CALCULATION OF SURFACE OVERHEATING OF THE WINDING OF ADDITIONAL RESISTORS IN ELECTRICAL MEASURING INSTRUMENTS

G. I. Fuks, A. S. Lyalyukov

1. The additional resistor is located in an unventilated case (Fig. 1).

The winding resistance \(R\) ohm, current \(I\) A, total external surface of the additional resistor \(F_1\ \mathrm{m}^2\), the inner and outer surfaces of the case \(F'_k\) and \(F''_k\ \mathrm{m}^2\), its thickness \(\Delta_k\ \mathrm{m}\), and the thermal conductivity coefficient \(\lambda_k\ \dfrac{\mathrm{W}}{\mathrm{m}\cdot\mathrm{degree}}\) are known.

It is necessary to determine the overheating of the winding surface \(\vartheta = t_1 - t_f^\circ\mathrm{C}\) under the steady-state thermal regime of the instrument.

1. On the basis of the general principles of heat-transfer theory, under the steady-state thermal regime the heat given off by the case to the surrounding medium, transmitted through the wall of the case, and arriving from the coil to the inner surface of the case is the same. This makes it possible to write three equations:

Fig. 1.

\[ P=\alpha''(t_3-t_f)F''_k,\ \mathrm{W}, \tag{1} \]

\[ P=\frac{\lambda_k}{\Delta_k}(t_2-t_3)\frac{F'_k+F''_k}{2},\ \mathrm{W}, \tag{2} \]

\[ P=\frac{\lambda_\mathrm{e}}{\delta_\mathrm{e}}(t_1-t_2)\sqrt{F_1\cdot F'_\mathrm{k}}+ \varepsilon C_0F_1\left[\left(\frac{T_1}{100}\right)^4-\left(\frac{T_2}{100}\right)^4\right], \]

or

\[ P=(t_1-t_2)\left(\frac{\lambda_\mathrm{e}}{\delta_\mathrm{e}}\sqrt{F_1\cdot F'_\mathrm{k}}+ \varepsilon C_0F_1\Theta\right),\ \text{W}. \tag{3} \]

Denoted:

\[ \alpha''=\alpha_\mathrm{k}+\alpha_\mathrm{l} \]

—the total coefficient of heat transfer by convection and radiation from the surface of the case, \(\dfrac{\text{W}}{\text{m}^2\cdot\text{deg}}\);

\(t_2\) and \(t_3\)—temperature of the inner and outer surfaces of the case, °C;

\[ \Theta=\frac{\left(\dfrac{T_1}{100}\right)^4-\left(\dfrac{T_2}{100}\right)^4}{T_1-T_2}, \]

where \(T_1\) and \(T_2\) are the temperatures \(t_1\) and \(t_2\) on the absolute scale, °K.

\(\lambda_\mathrm{e}\)—equivalent coefficient of thermal conductivity of the air layer between the coil and the case (with convection taken into account), \(\dfrac{\text{W}}{\text{m}\cdot\text{deg}}\);

\(\delta_\mathrm{e}\)—conditional equivalent thickness of the air layer between the coil and the case, m;

\(\sqrt{F_1\cdot F'_\mathrm{k}}\)—geometric-mean area of the equivalent air layer, m²;

\(C_0=5.7\)—radiation coefficient of an absolutely black body, \(\dfrac{\text{W}}{\text{m}^2\cdot\text{deg K}^4}\);

\(\varepsilon_1\) and \(\varepsilon''\)—degree of blackness of the coil and the case;

\[ \varepsilon= \frac{1}{\dfrac{1}{\varepsilon_1}+\dfrac{F_1}{F'_\mathrm{k}}\left(\dfrac{1}{\varepsilon''}-1\right)} \]

—the degree of blackness of the coil–case system.

In equation (3) the first term on the right-hand side represents the heat transferred by convection, and the second, by radiation.

On the basis of equations (1), (2), and (3), the following procedure can be outlined for finding \(t_1\) and \(\vartheta\).

From equations (1) and (2), respectively, we obtain

\[ t_3=t_f+\frac{P}{\alpha''\cdot F''_\mathrm{k}}. \tag{4} \]

\[ t_2=t_3+\frac{2P}{\dfrac{\lambda_\mathrm{k}}{\Delta_\mathrm{k}}\left(F'_\mathrm{k}+F''_\mathrm{k}\right)}. \tag{5} \]

Having found \(t_2\), one should use equation (3) to determine \(t_1\). It is first necessary to compute \(\delta'_\mathrm{e}\) and \(\lambda_\mathrm{e}\).

A. To determine \(\delta_\mathrm{e}\), let us consider the case of one body situated inside another, when the shape of both bodies is spherical (Fig. 2, \(a\)).

\[ \delta_\mathrm{e}=\frac{D-d_2}{2}. \tag{a} \]

The volume of air between the inner sphere and the spherical shell

\[ V=\frac{\pi}{6}\left(D^{3}-d_{2}^{3}\right), \tag{б} \]

and the sum of the surfaces \(F_{1}\) and \(F'_{\mathrm{k}}\)

\[ F_{1}+F'_{\mathrm{k}}=\pi\left(D^{2}+d_{2}^{2}\right). \tag{в} \]

The ratio of (б) and (в) gives

\[ \frac{V}{F_{1}+F'_{\mathrm{k}}} = \frac{1}{6}\cdot\frac{\left(D^{3}-d_{2}^{3}\right)}{\left(D^{2}+d_{2}^{2}\right)} = \frac{2}{6}\cdot \frac{D^{2}+Dd_{2}+d_{2}^{2}}{D^{2}+d_{2}^{2}} \cdot \frac{D-d_{2}}{2}. \tag{г} \]

From (г) one may write

\[ \delta_{\mathrm{e}}= \frac{D-d_{2}}{2} = 3\,\frac{D^{2}+d_{2}^{2}}{D^{2}+Dd_{2}+d_{2}^{2}} \cdot \frac{V}{F_{1}+F'_{\mathrm{k}}} = \]

\[ = K_{\phi}\cdot \frac{V}{F_{1}+F'_{\mathrm{k}}}, \tag{6} \]

where

\[ K_{\phi}=3\,\frac{D^{2}+d_{2}^{2}}{D^{2}+D\cdot d_{2}+d_{2}^{2}}. \tag{7} \]

Expressing \(d_{2}\) as a fraction of \(D\)

\[ \left(d_{2}=xD,\qquad x=\frac{d_{2}}{D},\qquad 0<x<1\right) \]

and substituting \(d_{2}=xD\) in (7), we obtain

\[ K_{\phi}=\frac{3(1+x^{2})}{(1+x+x^{2})}. \tag{8} \]

As is evident from (8), \(K_{\phi}\) is a geometrical factor characterizing the ratio of the dimensions of the additional resistor and the casing, a dimensionless quantity. If \(d_{2}\) and \(D\) are close in magnitude \((d_{2}\to D,\ x\to 1)\), then \(K_{\phi}\to 2\); if \(d_{2}\) is small in comparison with \(D\) \((d_{2}\to 0\) or \(D\to\infty,\ x\to 0)\), then \(K_{\phi}\to 3\). For intermediate values \(x=0.1;\ 0.2\ldots 0.9\), the value of \(K_{\phi}\) assumes intermediate values

Fig. 2.

\[ 2<K_{\phi}<3. \tag{9} \]

Consideration of the cases “cube within a cube” and “cylinder within a cylinder” (Fig. 2, б, в) leads to the same results.

Thus, for pairs of spherical, cubic, and cylindrical bodies, with a symmetrical arrangement of the resistor in the case, the equivalent thickness of the air interlayer \(\delta_{\mathrm{e}}\) can be found from the volume of air inside the case \(V\) and the sum of the surfaces \(F_1\) and \(F_k\) by means of relation (6); in this case \(K_{\mathrm{f}}\) is calculated by (8).

With a dissimilar shape of the resistor and the case (a cylinder in a cube or in a rectangular parallelepiped, etc.), deviation of \(K_{\mathrm{f}}\) from the indicated values is possible, but there is reason to assert that it will be insignificant, since in relation (8), for spherical, cubic, and cylindrical bodies, the influence of shape did not manifest itself.

B. The equivalent coefficient of thermal conductivity of the air interlayer is \(\lambda_{\mathrm{e}}=\varepsilon_k\lambda\), where \(\varepsilon_k\) is the convection coefficient. For plane (vertical and horizontal), cylindrical, and spherical interlayers, \(\varepsilon_k\) can be determined from the criterion equations [1]

\[ \varepsilon_k=0.105\,(\mathrm{Gr}\cdot\mathrm{Pr})_m^{0.3} \quad \text{for } 10^3<\mathrm{Gr}\cdot\mathrm{Pr}<10^6, \tag{d} \]

\[ \varepsilon_k=0.40\,(\mathrm{Gr}\cdot\mathrm{Pr})_m^{0.2} \quad \text{for } 10^6<\mathrm{Pr}\cdot\mathrm{Gr}<10^{10}. \]

Here

\[ \mathrm{Gr}=\frac{\delta_{\mathrm{e}}^{3} g \vartheta'}{\nu^2}\,\beta \quad \text{— Grashof criterion;} \]

\[ \mathrm{Pr}=\frac{\nu}{a} \quad \text{— Prandtl criterion;} \]

\[ \vartheta'=t_1-t_2 \quad \text{— temperature difference at the walls of the gas interlayer} \]
\[ \text{(in the case considered, the resistor and the inner surface of the case), }{}^\circ\mathrm{C}; \]

\[ g=9.81 \quad \text{— acceleration due to gravity, } \mathrm{m/sec}^2; \]

\[ \nu \quad \text{— coefficient of kinematic viscosity of air, } \mathrm{m}^2/\mathrm{sec}; \]

\[ a \quad \text{— coefficient of temperature diffusivity of air, } \mathrm{m}^2/\mathrm{h}; \]

\[ \beta \quad \text{— coefficient of volume expansion; for gases} \]

\[ \beta=\frac{1}{T},\quad \frac{1}{{}^\circ K}. \]

The subscript \(m\) means that all physical constants entering into \(\mathrm{Gr}\) and \(\mathrm{Pr}\) are taken at the mean temperature of the gas interlayer

\[ t_m=\frac{t_1+t_2}{2}. \]

Equations (1), (2), and (3), and the considerations for determining \(\delta_{\mathrm{e}}\) and \(\lambda_{\mathrm{e}}\) set forth above, constitute the basis of the theoretical approach to determining the winding surface temperature \(t_1\).

1. When using equations (1), (2), and (3) to calculate \(t_1\), difficulties arise consisting in the fact that equations (1) and (3) include the quantities \(\alpha''\) and \(\lambda_{\mathrm{e}}\), which are determined through the sought temperatures \(t_3\) and \(t_1\).

A. Since \(\alpha''=\alpha_k+\alpha_l\). The \(\alpha_l\) entering here is determined as

\[ \alpha_l= \frac{ \varepsilon''\cdot C_0 \left[ \left(\frac{T_3}{100}\right)^4 - \left(\frac{T_f}{100}\right)^4 \right] }{ T_3-T_f }. \tag{e} \]

Thus, \(\alpha_l\) depends on the sought quantity \(t_3\). The value of \(\alpha_k\) is determined by means of the criterion equations [1].

\[ \mathrm{Nu}_f = 0.47\, \mathrm{Gr}_f^{0.25} \tag{ж} \]

\[ \mathrm{Nu}_m = 0.54\,(\mathrm{Gr}\cdot \mathrm{Pr})_m^{0.25}. \tag{3} \]

In the present case the Nusselt, Grashof, and Prandtl criteria are expressed as

\[ \mathrm{Nu}=\frac{\alpha_k\cdot D}{\lambda_k}; \qquad \mathrm{Gr}=\frac{D^3 g\,(t_3-t_f)}{\nu^2}\cdot \beta; \qquad \mathrm{Pr}=\frac{\nu}{a}. \]

The subscript \(f\) indicates that the physical parameters of the air entering into the criteria are determined at the temperature of the surrounding air, while the subscript \(m\) indicates at the mean temperature of the boundary layer of air,

\[ t_m=\frac{t_3+t_f}{2}. \]

Since \(t_3\) enters into the Grashof criterion, it is evident that \(\alpha_k\), just like \(\alpha_l\), in any case depends on the unknown quantity \(t_3\), which leads to the necessity of solving equation (4) by trial. Repeated calculations of \(\alpha''=\alpha_l+\alpha_k\) by means of (e) and (ж) or (3) are very cumbersome and labor-intensive. A reduction in bulk and labor, as well as the elimination of trial selection of \(t_3\), can be achieved by tabulating and nomographically representing expression (4). We rewrite equation (1) in the form

\[ \frac{P}{F_k''}=\alpha''(t_3-t_f)=P_k''=\alpha''\vartheta_3=f(\varepsilon'',\vartheta_3,t_f,D). \tag{10} \]

Here \(P\) and \(P_k''\) are, respectively, the total power of the resistor and the specific power dissipated on the case.

The blackness factor \(\varepsilon''\) for plastics and for surfaces coated with lacquers and oil paints of various colors may be taken as \(\varepsilon''=0.9\). If the ambient temperature is taken as \(t_f=20^\circ\mathrm{C}\), then

\[ P_k''=f_1(\vartheta_3,D). \tag{10′} \]

By specifying the temperature difference between the case and the surrounding medium \(\vartheta_3\) and assigning various values to \(D\), one can calculate the corresponding \(P_k''\). From these data a nomogram is constructed which makes it possible to find \(\vartheta_3\), if \(D\) and \(P_k''\) are known, and then to determine \(t_3=\vartheta_3-t_f\). Such a nomogram has been constructed and is given in Fig. 3.

When constructing the nomogram of Fig. 3, the calculation of \(\alpha_k\) was carried out by equation (3), and \(\alpha_l\) by equation (e).

B. As is seen from equation (3), taking into account (d) and the expression for \(\Theta\), the winding temperature \(t_1\) is determined through quantities that depend on the sought temperature \(t_1\) directly (\(\Theta\)) or in the form \(\vartheta'=t_1-t_2\) and

\[ t_m=\frac{t_1+t_2}{2} \]

(\(\varepsilon_k\) and \(\lambda_e=\varepsilon_k\cdot\lambda\)). Here also it becomes necessary to solve equation (3) by trial. A reduction in bulk and labor here likewise can be achieved by nomographically representing

\[ \Theta=f_2(T_1,T_2) \]

and

\[ \varepsilon_k=f_3(\mathrm{Gr}\cdot\mathrm{Pr})_m=f_4(\vartheta_e,\vartheta',t_m). \]

The nomogram for determining \(\Theta\) is given in Fig. 4. To determine \(\varepsilon_k\), several nomograms are constructed, differing in the range of \(\vartheta_e\) and in the region of values of \((\mathrm{Gr}\cdot\mathrm{Pr})\). One of these nomograms is given in Fig. 5.

If \(\vartheta_e\) is known and \(t_2\) has already been calculated, then, by specifying \(t_1'\), one finds

\[ \vartheta'=t_1'-t_2 \]

and

\[ t_m=\frac{t_1'+t_2}{2}, \]

and then, from the nomogram, \(\varepsilon_k\); then \(\lambda_e=\varepsilon_k\cdot\lambda\).

Fig. 3.

Upper graph:

• Ordinate: \(p'',\ \mathrm{mW/cm^2}\)
• Abscissa: \(P_k^{*},\ \mathrm{mW/cm^2}\)
• Curve parameter: \(\vartheta\)
• Curves labeled: \(50,\ 45,\ 40,\ 35,\ 30,\ 25,\ 20,\ 15\)
• Dashed construction line: \(\vartheta_3 = 37.2\)

Lower graph:

• Ordinate: \(P_k^{*},\ \mathrm{mW/cm^2}\)
• Abscissa: \(D,\ \mathrm{mm}\)
• Curve parameter: \(\vartheta =\)
• Curves labeled: \(10,\ 7.5,\ 5,\ 4,\ 3,\ 2,\ 1\)

\[ \theta = f(t_1, t_2) \]

Fig. 4.

Fig. 5.

Visible labels and annotations in the figure:

\[ \varepsilon_k = 2.105(\omega_1+\omega_2)t_m^{0.3} \]

\[ \frac{\xi_k}{\varepsilon_k} \]

\[ \frac{\delta_2}{\mu} \]

\[ \varepsilon_k \]

\[ \delta_3 \]

Curve labels:

\[ t_m=15,\ 20,\ 30,\ 40,\ 50,\ 60,\ 70,\ 80,\ 90,\ 100 \]

\[ U=10,\ 20,\ 30,\ 40 \]

Additional visible numerical labels:

\[ 1.0,\ 1.05,\ 1.1,\ 1.15,\ 1.2,\ 1.3,\ 1.4,\ 1.5,\ 1.6,\ 1.7,\ 1.8,\ 1.9,\ 2.0 \]

\[ 0.004,\ 0.005,\ 0.006,\ 0.007,\ 0.010,\ 0.001,\ 0.017 \]

\[ 6000,\ 8000 \]

and from equation (3) \(P\) is calculated. The selection of \(t_1\) is repeated until the \(P\) obtained from equation (3) coincides with the value of the actual heat release of the winding \(P = I^2R\).

1. In order to verify the agreement of the proposed calculation method with experiment, an experimental check was carried out. The arrangement for conducting the experiments is shown in Fig. 6.

Additional resistors (coil No. 1, \(d_2 = 10\) mm and \(l = 14.8\) mm, and coil No. 2, \(d_2 = 21.5\) mm and \(l = 22\) mm) were placed under a cylindrical Plexiglas casing coated with black oil paint. The dissipated power was determined from the voltage drop across the coil winding and its resistance. The temperature was measured on the upper generatrix of the cylindrical surface of the winding, in its middle part (the maximum temperature of the winding surface), by means of a copper–constantan thermocouple. The results of the experimental verification of the calculation method are characterized by Table 1.

1 — support;
2 — additional resistor;
3 — casing;
4 — rheostats;
5 — voltmeter;
6 — switch;
7 — PP potentiometer;
8 — cold-junction thermostat.

Fig. 6.

On the basis of the table, a graph was constructed of the discrepancy between the calculated and experimental overheating values as a function of \(\dfrac{F_1}{F_k}\), from which it was established that, for

\[ \frac{F_1}{F_k} > 0.08, \]

the agreement between calculation and experiment is satisfactory (the discrepancies do not exceed 14–15%). At smaller values of \(\dfrac{F_1}{F_k}\), the discrepancies increase sharply. The reason for this, in our opinion, is that as the size of the casing increases, me-

the character of the heat-exchange processes changes—from heat exchange in a closed volume there is a transition to heat exchange in an unbounded volume.

Table 1

Discrepancy of calculated overheatings from experimental ones (in %)

I — the current regime corresponds to an overheating of \(\vartheta \approx 80^\circ\) under conditions of free cooling;
II — the regime is the same, \(\vartheta \approx 40^\circ\).
Note: * — casing with \(\delta = 44\) mm and \(h = 65\) mm.

REFERENCES

1. M. A. Mikheev. Fundamentals of Heat Transfer. 1956.