THERMOPHYSICAL CHARACTERISTICS OF A SINGLE-CORE CABLE

V. I. NAPLEKOV

(Presented by the scientific seminar of the Department of ATPPP)

In the vulcanization of wires and cables with rubber insulation, the determining factor is the thermal process associated with heating the sheath. To obtain calculated results for the vulcanization time of a cable product, it is necessary to know the nature of the temperature variation at the calculation point. As the calculation point, from the conditions for calculating vulcanization regimes, we take the point located at the lower boundary of the sheath. Wires and cables are complex composite bodies, and obtaining analytical solutions for their temperature distribution is difficult. This problem can be solved by reducing the solution for the temperature distribution of a complex body to the temperature distribution of a fictitious classical body with equivalent thermophysical characteristics. To obtain the equivalent thermophysical characteristics, it is necessary to find the temperature distribution for the complex body. One possible method for solving this problem is a numerical method. To formulate the computational equations, we shall use the method of elementary balances. Let us consider the heating problem as applied to a single-core cable, which has the simplest construction, easily formalized by virtue of symmetry: a metal core covered with a rubber insulation layer uniform in thickness (Fig. 1).

Vulcanization of the sheath takes place in an atmosphere of saturated steam with a large heat-transfer coefficient. Therefore the temperature on the cable surface may be regarded as constant (boundary conditions of the first kind), and in deriving the computational equations by the method of elementary balances only Fourier’s equation may be used:

$$ q = \lambda \operatorname{grad} t. $$

The initial equation for calculating the temperature in the rubber sheath will have the form:

$$ \frac{\partial t}{\partial z} = a_p \left( \frac{\partial^2 t}{\partial r^2} + \frac{1}{r}\cdot \frac{\partial t}{\partial r^2} \right). \tag{1} $$

Let us write it in finite differences. For a point we shall have

$$ \frac{ t_{R_i,\tau+\Delta z}-t_{R_i,\tau} }{ \Delta z } = a_p \left[ \frac{ \left(t_{R_i+\Delta R,\tau}+t_{R_i,\tau}\right) - \left(t_{R_i,\tau}-t_{R_i-\Delta R,\tau}\right) }{ \Delta R^2 } + \frac{1}{R_i} \left( \frac{ t_{R_i+\Delta R,\tau}-t_{R_i-\Delta R,\tau} }{ 2\Delta R } \right) \right]. \tag{2} $$

From the stability condition, which is written in the following form:

\[ \frac{2a_{\rho}\Delta\tau}{\Delta R^2}\leqslant 1, \]

we find the limiting value of the time step

\[ \Delta\tau=\frac{\Delta R^2}{2a_{\rho}}. \tag{3} \]

Substituting (3) into equation (2) and carrying out certain transformations, we obtain an expression for the current value of the temperature at any point of the sheath of the computational cable model.

Fig. 1. Computational cable design

Fig. 2. Dependence of the parameters \(\varphi\) and \(\left(\frac{r}{R}\right)_{\rho}\) on \(\frac{R_{\mathrm{m}}}{R}\) for various \(\xi\)

\[ t_{R_i,\tau+\Delta\tau} = \frac{t_{R_i+\Delta R,\tau}+t_{R_i-\Delta R,\tau}}{2} + \frac{1}{4n_{\mathrm{d}}\dfrac{R_i}{R}} \left(t_{R_i+\Delta R,\tau}-t_{R_i-\Delta R,\tau}\right), \tag{4} \]

where

\[ t_{R_i,\tau+\Delta\tau} \]
is the calculated temperature value;

\[ t_{R_i\pm\Delta R,\tau} \]
is the temperature value, respectively, in the layer above and in the layer below the point being calculated;

\[ \frac{R_i}{R} \]
is the relative position of the computational layer;

\[ n_{\mathrm{d}}=\frac{R}{\Delta R} \]
is the number of computational layers;

\[ \Delta R \]
is the thickness of the computational layer.

For the metallic core, the temperature gradient with respect to the coordinate

\[ \frac{R_i}{R} \]

is much smaller than the temperature gradient for the rubber insulation, and therefore it may be taken as equal to zero. Then the temperature of the metallic core can be found from an equation compiled on the basis of the heat balance:

\[ 2\pi R\lambda_{\rho} \frac{t_{R_{\mathrm{m}}+\Delta R,\tau}-t_{R_{\mathrm{m}},\tau}}{\Delta R} = C_{\mathrm{m}}\gamma_{\mathrm{m}}\pi R_{\mathrm{m}}^{2} \frac{t_{R_{\mathrm{m}},\tau+\Delta\tau}-t_{R_{\mathrm{m}},\tau}}{\Delta\tau}. \tag{5} \]

Substituting into equation (5) the value \(\Delta \tau\) from (3), expanding in it the value of the temperature diffusivity coefficient of rubber \(a_p=\dfrac{\lambda_p}{C_p\gamma_p}\), and carrying out some transformations, we obtain the temperature of the metal conductor:

\[ t_{R_{\mathrm{м}},\,\tau+\Delta\tau} = t_{R_{\mathrm{м}},\,\tau} + \frac{1}{n_{\mathrm{д}}\xi_i \dfrac{R_{\mathrm{м}}}{R}} \left(t_{R_{\mathrm{м}}+\Delta R,\,\tau}-t_{R_{\mathrm{м}},\,\tau}\right), \tag{6} \]

where: \(t_{R_{\mathrm{м}},\,\tau+\Delta\tau}\), \(t_{R_{\mathrm{м}},\,\tau}\) are the temperatures of the metal conductor, respectively, at the instants of time \(\tau+\Delta\tau\) and \(\tau\);

\(t_{R_{\mathrm{м}}+\Delta R,\,\tau}\) is the temperature of the adjacent layer of the rubber sheath at a point located above the metal conductor at a radial distance \(\Delta R\), at the instant of time \(\tau\);

\[ \xi_i=\frac{C_{\mathrm{м}}\gamma_{\mathrm{м}}}{C_p\gamma_p} \]

is a parameter characterizing the ratio of the volumetric heat capacities of the conductor and the insulation.

Using equations (4) and (6), heating curves for a single-core cable were obtained on a computer with the parameters varied within the following limits:

1. The number of divisions of the cylinder \(n_{\mathrm{д}}=10\). It was not changed during the calculations.

2. Variation of the ratio of the radius of the metal conductor to the total radius

\[ \frac{R_{\mathrm{м}}}{R}=0.2\div 0.9. \]

1. The following values of the parameter \(\xi_i\) were taken [1]:

a) for an aluminum conductor — \(\xi_1=1.48\);

b) for a copper conductor — \(\xi_2=2.02\);

c) for a steel conductor — \(\xi_3=2.20\).

The initial temperature distribution was assumed to be uniform over the cross section of the rubber sheath and the metal conductor.

Having the curves of the temperature variation of the composite body at the calculation point, it is necessary to find the equivalent thermophysical characteristics of a fictitious classical body that would make it possible to use the solution for temperature distributions for a classical body. One possible way is as follows [23]. It is necessary to select such a fictitious classical body (in our case it is more convenient to take an “infinite” cylinder as the classical body), with temperature diffusivity coefficient \(a_k\) and coordinate point \(\left(\dfrac{r}{R}\right)_p\), for which, at the stage of the regular regime, the temperature variation coincides sufficiently accurately with the temperature variation at the calculation point of the cable sheath. The change in relative temperature at the stage of the regular regime is determined by one exponential term of the series, which is the solution of the temperature distribution for any point of an “infinite” cylinder [1]:

\[ \Theta = 1.6\,I_0\left(2.405\,\frac{r}{R}\right) \exp\left(-5.78\,\frac{a_k\tau}{R^2}\right). \tag{7} \]

In calculation by the numerical method, real time is related to the computational steps by the following relation:

\[ \tau=\Delta\tau n. \tag{8} \]

Substituting (3), (8) into equation (7), we obtain

\[ \Theta = 1.6\,I_0\left(2.405\,\frac{r}{R}\right) \exp\left(-2.89\cdot 10^{-2}\varphi\cdot n\right), \tag{9} \]

$\varphi=\dfrac{a_k}{a_p}$ is a parameter characterizing the ratio of the coefficient of thermal diffusivity of the cable to the coefficient of thermal diffusivity of rubber,

$n$ is the number of the calculation step. The parameter $\varphi$ is determined at the stage of the regular regime, for two different values of $\Theta$, by the formula obtained from equation (7)

\[ \varphi=\frac{\ln \Theta_2-\ln \Theta_1}{2.89\cdot 10^{-2}(n_1-n_2)}. \tag{10} \]

From equation (7), at the stage of the regular regime one can determine the coordinate of the point of the fictitious classical body isothermal with the calculation point,

\[ \left(\frac{r}{R}\right)_p= \frac{J_0^{-1}\left[\Theta\exp\left(-2.89\cdot 10^{-2}\cdot\varphi\cdot n\right)\right]}{2.405}. \tag{11} \]

On the curves of variation of the relative temperature at the calculation point, obtained by the numerical method, we single out the section of the regular regime and, using formulas (10) and (11), find the calculated values of $a_k$ and $\left(\dfrac{r}{R}\right)_p$. The results of the calculations are presented in the graphs (Fig. 2).

The graphs make it possible, from the data $\dfrac{R_m}{R}$, $\xi_1$, $a_p$, to determine the equivalent value of the coordinate $\left(\dfrac{r}{R}\right)_p$ and $a_k$.

Having the equivalent thermophysical characteristics $\left(\dfrac{r}{R}\right)_p$ and $a_k$, one can use the solution for the temperature distribution for an “infinite” single-core cable, for which it is necessary to know the behavior of the temperature at the calculation point.

Conclusions

1. The article presents an algorithm for calculating, by a numerical method, the temperature at any point of the sheath and of the metal core of a single-core cable in the case of heating under boundary conditions of the first kind.

2. From the condition that the temperature-distribution curves coincide at the stage of the regular regime for the calculation point of the cable sheath and an “infinite” homogeneous cylinder, formulas have been obtained for determining the equivalent thermophysical characteristics.

3. Graphs have been obtained of the calculated values of $\varphi$ and $\left(\dfrac{r}{R}\right)_p$ as functions of the cable design and core material. The thermophysical characteristics found with the aid of these graphs make it possible, when determining the temperature at the calculation point of the cable sheath, to use the solution for a homogeneous “infinite” cylinder.

REFERENCES

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

2. A. A. Tatarnikov. Influence of copper cores on the heating of a rubber sheath. Cable Technology, No. 39, 1966.