Calculation of the Temperature Field of the Windings of Additional Resistors in Electrical Measuring Instruments
Abstract Generated abstract
This paper presents a method for calculating the temperature field in windings of additional resistors used in electrical measuring instruments, based on analytical relations for a short cylindrical winding with internal heat generation. Because direct use of these relations requires heat-transfer coefficients that depend on unknown surface temperatures, the authors generalize experimental data from 25 coil samples using dimensionless Pomerantsev and Biot criteria. Empirical power-law equations are derived that relate average overheating of the outer, inner, and end cooling surfaces to a geometric parameter of the winding, enabling subsequent determination of heat-transfer coefficients. Comparison with measurements for 15 selected specimens indicates that calculated temperature rises generally agree with experiment within about 15 percent.
Full Text
CALCULATION OF THE TEMPERATURE FIELD OF WINDINGS OF ADDITIONAL RESISTORS IN ELECTRICAL MEASURING INSTRUMENTS
L. S. Konovalova, A. S. Lyadikov
(Presented by Prof., Dr. G. I. Fuks)
The calculation is based on the equations [1]
\[ t(r,y)=\frac{q_v}{2\lambda \varepsilon} \left[ -\frac{r^2}{2}-r_0^2 \ln r-(\varepsilon-1)y^2+ \frac{\lambda\left(R_2^2-r_0^2\right)}{\alpha_{\mathrm{н}}R_2} +\frac{R_2^2-2r_0^2\ln R_2}{2} +\frac{\varepsilon-1}{3}L^2 \right]+t_{\mathrm{ж}}, \tag{1} \]
\[ r_0^2= \frac{ R_1R_2\left[2\lambda(\alpha_{\mathrm{н}}R_1+\alpha_{\mathrm{в}}R_2)+ \alpha_{\mathrm{в}}\alpha_{\mathrm{н}}\left(R_2^2-R_1^2\right)\right] }{ 2\lambda(\alpha_{\mathrm{н}}R_2+\alpha_{\mathrm{в}}R_1)+ 2\alpha_{\mathrm{н}}\alpha_{\mathrm{в}}R_1R_2\ln\frac{R_2}{R_1} }, \tag{2} \]
\[ \varepsilon= \frac{3\lambda\alpha_{\mathrm{т}}}{\alpha_{\mathrm{т}}L^2+3\lambda L} \left\{ \frac{R_2}{2\alpha_{\mathrm{н}}} \left[1-\left(\frac{r_0}{R_2}\right)^2\right] +\frac{R_2^2}{\lambda} \left[ \frac{1}{8}-\frac{1}{8}\left(\frac{R_1}{R_2}\right)^2 -\frac{1}{4}\left(\frac{r_0}{R_2}\right)^2 -\left(\frac{r_0}{R_2}\right)^2 \left(\frac{R_1}{R_2}\right)^2 \frac{\ln\frac{R_1}{R_2}}{2\left[1-\left(\frac{R_1}{R_2}\right)^2\right]} \right] \right\}+1 . \tag{3} \]
\(\varepsilon\) and \(r_0\) can be determined from nomograms, which are not presented here; this considerably simplifies the calculation.
In order to use (1)–(3), it is necessary to know the effective heat-transfer coefficients over the cooling surfaces of the winding, \(\alpha_{\mathrm{н}}, \alpha_{\mathrm{в}}, \alpha_{\mathrm{т}}\). Calculating \(\alpha_{\mathrm{н}}, \alpha_{\mathrm{в}}, \alpha_{\mathrm{т}}\) from known criterion equations requires knowledge of the temperatures of the corresponding surfaces, which in the problem posed are the unknown quantities. Usually such problems are solved by the method of successive approximations. The laborious operations of this method can be avoided if the experimental material on the overheating of the surface points of the winding is generalized and the generalized equations are used to calculate the average overheating of the winding surfaces.
The experimental procedure is described in [2]. On 25 samples of coils of additional resistors, the temperature was measured at 6 points (Fig. 1), which made it possible to find the average overheating values
Fig. 1
over the outer, inner, and end cooling surfaces of the winding
\[ \bar{\vartheta}_{\mathrm{n}}=\frac{\vartheta_1+\vartheta_3}{2},\qquad \bar{\vartheta}_{\mathrm{v}}=\frac{\vartheta_2+\vartheta_4}{2},\qquad \bar{\vartheta}_{\mathrm{t}}=\frac{\vartheta_3+\vartheta_4}{2}. \]
Writing the equations for \(\bar{\vartheta}_{\mathrm{n}}, \bar{\vartheta}_{\mathrm{v}}, \bar{\vartheta}_{\mathrm{t}}\) according to (1) in dimensionless form, we obtain
\[ \frac{1}{\mathrm{Po}_{\mathrm{n}}} = \frac{1}{4\varepsilon} \left[ \frac{1}{2\mathrm{Bi}_{\mathrm{n}}} \frac{R_2^2-r_0^2}{L^2} - \frac{1}{3}(\varepsilon-1) \right], \tag{4} \]
\[ \frac{1}{\mathrm{Po}_{\mathrm{t}}} = \frac{1}{4\varepsilon} \left[ \frac{1}{2} \frac{R_2^2-R_1^2}{L^2} + \left(\frac{r_0}{L}\right)^2 \ln\frac{R_1}{R_2} + \frac{1}{\mathrm{Bi}_{\mathrm{n}}} \frac{R_2^2-r_0^2}{L^2} - \frac{4}{3}(\varepsilon-1) \right], \tag{5} \]
\[ \frac{1}{\mathrm{Po}_{\mathrm{v}}} = \frac{1}{4\varepsilon} \left[ \frac{R_2^2-R_1^2}{L^2} + 2\left(\frac{r_0}{L}\right)^2 \ln\frac{R_1}{R_2} + \frac{1}{\mathrm{Bi}_{\mathrm{n}}} \frac{R_2^2-r_0^2}{L^2} - \frac{1}{3}(\varepsilon-1) \right], \tag{6} \]
where
\[ \mathrm{Po}_{\mathrm{n}}=\frac{q_v L^2}{\lambda \bar{\vartheta}_{\mathrm{n}}}, \qquad \mathrm{Po}_{\mathrm{v}}=\frac{q_v L^2}{\lambda \bar{\vartheta}_{\mathrm{v}}}, \qquad \mathrm{Po}_{\mathrm{t}}=\frac{q_v L^2}{\lambda \bar{\vartheta}_{\mathrm{t}}} \quad\text{are the Pomerantsev criteria,} \]
\[ \mathrm{Bi}_{\mathrm{n}}=\frac{\alpha_{\mathrm{n}}R_2}{\lambda} \quad\text{is the Biot criterion.} \]
Numerical calculations showed that, within the limits of the experimental material considered, the term on the right-hand side of equations (4)–(6) containing \((\varepsilon-1)\), as well as the term \(\left(\frac{r_0}{L}\right)^2 \ln \frac{R_1}{R_2}\), are relatively small, have different signs, and therefore do not exert a substantial influence on the result. Proceeding from the assumption that the changes in \(\varepsilon\) and \(\mathrm{Bi}_{\mathrm{n}}\) are small, and that \(r_0\) is close to \(R_1\), it may be expected that the criterion \(\mathrm{Po}\) is a function of the geometric complex
\[ \frac{R_2^2-R_1^2}{L^2}, \]
or
\[ \mathrm{Po}\approx f\left(\frac{d_2^2-d_1^2}{h^2}\right), \]
where
\[ d_1=2R_1,\qquad d_2=2R_2,\qquad h=2L. \]
Fig. 2. Vertical axis: \(P_{0\mathrm{н}}\). Horizontal axis: \(\dfrac{d_2^2 - d_1^2}{h^2}\).
Fig. 3. Vertical axis: \(P_{0\mathrm{в}}\). Horizontal axis: \(\dfrac{d_2^2 - d_1^2}{h^2}\).
From these considerations, for each coil the quantities \(\dfrac{d_2^2-d_1^2}{h^2}\) were calculated, as well as the criteria \(\mathrm{Po}_{\mathrm{n}}, \mathrm{Po}_{\mathrm{z}}, \mathrm{Po}_{\mathrm{t}}\) for all temperature regimes, which were then averaged (for any coil the deviation of the mean \(\mathrm{Po}\) criterion from the true values, calculated from the greatest and least overheatings, did not exceed 2–3%). The results of the calculations are presented graphically in Figs. 2, 3, 4.
Fig. 4
The equations of the curves \(\mathrm{Po} \simeq f\left(\dfrac{d_2^2-d_1^2}{h^2}\right)\), obtained with the aid of an electronic digital computer, have the form
\[ \mathrm{Po}_{\mathrm{n}} = 1.74\left(\frac{d_2^2-d_1^2}{h^2}\right)^{-0.75}, \tag{7} \]
\[ \mathrm{Po}_{\mathrm{z}} = 1.56\left(\frac{d_2^2-d_1^2}{h^2}\right)^{-0.775}, \tag{8} \]
\[ \mathrm{Po}_{\mathrm{t}} = 1.83\left(\frac{d_2^2-d_1^2}{h^2}\right)^{-0.8}. \tag{9} \]
The limits of applicability of (7)–(9) are as follows:
\[ 0.5 < \mathrm{Po} < 23, \qquad 0.033 < \frac{d_2^2-d_1^2}{h^2} < 4.3. \]
Equations (7)–(9) make it possible to find the mean overheatings over the cooling surfaces of the coil and then, by known criterion ...
by the equations to determine the convective part \(\alpha_{\mathrm{nk}}, \alpha_{\mathrm{tk}}\), as well as the complete values \(\alpha_{\mathrm{n}}\) and \(\alpha_{\mathrm{t}}\), taking into account that the latter represent the sum of the radiant and convective components. The heat dissipated from the inner surface of the coil is calculated from the balance equation
\[ Q_{\mathrm{v}} = Q - (Q_{\mathrm{n}} + Q_{\mathrm{t}}), \tag{10} \]
after which \(\alpha_{\mathrm{v}}\) is found without difficulty.
For 15 arbitrarily selected specimens, a comparison was made of the temperature rises calculated for points 1, 2, 3, 4 and by equations (1)–(3) with the corresponding experimental data; the heat-transfer coefficients in this case were determined from the average temperature rises calculated using equations (7)–(9), and the thermal-conductivity coefficients from the data of [3]. As a result of this comparison it turned out that the calculation errors lie mainly within \(\pm 15\%\). Naturally, it is difficult to expect higher calculation accuracy for real resistors, taking into account the approximate nature of the solution (1)–(3), the approximation of the experimental data by equations (7)–(8), and the fluctuations of the experimental data on the thermal-conductivity coefficients of windings associated with unequal winding density and with deviations of winding wires from their nominal values.
Conclusions
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Relations (1)–(3), when reliable initial data on \(\alpha_{\mathrm{n}}, \alpha_{\mathrm{v}}, \alpha_{\mathrm{t}}\), and \(\lambda\) are used, make it possible to calculate with sufficient accuracy the temperature field of windings of electrical coils.
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Generalization of experimental data on the average temperatures of heat-emitting surfaces of electrical coils in the form of equations (7)–(9) makes it possible to use such equations for calculating the averaged temperatures of the cooling surfaces of a winding and for the subsequent determination of \(\alpha_{\mathrm{n}}, \alpha_{\mathrm{v}}, \alpha_{\mathrm{t}}\).
References
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Lyalikov A. S., Konovalova L. S. Temperature field of a short tube with internal heat sources. IFZh, vol. XV, no. 3, 1968.
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Lyalikov A. S., Zagromov Yu. A., Ershova L. S. Experimental data on the dissipation power of additional resistors of electrical measuring instruments (under conditions of free convection). Izv. TPI, vol. 137, 1965.
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Konovalova L. S., Lyalikov A. S. Thermal-conductivity coefficients of windings made of PEMS-grade wire. In the present collection.