EFFECT OF CHANGES IN THE TEMPERATURE COEFFICIENT “\(K\)” ON THE MAGNITUDE OF THE VULCANIZATION EFFECT AT THE CALCULATION POINT OF THE CABLE SHEATH

A. A. Tatarnikov

(Presented by the Department of ATPPP)

In [1, 2] a calculation is given of the magnitude of the vulcanization effect \(E_\tau\) obtained by the cable sheath at the calculation point, under the condition that the value of the temperature coefficient is \(K = 2\). In reality, as noted in [3], the value of \(K\) depends on temperature. This dependence is determined by the Arrhenius equation [3]:

\[ K=\frac{I_{t+5}}{I_{t-5}}=\exp\left[\frac{10E}{R\,(T+5)(T-5)}\right], \tag{1} \]

where \(I\) is, respectively, the vulcanization intensity at temperatures \(t+5^\circ\text{C}\) and \(t-5^\circ\text{C}\);

\(E\) is the activation energy;

\(R\) is the universal gas constant.

A calculation of the values of \(K\) by equation (1) shows that the change in \(K\) over the temperature interval \(120\text{--}160^\circ\text{C}\) does not exceed 7%. In the same work it is noted that a number of recent experimental studies confirm the constancy of the temperature coefficients of vulcanization in the temperature range \(120\text{--}160^\circ\text{C}\) for certain types of rubber. Therefore, the problem of determining the influence of a change in \(K\) on the magnitude of the vulcanization effect must be solved for two cases:

1. The temperature coefficient of vulcanization is \(f(t)\).
2. The temperature coefficient of vulcanization is a constant, but its value is not equal to two.

1. \(K=f(t)\)

A direct solution of the problem of determining the magnitude of the vulcanization effect \(E\) for \(K=f(t)\) is very complicated, since in this case it is necessary to calculate the integral

\[ E=\int_0^\tau \exp\left[ \frac{10E}{R\,(t+278)(t-273)} \cdot \frac{t-t_b}{10} \right]\,d\tau, \tag{2} \]

where \(t\) is the current value of the temperature, determined by the solution of the problem of heating an infinite cylinder [1].

If (2) is calculated for changes in values within the interval \(120\text{--}160^\circ\text{C}\), then equation (1) can, with sufficient accuracy, be represented in the form

of a linear dependence. Taking as the starting point the value \(K_0=2\) at \(t=138^\circ\mathrm{C}\), (1) can be represented in the form

\[ K=K_0-a(t-138), \tag{3} \]

where \(a=0.00665\). The error in determining \(K\) from equation (3) does not exceed 0.7%. In this case integral (2) can be represented in the form

\[ E=\int_0^\tau (K_0+\Delta K)^{\frac{t-t_b}{10}}\,d\tau \]

or

\[ E=\int_0^\tau K_0\left(1+\frac{\Delta K}{K_0}\right)^{\frac{t-t_b}{10}}\cdot d\tau . \tag{4} \]

The quantity \(\dfrac{\Delta K}{K_0}\) does not exceed 0.15. Therefore the expression \(\left(1+\dfrac{\Delta K}{K_0}\right)^{\frac{t-t_b}{10}}\) can be expanded in a series, in which we restrict ourselves to the first power of \(\dfrac{\Delta K}{K_0}\). In this case the error will not exceed 5%.

Then integral (4) can be represented in the form

\[ E=\int_0^\tau K_0^{\frac{t-t_b}{10}}\cdot d\tau+ \int_0^\tau \frac{\Delta K}{K_0}\cdot \frac{t-t_b}{10}\cdot K_0^{\frac{t-t_b}{10}}\cdot d\tau . \]

Substituting \(\Delta K\) from (3), we obtain

\[ E=\int_0^\tau K_0^{\frac{t-t_b}{10}}\cdot d\tau+ \frac{a}{10K_0}\cdot \int_0^\tau (t-138)(t-t_b)\cdot K_0^{\frac{t-t_b}{10}}\cdot d\tau . \tag{5} \]

The quantity \(E\) is equal to the sum of two integrals. The first determines the magnitude of the vulcanization effect for \(K_0=2\), and the second integral determines the magnitude of the correction for \(K=f(t)\).

The first integral can be transformed into the dimensionless magnitude of the vulcanization effect [2]. By analogous transformations, the integral determining the correction can be transformed to the form

\[ \Delta E_a^k=\Delta E\,\frac{a_k}{R^2} =\frac{a}{10K_0}\cdot K_0^{\frac{t_n-t_b}{10}} \cdot\left[ \bigl((t_n-138)t_n-t_b\bigr) \int_0^{\mathrm{Fo}} (K_0^{-\Theta})^{\frac{\Delta t}{10}}\cdot d\mathrm{Fo} \right. \]

\[ \left. -(2t_n-128-t_b)\cdot \int_0^{\mathrm{Fo}} \Theta\cdot \Delta t\,(K_0^{-\Theta})^{\frac{\Delta t}{10}}\cdot d\mathrm{Fo} + \int_0^{\mathrm{Fo}} \Theta^2\cdot \Delta t^2\,(K_0^{-\Theta})^{\frac{\Delta t}{10}}\cdot d\mathrm{Fo} \right], \tag{6} \]

where \(\Delta E_a^k\) is the dimensionless correction to the magnitude of the vulcanization effect for \(K=f(t)\);

\(t_n\) is the temperature of saturated steam in the vulcanization tube;

\(t_b\) is the temperature of vulcanization of rubber specimens in the laboratory;

\(\Theta\) is the relative temperature at the calculation point of the cable sheath;

\[ \Delta t=t_n-t_{\mathrm{n}}; \]

\(t_{\mathrm{n}}\) is the initial temperature of the cable;

\(\mathrm{Fo}\) is the Fourier criterion determining the residence time of the cable in the vulcanization tube.

Analogously, for the case \(K=\mathrm{const}\), but \(K \ne 2\), one can obtain the equation for the dimensionless correction \(\Delta E_a^K\) in the following form:

\[ \Delta E_a^K = \frac{\Delta K}{10K_0}\cdot K_0^{\frac{t_n-t_b}{10}} \left[ (t_n-t_b)\cdot \int_0^{\mathrm{Fo}} \left(K_0^{-\Theta}\right)^{\frac{\Delta t}{10}}\cdot d\mathrm{Fo} - \int_0^{\mathrm{Fo}} \Theta\cdot \Delta t \cdot \left(K_0^{-\Theta}\right)^{\frac{\Delta t}{10}}\cdot d\mathrm{Fo} \right]. \tag{7} \]

The notation in (7) is the same as in (6).

To obtain a clearer picture of the influence of changes in \(K\), calculations were performed for the relative magnitude of the correction \(\dfrac{\Delta E_a^K}{E_T}\) according to equations (6) and (7).

\(E_T\) is the magnitude of the vulcanization effect obtained by the cable sheath at the calculation point in the vulcanization tube when \(K=2\) [2].

Fig. 1. Graphs of the values \(\dfrac{\Delta E_a^K}{E_T}\):
a — solid line, for \(K=f(t)\);
b — dashed line, for \(K=\mathrm{const}\ne 2\).

Integration of expressions (6) and (7) was carried out by a numerical method. The results of the calculations are presented in the graphs in Fig. 1 for the values \(\dfrac{r}{R}=0.5\), \(\Delta t_{\min}=40^\circ\mathrm{C}\), and \(\Delta t_{\max}=150^\circ\mathrm{C}\).

Conclusions

1. The magnitude of the relative correction depends on the operating conditions of the units, the cable design, and the position of the calculation point.

1. As follows from the graphs in Fig. 1, the magnitude of the correction reaches:

a) for \(K = f(t)\), \(\sim 40\%\).

b) for \(K = \mathrm{const} \ne 2\), \(\sim 30\%\).

A correction of this magnitude cannot be neglected when calculating technological regimes.

1. Expressions (6) and (7) make it possible, in a comparatively simple way, to construct calculation nomograms for determining

\[ \frac{\Delta E_a^K}{E_\tau}. \]

LITERATURE

1. P. I. Chudikov. Graph-analytical method for calculating the vulcanization time of hose sheaths of cables on ANV. Cable Engineering, p. 38, issue 3/10, 1959.

2. A. A. Tatarnikov. Calculation of the vulcanization time of wires and cables using nomograms. Collection of materials of the 11th sectoral scientific and technical conference “Improvement of the Production of Wires and Cables with Rubber Insulation,” p. 206, 1964.

3. V. O. Fogel. “Nomographic method for calculating the vulcanization duration of rubber-technical plates,” Kauchuk i rezina, No. 3, p. 15, 1963.