Determination of the Magnitude of the Vulcanization Effect Obtained by a Cable Sheath in a Cooling Device
Abstract Generated abstract
This paper develops a method for estimating the additional vulcanization effect received by a rubber cable sheath while it passes through the cooling device after continuous vulcanization. Treating the sheath in first approximation as a homogeneous rubber cylinder with an initially parabolic temperature distribution, the author uses heat conduction relations with prescribed surface temperature to express the post-vulcanization effect as a dimensionless integral over the Fourier cooling criterion. By approximating the relevant cooling curves with straight lines over the practical cooling interval, the paper derives analytical formulas and graphical coefficients for calculating the effect under different operating conditions. The resulting dependence is intended to support assessment of cooling-device vulcanization, productivity improvements through reduced vulcanization-tube residence time, and selection of cooling-water temperature for more uniform sheath vulcanization.
Full Text
DETERMINATION OF THE MAGNITUDE OF THE VULCANIZATION EFFECT OBTAINED BY THE CABLE SHEATH IN A COOLING DEVICE
A. A. Tatarnikov
(Presented by the Department of ATPS)
Cables with a rubber sheath, manufactured on continuous-vulcanization units, after the vulcanization tube enter a cooling device. The principal purpose of the cooling device is to cool the cable sheath after vulcanization in order to give it the mechanical hardness necessary for winding the cable without damage onto the take-up drum.
However, during the cooling period the inner layers of the cable sheath undergo additional vulcanization. This process of post-vulcanization of the inner layers of the cable sheath can be characterized by the magnitude of the vulcanization effect \(\Delta E_{ox}\), equal to [1]
\[ \Delta E_{ox}=\int_{0}^{\tau_{ox}} K^{\frac{t(\tau)_{ox}-t_b}{10}}\,d\tau, \tag{1} \]
where \(t(\tau)_{ox}\) is the current value of the temperature of the cable sheath in the corresponding layer;
\(t_b\) is the vulcanization temperature of rubber specimens under laboratory conditions;
\(K=2\) is the temperature coefficient of vulcanization;
\(\tau_{ox}\) is the residence time of the cable in the cooling device.
The total magnitude of the vulcanization effect obtained at the calculated point of the cable sheath will be equal to
\[ E_a = E_t + \Delta E_{ox}, \tag{2} \]
where \(E_a\) is the specified magnitude of the vulcanization effect at the calculated point of the cable sheath;
\(E_t\) is the magnitude of the vulcanization effect obtained by the sheath in the vulcanization tube.
It follows from (2) that, owing to the magnitude \(\Delta E_{ox}\), it is possible to reduce \(E_t\), and thereby shorten the residence time of the cable in the vulcanization tube, i.e., increase the productivity of the unit.
To determine \(\Delta E_{ox}\) in the first approximation, let us assume that a homogeneous rubber cylinder is vulcanized in the vulcanization tube. At entry into the cooling device the temperature distribution over the cross section of the cylinder is parabolic.
We shall assume that cooling occurs under boundary conditions of the first kind, i.e., the temperature of the surface of the cable sheath is equal to the temperature of the cooling water, \(t_c=\text{const}\).
We take the cable velocity to be \(V_k=\mathrm{const}\). In this case the solution for cooling of the cylinder will have the form [2]
\[ \begin{aligned} t(r,\tau)-t_c &=(t_n-t_c)\sum_{n=1}^{\infty} A_n I_0\!\left(\mu_n\cdot\frac{r}{R}\right) \exp(-\mu_n^2\cdot \mathrm{Fo}_{o.x}) \\ &\quad -4(t_n-t_{\mathrm{c}})\sum_{n=1}^{\infty}\frac{A_n}{\mu_n^2}\cdot I_0\!\left(\mu_n\frac{r}{R}\right)\exp(-\mu_n^2\cdot \mathrm{Fo}_{o.x}), \end{aligned} \tag{3} \]
where \(t_n\) is the temperature of the surface of the cable sheath in the vulcanization tube, equal to the temperature of the heating saturated steam;
\(t_{\mathrm{c}}\) is the temperature at the center of the rubber cylinder at the moment it enters the cooling device.
Equation (3) consists of two parts. The first part of (3) characterizes the cooling of a homogeneous cylinder for a uniform initial temperature distribution over the cross section, equal to \(t_n\). The second part of (3) takes into account the initial nonuniform temperature distribution in the cylinder during the initial period of cooling.
The value of the temperature at the center of the cable \(t_{\mathrm{c}}\) depends on the time the cable remains in the vulcanization tube \(t_n\) and on the initial temperature distribution of the cylinder \(t_{\mathrm{n}}\) entering the vulcanization tube.
For values \(\mathrm{Fo}_{v}^{k}=0.3\) and higher, the value \(t_{\mathrm{c}}\) can be determined from the formula
\[ t_{\mathrm{c}}=t_n-(t_n-t_{\mathrm{n}})\, \frac{2}{\mu_1 I_1(\mu_1)}\cdot \exp(-\mu_n^2\cdot \mathrm{Fo}_{v}^{k}), \tag{4} \]
where \(t_{\mathrm{n}}\) is the initial temperature over the cable cross section;
\(\mathrm{Fo}_{v}^{k}\) is the value of the Fourier criterion for the time the cable remains in the vulcanization tube.
Substituting (4) into (3) and replacing the constants by their numerical values, we obtain
\[ \begin{aligned} t(r,\tau)-t_c &=(t_n-t_c)\sum_{n=1}^{\infty} A_n\cdot I_0\!\left(\mu_n\cdot\frac{r}{R}\right) \exp(-\mu_n^2\cdot \mathrm{Fo}_{o.x}) -6.4(t_n-t_{\mathrm{n}})\times \\ &\quad \times \exp(-5.76\cdot \mathrm{Fo}_{v}^{k}) \sum_{n=1}^{\infty}\frac{A_n}{\mu_n^2}\cdot I_0\!\left(\mu_n^2\cdot\frac{r}{R}\right) \exp(-\mu_n^2\cdot \mathrm{Fo}_{o.x}). \end{aligned} \tag{4′} \]
The quantity \(\Delta E_{o.x}\), defined by integral (1), can, after transformations [2], be reduced to a dimensionless quantity of the vulcanization effect, determined by the integral
\[ \Delta E_{o.x}^{a}=\Delta E\,\frac{a}{R^2} = K^{\frac{t_b-t_c}{10}}\cdot \int_{0}^{\mathrm{Fo}_{o.x}} \left(K^{\Theta_1}\right)^{\frac{\Delta t}{10}}\,d\mathrm{Fo}, \tag{5} \]
where
\[ \Theta_1=\frac{t(r_1,\tau)-t_c}{t_n-t_{\mathrm{n}}} \]
is determined from (4′).
\[ \Delta E_{o.x}^{a} \]
is the dimensionless value of the effect in the cooling device.
We represent \(\Theta_1\) in the form
\[ \Theta_1=\alpha\Theta-\Delta\Theta, \tag{6} \]
where
\[ \alpha=\frac{t_n-t_c}{t_n-t_{\mathrm{n}}}. \]
\(\Theta\) is the value of the relative temperature during cooling of an infinite cylinder with a uniformly distributed initial temperature and boundary conditions of the first kind.
\(\Delta \Theta\) is the second term in equation (3), divided by \(t_n - t_n\); \(\Delta t = t_n - t_n\).
Substituting (6) into (5), we obtain
\[ \Delta E_{o.x}^{a} = K^{\frac{t_b-t_c}{10}} \cdot \int_{0}^{\mathrm{Fo}_{o.x}} \left(K^{\Theta}\right)^{\left(x-\frac{\Delta \Theta}{\Theta}\right)\frac{\Delta t}{10}} \, d\tau . \tag{7} \]
The process of vulcanization of the cable sheath in the cooling device ends at \(\mathrm{Fo}_{o.x}=0.02\text{--}0.08\). Therefore, the value \(\dfrac{\Delta \Theta}{\Theta}\) over this interval \(\mathrm{Fo}_{o.x}\) may be taken as constant and equal to \(\left(\dfrac{\Delta \Theta}{\Theta}\right)\mathrm{Fo}_{o.x}^{0}\). For the value of \(\Theta\) from (4′), it is not possible to evaluate the integral analytically. But if, over the section of variation \(\mathrm{Fo}_{o.x}=0.02\text{--}0.08\), the cooling curve is represented as a straight line (Fig. 1),
Fig. 1. Approximation of the cooling curves of an infinitely homogeneous cylinder over the interval \(\mathrm{Fo}_{o.x}=0.02\text{--}0.1\)
then this integral can be evaluated. From Fig. 1 the cooling curve may be represented in the form
\[ \Theta = 1 - a(\mathrm{Fo}-\mathrm{Fo}_{0}), \tag{8} \]
where \(a\) is the tangent of the angle of inclination of the straight line;
\(\mathrm{Fo}_{0}\) is the value of the criterion \(\mathrm{Fo}\) at which the value \(\Theta=1\).
Moreover, we assume that for \(\mathrm{Fo}<\mathrm{Fo}_{0}\), \(a(\mathrm{Fo}-\mathrm{Fo}_{0})=0\). Substituting \(\Theta\) from (8) into (5) and integrating, after certain transformations we finally obtain an analytical expression for \(\Delta E_{o.x}^{a}\):
\[ \Delta E_{o.x}^{a} = K^{\left(\frac{t_n-t_b}{10}-\frac{\Delta t}{10}\cdot\frac{\Delta \Theta}{\Theta}\right)} \cdot \left\{ \mathrm{Fo}_{0} + \frac{10}{a\Delta t\left(x-\frac{\Delta \Theta}{\Theta}\right)\ln K} \left[ 1-\exp \times \right. \right. \]
\[ \left. \left. \times \left[ -a\cdot\frac{\Delta t}{10} \left(x-\frac{\Delta \Theta}{\Theta}\right) \ln K\cdot(\mathrm{Fo}-\mathrm{Fo}_{0}) \right] \right] \right\}. \tag{9} \]
For \(\mathrm{Fo}_{0x} \ge 0.1\), expression (9) is simplified and will have the form
\[ \Delta E_{0x}^{a} \approx K^{\left(\frac{t_{\mathrm{н}}-t_{\mathrm{в}}}{10}-\frac{\Delta \vartheta}{\Theta}\right)} \left[ \mathrm{Fo}_{0}+ \frac{10}{\ln K \cdot a \cdot \Delta t\left(a-\frac{\Delta \Theta}{\Theta}\right)} \right]. \tag{10} \]
Figure 2 gives graphs for determining the values of \(a\) and \(\mathrm{Fo}_{0}\) as functions of \(\dfrac{r}{R}\).
Fig. 2. Graphs for determining the coefficients \(a\) and \(\mathrm{Fo}_{0}\) in equations (9) and (10): a) \(\mathrm{Fo}_{0}\), b) \(a\)
Conclusions
The obtained dependence for \(\Delta E_{0x}^{a}\) makes it possible:
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To quantitatively determine the magnitude of \(\Delta E_{0x}\) under various operating conditions of the unit.
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To analyze the influence of various factors on the value of \(\Delta E_{0x}\).
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To evaluate the influence of vulcanization of the cable sheath in the cooling device on the productivity of the unit.
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To choose correctly the temperature of the cooling water at which the vulcanization of the cable sheath over its cross section will be most uniform.
References
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F. F. Koshelev, N. S. Klimov. General Technology of Rubber, Moscow, GNTIKhL, p. 400, 1958.
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A. A. Tatarnikov. Calculation of the vulcanization time of wires and cables by means of nomograms. In: Proceedings of the 11th industry scientific and technical conference “Improvement of the Production of Wires and Cables with Rubber Insulation,” p. 209, 1964.