On the Question of Determining from Experiment the Calculated Values of the Thermal Diffusivity Coefficient \(a_k^{p}\) and \(\left(\dfrac{r}{R}\right)_p\) of a Cable Product with a Rubber Sheath

A. A. Tatarnikov

(Presented by the scientific seminar of the ATPPP Department)

The vulcanization time of the rubber sheath of a cable product is determined by the process of its heating. One of the approximate analytical methods for calculating the nonstationary process of heating the sheath of a cable product, which is a complex composite body, may be the method of replacing it by an equivalent infinite cylinder with calculated values of the thermal diffusivity coefficient \(a_k^{p}\) and of the position of the point \(\left(\dfrac{r}{R}\right)_p\), for which the temperature is determined [1, 2].

The calculated value of the thermal diffusivity coefficient and \(\left(\dfrac{r}{R}\right)_p\) can be determined rather simply by the regular-regime method for various types of cable products. However, if, for determining \(a_k^{p}\), the place where the thermocouple is embedded in the cable specimen is of no importance, then, in order to determine \(\left(\dfrac{r}{R}\right)_p\), it is necessary to embed the thermocouple at the point for which the value of \(\left(\dfrac{r}{R}\right)_p\) is determined. Embedding a thermocouple at a specified point of the rubber sheath of a cable presents great difficulties. The actual position of the hot junction of the thermocouple \(\left(\dfrac{r}{R}\right)_{\mathrm{д}}\) in the cable sheath can be determined only after the experiment, by destroying the sheath, and with a large error. Consequently, it is difficult to establish with sufficient accuracy the dependence

\[ \left(\frac{r}{R}\right)_p = f\left[\left(\frac{r}{R}\right)_{\mathrm{д}}\right], \tag{1} \]

which is necessary for calculating the temperature at various points of line AB of the rubber sheath of the cable (Fig. 1).

Therefore, the solution of the following problem is of practical interest. Find the dependence (1) for various points of line AB (Fig. 1) of the cable sheath, using data obtained from the cooling or heating curve of the copper conductor of the cable. Such a method for determining the dependen-

…of (1) greatly facilitates carrying out experiments to determine \(a_k^{p}\) and \(\left(\dfrac{r}{R}\right)_p\), since the position of the thermocouple is determined uniquely.

Let there be two heating curves of the cable with a definite value of \(a_k^{p}\), and curve II for \(\left(\dfrac{r}{R}\right)\). Here curve I is obtained for \(\left(\dfrac{r}{R}\right)_p\), and curve II for \(\left(\dfrac{r}{R}\right)_d\). For the regular regime, under the condition \(\vartheta_2=\vartheta_1\) (Fig. 2), one may write

\[ 1-\frac{2J_0\left[\mu_1\left(\dfrac{r}{R}\right)_d\right]} {\mu_1J_1(\mu_1)} \exp\left(-\mu_1^2 Fo_2\right) = 1-\frac{2J_0\left[\mu_1\left(\dfrac{r}{R}\right)_p\right]} {\mu_1J_1(\mu_1)} \exp\left(-\mu_1^2 Fo_1\right) \]

or

\[ J_0\left[\mu_1\left(\frac{r}{R}\right)_p\right] = J_0\left[\mu_1\left(\frac{r}{R}\right)_d\right] \exp\left(\mu_1^2 \Delta Fo\right), \tag{2} \]

where \(\Delta Fo=Fo_1-Fo_2\).

Fig. 1

Fig. 2

The dependence \(J_0\left(\mu_1\dfrac{r}{R}\right)\) for \(0.5\leq \mu_1\dfrac{r}{R}\leq 2.4\) may be represented in the form

\[ J_0\left(\mu_1\frac{r}{R}\right) = 1.35\left(1-\frac{r}{R}\right). \tag{3} \]

Substituting (3) into (2), we obtain

\[ \left(\frac{r}{R}\right)_p = 1- \left[ 1-\left(\frac{r}{R}\right)_d \right] \exp\left(\mu_1^2\Delta Fo\right). \tag{4} \]

Equation (4) gives the desired relation between \(\left(\dfrac{r}{R}\right)_p\) and \(\left(\dfrac{r}{R}\right)_d\), in which the value \(\Delta Fo\) remains unknown.

To determine \(\Delta Fo\), let us consider the heating of the cable at two different points: in the core and at some point on the line \(AB\) in Fig. 3. As in the first case, we shall have heating curves for these points at \(\left(\dfrac{r}{R}\right)_p\) and \(\left(\dfrac{r}{R}\right)_d\). For the regular regime, for these curves one may obtain the following equality:

\[ \vartheta_2' - \vartheta_1' = \frac{ 2J_0\left[\mu_1\left(\frac{r}{R}\right)_{\mathrm{d}}^{\mathrm{ж}}\right] }{ \mu_1 J_1(\mu_1) } e^{-\mu_1^2 Fo_2'}\left(e^{\mu_1^2 \Delta Fo_{\mathrm{ж}}}-1\right), \tag{5} \]

where \(\Delta Fo_{\mathrm{ж}} = Fo_1' - Fo_2'\).

Similarly, for

\[ \vartheta_2'' - \vartheta_1'' = \frac{ 2J_0\left[\mu_1\left(\frac{r}{R}\right)_{\mathrm{д}}\right] }{ \mu_1 J_1(\mu_1) } e^{-\mu_1^2 Fo_2''}\left(e^{\mu_1^2 \Delta Fo}-1\right). \tag{6} \]

Under the condition that

\[ \vartheta_2''-\vartheta_1''=\vartheta_2'-\vartheta_1' \]

from (5) and (6), after transformations, we obtain that \(\Delta Fo=\Delta Fo_{\mathrm{ж}}\).

Fig. 3

Then equation (4) takes the form

\[ \left(\frac{r}{R}\right)_{\rho} = 1-\left[ 1-\left(\frac{r}{R}\right)_{\mathrm{д}} \right]\exp\left(\mu_1^2 \Delta Fo_{\mathrm{ж}}\right). \tag{7} \]

To use equation (7) in calculations, it is necessary to determine \(\Delta Fo_{\mathrm{ж}}\) experimentally. To calculate \(\Delta Fo_{\mathrm{ж}}\) from experiment, we shall use the heating curve of the copper conductor of the cable (curve I, Fig. 4). From curve I of Fig. 4, under regular conditions, the value of \(a_k^{\mathrm{p}}\) is determined from equation [3]

\[ a_k^{\mathrm{p}}= \frac{ 2.3\,[\lg(1-\vartheta_1)-\lg(1-\vartheta_2)]\cdot 60 }{ \frac{\mu_1^2}{R^2}(\tau_2-\tau_1) }, \tag{8} \]

where \(\tau\) is in minutes.

From (8) we determine the value \(Fo_2\)

\[ Fo_2= \frac{a_k^{\mathrm{p}}}{R^2}\cdot\frac{\tau_2}{60} = \frac{ 2.3\lg\frac{1-\vartheta_1}{1-\vartheta_2} }{ \mu_1^2\left(1-\frac{\tau_1}{\tau_2}\right) }. \tag{9} \]

The value \(Fo_1\) is determined from the expression

\[ 1-\vartheta_2= \frac{ 2 I_0\left[\mu_1\left(\frac{r}{R}\right)_{\!d}^{\,\zh}\right] }{ \mu_1 J_1(\mu_1) } \exp(-\mu_1^2 Fo_1). \]

Whence

\[ Fo_1=-\frac{2.3}{\mu_1^2}\lg \frac{(1-\vartheta_2)\mu_1 J_1(\mu_1)} { 2 I_0\left[\mu_1\left(\frac{r}{R}\right)_{\!d}^{\,\zh}\right] }. \tag{10} \]

Then, taking into account the numerical values of the functions and expressions (9) and (10), we obtain

\[ \Delta Fo_{\zh}=Fo_2-Fo_1 =0.4\left[ \frac{ \lg\frac{1-\vartheta_1}{1-\vartheta_2} }{ 1-\frac{\tau_1}{\tau_2} } +\lg\frac{1-\vartheta_2}{1-\left(\frac{r}{R}\right)_{\!d}^{\,\zh}} -0.3354 \right]. \tag{11} \]

Equation (11) contains only data that are taken from the heating curve of the copper conductor of the cable under regular conditions. Thus, the problem posed above has been solved.

Fig. 4

Conclusions

1. The proposed method simplifies the conducting of experiments to determine the thermophysical characteristics of a cable as a complex body.

2. The value \(\Delta Fo_{\zh}\) in equation (7) determines the influence of the copper conductors on the heating process of the cable sheath.

3. Equation (7) determines the relationship between the actual values \(\left(\frac{r}{R}\right)_d\) and the calculated values \(\left(\frac{r}{R}\right)_p\) at different points of the cable sheath along line AB (Fig. 1).

4. The function \(\left(\frac{r}{R}\right)_p=f\left[\left(\frac{r}{R}\right)_d\right]\) within the limits \(0.5 \leq \left(\frac{r}{R}\right)_d \leq 1\) is a straight line, the slope of which depends on the value of \(\Delta Fo_{\zh}\).

Literature

1. A. V. Lykov. Theory of Heat Conduction. Moscow, GITTL, p. 207, 1952.

2. A. I. Veinik. Approximate Calculation of Heat-Conduction Processes. GEI, p. 160, 1959.

3. G. M. Kondrat’ev. Regular Thermal Regime. GITTL, p. 29, 1954.