A SIMPLIFIED METHOD FOR CALCULATING THE SURFACE TEMPERATURE RISE OF THE WINDING OF ADDITIONAL RESISTORS IN ELECTRICAL MEASURING INSTRUMENTS

A. S. LYALIKOV, G. I. FUKS

From the equations of steady-state heat exchange of an additional resistor (coil) located in the instrument housing,

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

\[ P=\frac{\lambda_{\mathrm{k}}}{\Delta_{\mathrm{k}}}(t_2-t_3)\frac{F'_{\mathrm{k}}+F''_{\mathrm{k}}}{2}, \tag{2} \]

\[ P=\frac{\lambda_{\mathrm{e}}}{\delta_{\mathrm{e}}}(t_1-t_2)\sqrt{F_1\cdot F'_{\mathrm{k}}} +\varepsilon C_0 F_1\left[\left(\frac{T_1}{100}\right)^4-\left(\frac{T_2}{100}\right)^4\right] \]

or

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

it follows that

\[ t_3-t_f=\frac{P}{\alpha''F''_{\mathrm{k}}}, \tag{1'} \]

\[ t_2-t_3= \frac{P}{ \frac{\lambda_{\mathrm{k}}}{\Delta_{\mathrm{k}}}\frac{(F'_{\mathrm{k}}+F''_{\mathrm{k}})}{2} }, \tag{2'} \]

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

Summing the right- and left-hand sides and solving with respect to \(P\), we obtain

\[ P= \frac{t_1-t_f}{ \frac{1}{\alpha''F''_{\mathrm{k}}} + \frac{1}{ \frac{\lambda_{\mathrm{k}}}{\Delta_{\mathrm{k}}}\cdot \frac{(F'_{\mathrm{k}}+F''_{\mathrm{k}})}{2} } + \frac{1}{ F'_{\mathrm{k}}\left( \frac{\lambda_{\mathrm{e}}}{\delta_{\mathrm{e}}}\sqrt{\frac{F_1}{F'_{\mathrm{k}}}} +\varepsilon C_0\frac{F_1}{F'_{\mathrm{k}}}\Theta \right) } }. \tag{4} \]

Taking into account that \(F'_{\mathrm{k}}\) and \(F''_{\mathrm{k}}\) are close in magnitude, one may assume

\[ F'_{\mathrm{k}}\approx F''_{\mathrm{k}}\approx \frac{F'_{\mathrm{k}}+F''_{\mathrm{k}}}{2}=F_{\mathrm{k.avg}}. \]

Then

\[ P= \frac{(t_1-t_f)F_{\mathrm{k.cp}}} {\dfrac{1}{\alpha''}+\dfrac{\Delta_{\mathrm{k}}}{\lambda_{\mathrm{k}}}+ \dfrac{1}{\dfrac{\lambda_{\mathrm{e}}}{\delta_{\mathrm{e}}} \sqrt{\dfrac{F_1}{F'_{\mathrm{k}}}}+\varepsilon C_0\dfrac{F_1}{F'_{\mathrm{k}}}\Theta}} =K_{1-f}\cdot\vartheta\cdot F_{\mathrm{k.cp}} . \tag{5} \]

Here

\[ \begin{aligned} K_{1-} &= \frac{1} {\dfrac{1}{\alpha''}+\dfrac{\Delta_{\mathrm{k}}}{\lambda_{\mathrm{k}}}+ \dfrac{1}{\dfrac{\lambda_{\mathrm{e}}}{\delta_{\mathrm{e}}} \sqrt{\dfrac{F_1}{F'_{\mathrm{k}}}}+\varepsilon C_0\dfrac{F_1}{F'_{\mathrm{k}}}\Theta}} \\[6pt] &= \frac{1} {\dfrac{1}{\alpha''}+\dfrac{\Delta_{\mathrm{k}}}{\lambda_{\mathrm{k}}}+\dfrac{1}{\alpha_{1-\mathrm{k}}}} \end{aligned} \tag{6} \]

and

\[ \alpha_{1-\mathrm{k}}= \frac{\lambda_{\mathrm{e}}}{\delta_{\mathrm{e}}} \sqrt{\frac{F_1}{F'_{\mathrm{k}}}} +\varepsilon C_0\frac{F_1}{F'_{\mathrm{k}}}\Theta . \tag{7} \]

Expression (6) represents the conditional heat-transfer coefficient for the coil—case—ambient system, expressed per unit mean area of the case \(F_{\mathrm{k.cp}}\), while (7) is the conditional coefficient of heat transfer from the coil to the case, likewise referred to \(F_{\mathrm{k.cp}}\); \(\alpha_{1-\mathrm{k}}\) accounts for heat transfer from the coil to the case by conduction, convection, and radiation. The notation for the other quantities is given in [1].

From (5), the surface temperature rise of the winding of the additional resistor is

\[ \vartheta=\frac{P}{K_{1-f}\cdot F_{\mathrm{k.cp}}} =\frac{P''_{\mathrm{k}}}{K_{1-f}}, \tag{8} \]

where \(P''_{\mathrm{k}}\) is the specific power per unit surface area of the case.

The entire burden of calculating the temperature rise is reduced to determining the heat-transfer coefficient of the system \(K_{1-f}\), and in essence to determining the conditional heat-transfer coefficient \(\alpha_{1-\mathrm{k}}\), which enters into \(K_{1-f}\).

In this connection let us analyze equation (7).

A. When \(\dfrac{F_1}{F'_{\mathrm{k}}}\to 0\) (a large case and a small coil), \(\alpha_{1-\mathrm{k}}\to 0\); in this case

\[ K_{1-f}= \frac{1} {\dfrac{1}{\alpha''}+\dfrac{\Delta_{\mathrm{k}}}{\lambda_{\mathrm{k}}}+\dfrac{1}{0}} \quad \text{also} \to 0 . \]

B. When \(\dfrac{F_1}{F'_{\mathrm{k}}}\to 1\) (the dimensions of the coil and case are close),

\[ \alpha_{1-\mathrm{k}}= \left(\frac{\lambda_{\mathrm{e}}}{\delta_{\mathrm{e}}} +\varepsilon C_0\Theta\right)\to\infty, \quad \text{since } \delta_{\mathrm{e}}\to 0 . \]

It may seem that the case \(\alpha_{1-\mathrm{k}}\to\infty\) can have no meaning at all. Here it is appropriate to note that \(\alpha_{1-\mathrm{k}}\), as a heat-transfer coefficient, is a conditional quantity, whereas \(\dfrac{1}{\alpha_{1-\mathrm{k}}}\), as the thermal resistance to the transfer of heat from the coil to the case, has a real meaning.

Then, both mathematically and physically, the tendency \(\alpha_{1-\mathrm{k}} \to \infty\) as \(\dfrac{F_1}{F'_{\mathrm{k}}} \to 0\) means that the thermal resistance in heat transfer from the coil to the case tends to zero, and consequently,

\[ K_{1-f}=\frac{1}{\dfrac{1}{\alpha''}+\dfrac{\Delta_{\mathrm{k}}}{\lambda_{\mathrm{k}}}} . \tag{9} \]

On the basis of (9), when \(F_1/F'_{\mathrm{k}} \to 1\), the heat-transfer coefficient \(K_{1-f}\), with accuracy up to the value of the thermal resistance of the case \(\dfrac{\Delta_{\mathrm{k}}}{\lambda_{\mathrm{k}}}\), is equal to the heat-transfer coefficient of the case reduced to the size of the coil, i.e., to the heat-transfer coefficient of the coil in an unbounded volume,

\[ K_{1-f}=\alpha''=\alpha_{\mathrm{coil}}=(\alpha_{\mathrm{k}}+\alpha_{\mathrm{l}})_{\mathrm{coil}} . \tag{10} \]

This means that, together with the value \(\alpha_{1-\mathrm{k}}\to 0\) as \(\dfrac{F_1}{F_{\mathrm{k}}}\to 1\), the thermal resistance, and consequently also the temperature rise of the winding \(\vartheta\), are completely determined by the value \(\alpha_{\mathrm{coil}}\) under conditions of free convection.

If one uses a diagram with two ordinates \(\alpha_{1-\mathrm{k}}\) and \(\alpha_{1-\mathrm{k}}\,(K_{1-f})\) and the abscissa \(\dfrac{F_1}{F'_{\mathrm{k}}}\), then the results of the analysis given above make it possible to plot only the limiting values of \(\alpha_{1-\mathrm{k}}\): when \(\dfrac{F_1}{F'_{\mathrm{k}}}\to 0\), \(\alpha_{1-\mathrm{k}}\to 0\), and when \(\dfrac{F_1}{F_{\mathrm{k}}}\to 1\), \(\alpha_{1-\mathrm{k}}\to \infty\) \((K_{1-f}\to \alpha_{\mathrm{coil}})\) (Fig. 1, \(A\) and \(B\)).

Fig. 1.

To clarify the nature of the dependence \(\alpha_{1-\mathrm{k}}=f\!\left(\dfrac{F_1}{F_{\mathrm{k}}}\right)\) for values of \(\dfrac{F_1}{F_{\mathrm{k}}}\) different from 0 and 1, experiments were carried out to determine the temperature rise of 3 additional resistors (8PA.732.022, 8PA.732.024

and 8PA.732.032), placed in a housing made of plexiglass of thickness \(\Delta_k=2\) mm, coated with black oil paint, with various values of \(\dfrac{F_1}{F_k}\). The experiments were carried out in two modes for each combination of coil and housing. The arrangement and procedure for carrying out these experiments are described in [1].

From the experimental data the values

\[ K_{1-f}=\left(\frac{P}{\vartheta\cdot F_{k.\mathrm{av}}}\right)^{\mathrm{ex}} \]

were determined, and then, from expression (6), the values \(\alpha_{1-k}^{\mathrm{ex}}\) were calculated; these are plotted on the graphs

\[ \alpha_{1-k}=f\left(\frac{F_1}{F_k}\right) \]

(Figs. 2, 3, and 4—circles). At the same time, for conditions corresponding to the conditions under which the experiments were carried out, \(\alpha_{1-k}\) was calculated from equation (7), the results of which are also plotted in Figs. 2, 3, and 4 (crosses).

Consideration of the graphs makes it possible to note that:

a) the character of the experimental dependence

\[ \alpha_{1-k}=f\left(\frac{F_1}{F_k}\right) \]

is in good agreement with the form of this dependence obtained by calculation from (7);

b) in the region of small values of \(\dfrac{F_1}{F_k}\) there is a noticeable deviation of the calculated curve from the experimental one; the cause of this may be a change in the character of heat exchange—from heat exchange in a closed volume to heat exchange in an unbounded volume—and, in connection with this, the inadmissibility of calculating this region by equations used for calculating heat exchange in a closed volume;

c) a substantial deviation of the calculated curve from the experimental one is also observed with small gaps between the coil and the housing (the latter ascending portions in all cases were obtained when the gaps were reduced to dimensions smaller than 5 mm). But such a dense arrangement of coils in a housing is practically not used;

d) the experimental curves

\[ \alpha_{1-k}=f\left(\frac{F_1}{F_k}\right) \]

start from the origin and, in the region \(\dfrac{F_1}{F_k}<0.4\), are lines close to straight lines, whose extensions pass near the limiting point \(B\) with ordinate \(\alpha_{\mathrm{cat}}\). The existence of such a relationship between \(\alpha_{1-k}\) and the heat-transfer coefficient of the coil under conditions of free cooling \(\alpha_{\mathrm{cat}}\) makes it possible to calculate \(\alpha_{1-k}\) from the proportion

\[ \frac{\alpha_{1-k}}{F_1/F_k}=\frac{\alpha_{\mathrm{cat}}}{1}, \]

\[ \alpha_{1-k}=\frac{F_1}{F_k}\cdot \alpha_{\mathrm{cat}}, \tag{11} \]

and to calculate the surface overheating of the winding of single multiplier resistors by a method whose essence is as follows:

1. The mean overheating of the coil under conditions of free cooling is determined:

\[ \vartheta_{\mathrm{sv}}=\frac{1}{\alpha_{\mathrm{cat}}}\cdot\frac{P}{F_1} =\frac{P'_{\mathrm{cat}}}{\alpha_{\mathrm{cat}}}. \tag{12} \]

Fig. 2. Coil 8PA.732.022: \(F_1 = 0.597 \cdot 10^{-3}\ \text{m}^2;\ d_2 = 10\ \text{mm};\)
\(P_1 = 0.858\ \text{W};\ \vartheta_{\text{sv}} = 72.5^\circ\text{C};\ \alpha_{\text{klt}} = 19.82\ \text{W}/(\text{m}^2\cdot\text{deg});\)
\(P_2 = 0.354\ \text{W};\ \vartheta_{\text{ov}} = 36.2^\circ\text{C};\ \alpha_{\text{klt}} = 16.38\ \text{W}/(\text{m}^2\cdot\text{deg}).\)

Fig. 3. Coil 8PA.732.024.

Visible labels in the graph: \(P_1\), \(P_2\), \(B_1\), \(B_2\), \(d_{\text{cat}}/2 = 6.75\), \(d_{\text{cat}}/2 = 6.65\), \(F_1\), \(F_2\), \(0.1\), \(0.2\), \(0.3\), \(0.4\), \(0.5\), \(5\), \(10\), \(15\); axis/unit labels include \(\dfrac{\text{W}}{\text{m}^2\cdot\text{deg}}\) and \(d_{1x}\).

\(P_1 = 2.85\ \text{W};\qquad \vartheta_{\text{sv}} = 98.1^\circ\text{C};\qquad F_1 = 1.485\cdot 10^{-3}\ \text{m}^2;\qquad d_2 = 22\ \text{mm};\)

\[ z_{\text{cat}} = 19.5\,\frac{\text{W}}{\text{m}^2\text{deg}} \]

\(P_2 = 1.17\ \text{W};\qquad \vartheta_{\text{sv}} = 49.4^\circ\text{C};\)

\[ z_{\text{cat}} = 15.9\,\frac{\text{W}}{\text{m}^2\text{deg}} \]

Fig. 4. Coil 8PA.732.032:

\[ F_1 = 7.319 \cdot 10^{-3}\ \text{m}^2;\qquad d_e = 37\ \text{mm}; \]

\[ P_1 = 7.50\ \text{W};\qquad \vartheta_{\mathrm{sv}} = 65^\circ\mathrm{C};\qquad \lambda_{\mathrm{cat}} = 15.7\ \frac{\text{W}}{\text{m}^2\cdot \text{deg}}; \]

\[ P_1 = 3.32\ \text{W};\qquad \vartheta_{\mathrm{sv}} = 34.1^\circ\mathrm{C};\qquad \alpha_{\mathrm{cat}} = 13.4\ \frac{\text{W}}{\text{m}^2\cdot \text{deg}}. \]

Visible labels in the graph: \(B_1\), \(B_2\), \(R_1\), \(R_2\); vertical axis \(\alpha_{\mathrm{cat}}\), \(\frac{\text{W}}{\text{m}^2\cdot\text{deg}}\); horizontal scale \(0.1\)–\(0.6\).

— from \(P'_{\text{coil}}\) and the diameter of the winding \(d_2\), using the nomogram in Fig. 5

\[ \left(P'_{\text{coil}}=\frac{P}{F_1}\text{ is the specific power of the coil}\right). \]

1. From \(\vartheta_{\text{sv}}\) and \(P'_{\text{coil}}\), by (12), \(\alpha_{\text{coil}}\) is calculated:

\[ \alpha_{\text{coil}}=\frac{P'_{\text{coil}}}{\vartheta_{\text{sv}}}. \tag{13} \]

1. One finds

\[ \alpha_{1-\text{k}}=\frac{F_1}{F_{\text{k}}}\alpha_{\text{coil}}. \]

1. The temperature rise of the case \(\vartheta_3\) is found using the nomogram [1] for the case, analogous to the nomogram in Fig. 5 for the coil—by the dia-

Fig. 5.

meter of the case \(D\) and the specific power on the case

\[ P''_{\text{k}}=\frac{P}{F''_{\text{k}}}. \]

1. From \(\vartheta_3\) and \(P''_{\text{k}}\), \(\alpha''\) is calculated:

\[ \alpha''=\frac{P''_{\text{k}}}{\vartheta_3}. \tag{14} \]

1. The conditional heat-transfer coefficient is determined:

\[ K_{1-f}=\frac{1}{\dfrac{1}{\alpha''}+\dfrac{\Delta_k}{\lambda_k}+\dfrac{1}{\alpha_{1-k}}}. \]

1. The winding temperature rise is calculated

\[ \vartheta=\frac{P''_k}{K_{1-f}},\ ^\circ\mathrm{C}. \tag{15} \]

The results of an experimental check of the agreement between the observed temperature rises and those calculated by the method set forth are given in Table 1.

Table 1

Discrepancy between calculated temperature rises and experimental ones (in %)

For slots \(\delta=2\) mm the results are not given, since the task of covering by calculation the region of slots \(\delta<5\) mm was not set.

Current regime I corresponds to a temperature rise \(\vartheta \simeq 80^\circ\mathrm{C}\) under free-cooling conditions; regime II is the same, \(\vartheta \simeq 40^\circ\mathrm{C}\). Coil 8PA.732.032 is sectioned.

It is evident from the table that the abbreviated method agrees satisfactorily with the experimental data.

The more substantial errors in calculating the temperature rise of the sectioned coil 8PA.732.032, and the occurrence of errors for each of the coils only in the positive or negative region, indicate the need to refine the position of the point \(z_{\mathrm{kat}}=(z_k+z_l)_{\mathrm{kat}}\) on the diagram \(z_{1-k}=f\left(\dfrac{F_l}{F_k}\right)\).

REFERENCES

1. G. I. Fuks, A. S. Lyatikov. Calculation of the surface temperature rise of the winding of additional resistors of electrical measuring instruments. In the present collection.