solution of the problem is impossible, since an equation of thermal equilibrium taking into account the actual temperature for the entire annealed section cannot be formulated, because even in the established regime, with
\(\sigma_1=\mathrm{const}\), \(\sigma_2=\mathrm{const}\), \(L_1=\mathrm{const}\), \(L_2=\mathrm{const}\), \(v=\mathrm{const}\), all points of the wire are under different conditions (a nonlinear temperature distribution along the length of the annealing section). Solving the problem by numerical methods, however, requires a large expenditure of time and a large amount of graphical work.

Therefore the authors chose a path that makes it possible to take into account the operating features of the constructed installation for the electrical annealing of wire. On an installation operating under production conditions, curves of the change in annealing current density in the two arms were oscillographically recorded when a voltage of constant amplitude was applied; these are presented in Fig. 1. The character of the curves \(\sigma_1(t)\) and \(\sigma_2(t)\) is determined mainly by the lengths of the annealing sections and by the speed of motion of the wire.

Fig. 1. Change in current densities in the two annealing arms.

The established values of the current densities in both sections as a function of the annealing voltage are determined from the formulas derived in [1]:

\[ \sigma_1=\sqrt{\frac{v}{L_1\cdot k_1}\cdot \ln\left(1+\frac{\Theta_{k1}}{\tau_0+\tau_{\mathrm{H}1}}\right)}, \tag{1} \]

\[ U=\frac{v\cdot \gamma \cdot c \cdot \Theta_{k1}}{\sigma_1}, \tag{2} \]

\[ \frac{L_1}{L_2}= \frac{ \Theta_{k2}^{2}\cdot \ln\left(1+\frac{\Theta_{k1}}{\tau_0+\tau_{\mathrm{H}1}}\right) }{ \Theta_{k1}^{2}\cdot \ln\left(1+\frac{\Theta_{k2}}{\tau_0+\tau_{\mathrm{H}2}}\right) }, \tag{3} \]

\[ \sigma_2=\sqrt{\frac{v}{L_2\cdot k_1}\cdot \ln\left(1+\frac{\Theta_{k2}}{\tau_0+\tau_{\mathrm{H}2}}\right)}. \tag{4} \]

The adopted notation is:

\[ k_1=\frac{1}{c\gamma\cdot \eta_{\mathrm{H}}(235+\tau_{\mathrm{H}})}; \tag{5} \]

$\sigma_1$ and $\sigma_2$ are the current densities in the first and second sections;

$L_1$ and $L_2$ are the lengths of the first and second sections;

$\Theta_{k1}$ and $\Theta_{k2}$ are the final overheating of the wire above the ambient medium in the first and second sections;

$\tau_{n1}$ and $\tau_{n2}$ are the initial temperature of the wire in the first and second sections;

\[ \tau_{n2} = \tau_{n1} + \Theta_{n2}; \]

$U$ is the annealing voltage;

$v$ is the speed of motion of the wire;

$\gamma$ is the specific weight of the wire material;

$c$ is the specific heat capacity of the wire material;

$\eta_n$ is the specific electrical conductivity of the wire material at

\[ \tau = \tau_n. \]

Figure 2 gives the dependences $\sigma_{1\,\mathrm{st}} = f(U)$ and $\sigma_{2\,\mathrm{st}} = f(U)$ for $L_1 = 2.47$ m, $L_2 = 1.29$ m, $v = 12.2$ m/s for copper wire.

Fig. 2. Dependences of the steady-state values of current densities on annealing voltages: a) for the second annealing section; b) for the first annealing section.

To determine the final overheating of the wire for a known law of variation of the annealing-current densities in time, we use the heat-balance equation for an elementary section of the wire in the following form [1]:

\[ \frac{\sigma^2}{\gamma \cdot \eta_g} \cdot dt = c \cdot d\Theta, \tag{6} \]

where

\[ \eta_g = \frac{\eta_n (235 + \tau_n)} {(235 + \tau_n + \Theta)}; \tag{7} \]

$\eta_g$ and $\eta_n$ are, respectively, the specific electrical conductivity of the wire material in the hot and cold states;

$\Theta$ is the current overheating of the wire above the ambient medium.

Rewrite expression (6) taking (7) into account:

\[ dt \frac{\sigma^2}{\gamma \cdot \eta_{\mathrm{h}}(235+\tau_{\mathrm{h}})} \cdot (235+\tau_{\mathrm{h}}+\Theta)=c \cdot d\Theta , \tag{8} \]

Taking (5) into account, we obtain

\[ \sigma^2 k_1(235+\tau_{\mathrm{h}}+\Theta)=\frac{d\Theta}{dt}. \tag{9} \]

We solve the problem on a modeling installation of the MNB-1 type. The setup of the problem is divided into two stages: the first is the setup of the dependences \(\sigma_1=f(U,t)\) and \(\sigma_2=f(U,t)\), and the second is the setup of differential equation (9) for two annealing sections. The complete structural diagram of the model is shown in Fig. 3.

Links 1 and 2, modeling the dependences \(\sigma_1=f(U,t)\) and \(\sigma_2=f(U,t)\), are constructed by the second method of setting up the problem, i.e., according to the structural diagram of the links.

Fig. 3. Structural diagram of the model.

According to the known method [3], we determine the transfer functions of the links whose transient processes are presented in Fig. 1: for the first annealing section

\[ W_{1(p)}=\frac{\sigma_{1(p)}}{U_{(p)}}=k'_1 \frac{0.0981p+1}{0.0644p+1}; \]

for the second annealing section

\[ W_{2(p)}=\frac{\sigma_{2(p)}}{U_{(p)}}=k'_2 \frac{0.118p+1}{0.06p+1}, \]

where

\[ k'_1=\frac{\sigma_{1\,\mathrm{st}}}{U}; \qquad k'_2=\frac{\sigma_{2\,\mathrm{st}}}{U}. \]

The coefficients \(k'_1\) and \(k'_2\) are introduced into the structure of the model by nonlinear blocks NB\(_1\) and NB\(_3\), which reproduce the dependences \(\sigma_{1\,\mathrm{st}}=f(U)\) and \(\sigma_{2\,\mathrm{st}}=f(U)\).

According to equation (9), for the two annealing sections we compose links 3 and 4, giving the dependences: $\Theta_1=f(\sigma_1,t)$ and $\Theta_2=f(\sigma_2,\tau_{\mathrm{н}2},t)$.

A special feature of the electrical annealing of wire in two sections is that the initial temperature of the wire in the second section depends on the final temperature in the first section. In [1] it is shown that the heat transfer of the wire during electrical annealing may be neglected; therefore we take $\Theta_{\mathrm{н}2}=\Theta_{\mathrm{к}1}$, and in the model we feed the output of block 4 as the initial condition to the input of block 9.

The adopted scales for setting up the model are:

\[ M_t=0{,}05;\quad M_u=0{,}5;\quad M_\sigma=9{,}18; \]

\[ M_\Theta=10;\quad M_\nu=8420;\quad M_{\sigma^2\Theta}=5{,}05\cdot10^6. \]

The values of the model coefficients, determined from the adopted scales and the installation parameters, are:

\[ R_{11}=0{,}844;\quad R_{12}=1{,}0;\quad R_{21}=1{,}0;\quad R_{31}=0{,}267; \]

\[ R_{32}=0{,}51;\quad R_{41}=R_{91}=k_1\cdot M_{\sigma^2}\cdot M_t\cdot100=0{,}834; \]

\[ R_{42}=R_{92}=k_1(235+\tau_{\mathrm{н}})\frac{M_{\sigma^2}\cdot M_t}{M_\Theta}=0{,}213; \]

\[ R_{51}=1{,}07;\quad R_{52}=1{,}0;\quad R_{61}=1{,}0; \]

\[ R_{71}=0{,}41;\quad R_{72}=0{,}454;\quad R_{81}=1{,}0. \]

Since the equation of heat balance is compiled for an elementary section, the model provides switching:

a) at the moment when the elementary section enters the annealing zone;
b) at the moment when the section passes from the first annealing arm into the second;
c) “freezing” of the solution at the moment when the section leaves the annealing zone.

For this purpose, three auxiliary control circuits were used, each of which consists of an operational amplifier operating in integration mode and a block of special nonlinearities operating in a programmed mode (Fig. 4). The output element of the control circuits is a relay, whose contacts perform switching in the main model of the problem.

Fig. 4. Auxiliary model-control circuit.

Relay $P'$ serves to determine the dependence $\Theta=f(U,t)$ for points of the wire that, at the moment the voltage is switched on, are located before the annealing section. Its time delay corresponds to the time of motion of the point under study up to the beginning of the first annealing section. Relay $P''$ (its contacts are not shown in Fig. 3) transfers block 9 from the “initial conditions” mode to the “integration” mode at the moment when the point passes from the first annealing section to the second. Relay $P'''$ (its contacts are also not shown in Fig. 3) serves to “freeze” the result of solving the problem. The time delays of all relays are strictly coordinated for each point under study. By the adopted method one can determine the heating curve of any point of the wire when voltage is applied to the annealing sections.

Figure 5 presents the curves $\Theta=f(t)$, obtained on the model for two points of the wire when a voltage $U=26$ V is applied to the annealing sections.

Curve 1 belongs to the point of the wire that, when the voltage is switched on, is heated to the very highest temperature; curve 2 belongs to the point that enters the annealing section after the completion of the transient processes of the current densities and of the corresponding established annealing regime. Curve 2 can also be regarded as the distribution of overheating along the length of the annealed section of the wire in the steady-state regime with unchanged installation parameters.

In Fig. 6 are shown curves of the change in the final temperature of the wire when three values of the annealing voltage are applied, constructed from the family of curves \(\Theta = f(t)\) obtained on the model for different points of the wire.

As shown above, the value of the final temperature of the wire is the principal characteristic determining the quality of annealing; therefore, on the basis of Fig. 6 the following conclusions can be drawn.

Fig. 5. Change in the overheating of two points of the wire when a constant annealing voltage is applied \(U = 26\) V: 1) for the point located at \(t = 0\) at the beginning of the annealing section; 2) for the point located at \(t = 0\) 3 meters from the beginning of the annealing section.

The application to the wire of a constant voltage, even less than the nominal value (\(U_{\text{nom}} = 26\) V), at an unchanged speed of its motion leads to scrap. Thus, at \(U = 24\) V, \(\Theta_{\text{final,max.}} = 720^\circ\mathrm{C}\), whereas according to technological requirements and experimental data, for quali—

Fig. 6. Change in the final overheating of the wire over time when constant voltages are applied: 1) \(U = 28\) V; 2) \(U = 26\) V; 3) \(U = 24\) V.

for proper annealing, the final overheating of the wire must be no higher than 620°C.

At the moment the voltage is switched on, the temperature of the wire over the entire length of the section being annealed rises sharply; the wire elongates and falls off the contact rollers.

The steady-state regime is reached after 0.49 sec, which at \(v = 12.2\) m/sec corresponds to 6 m of wire.

It follows from the foregoing that:

a) the application of mathematical modeling to the study of the thermal regime during electric annealing of wire makes it possible, in a sufficiently simple way, to determine the temperature of the wire in steady-state and transient regimes;

b) in order to eliminate unannealed ends and local burn-through of the wire, the annealing voltage must be switched on at the moment the machine is started and must vary according to the law established in [1].

REFERENCES

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

2. K. D. Kolesnikov. Theory and basic principles of constructing systems for automatic control of combined processes of drawing and annealing with alternating current. In the collection Comprehensive Mechanization and Automation of Cable Production, based on materials of the 2nd All-Union Conference, Tashkent, 1962.

3. I. N. Pechorina. Calculation of Automatic Control Systems. Moscow–Sverdlovsk, Mashgiz, 1962.