On Heating of a Shunt under the Action of Overload Currents
Abstract Generated abstract
The paper addresses the calculation of maximum heating in direct current shunts subjected to overload currents, a quantity needed for assessing thermal stability and resistance stability in manganin elements. It develops a one dimensional transient heat conduction model for shunt plates, justified by small Biot numbers, including heat removal to the surrounding medium and to terminals treated as constant temperature boundaries. Using separation of variables, the work derives expressions for the temperature field and for the maximum temperature at the plate midpoint under both initially cold and preheated conditions. The example shows that simplified adiabatic estimates may be adequate in some cases but can seriously understate temperatures when overloads act on a shunt already heated by rated current.
Full Text
On the Heating of a Shunt Under the Action of Overload Currents
L. G. Fuks
(Presented by the Department of Boiler Construction and Boiler Installations)
In calculations for many electrical-engineering devices it is necessary to determine the maximum heating temperatures under the action of overload currents. These temperatures make it possible to assess the thermal stability of the apparatus. In certain technically important cases, knowledge of the maximum temperatures is important for clarifying the stability of materials over time.
Below a method is proposed for determining the maximum heating temperature of direct-current shunts.
As is known, shunts with manganin plates or rods do not permit high heating because of the possible disturbance of the stability of the resistance. The operating temperatures of the most strongly heated points of a shunt are usually limited to the range 150–185°C. Under these conditions the instrument can remain in operation for a long time without any noticeable increase in the measurement error associated with maintaining the constancy of the resistance.
The simplest determination of the heating temperature is the calculation based on neglecting heat transfer from the plates during the action of overload currents. It is assumed that all the supplied heat goes into increasing the heat content of the plates and that during its action the overload current retains a constant value. In other words, such a simplification of the calculation is based on neglecting the removal of heat into the terminals and heat transfer to the surrounding medium. This method was used, for example, in work [1].
Let us give a solution of the problem of heating a shunt with allowance for heat removal from the plates. For this purpose we shall consider a simplified diagram of the shunt, Fig. 1.
The coefficient of thermal conductivity of manganin is of the order of \(21—25.5 \dfrac{\mathrm{W}}{\mathrm{m}\cdot{}^\circ\mathrm{C}}\). In shunts of ordinary designs, the coefficient of heat transfer by convection has, on average, a value of about \(11.6 \dfrac{\mathrm{W}}{\mathrm{m}^2{}^\circ\mathrm{C}}\).
The width of the plates varies within the range 16–40 mm. The value of the Biot criterion, calculated with respect to the width of the plate, will lie within the limits
\[ \mathrm{Bi}=0.008—0.02. \]
For such small values of the Biot criterion it is permissible to neglect
the spread of heat over the width of the plate, assuming that the temperature of the plates varies only along their length.
The heat balance of an elementary volume \(f\,dx\), released in the plate during the time \(d\tau\), after cancellations and transformations gives the differential equation of the temperature field of the plate
\[ \frac{1}{a}\vartheta'_{\tau}=\vartheta''_{xx}-m^{2}\vartheta+\frac{q'_v}{\lambda}. \tag{1} \]
Here
\(a\) — coefficient of thermal diffusivity,
\(\lambda\) — coefficient of thermal conductivity,
\(c\) — heat capacity,
\(\gamma\) — specific weight,
\(\vartheta\) — excess of the plate temperature over the ambient temperature,
\(\tau\) — time,
\(q'_v\) — volumetric heat generation.
In equation (1) the known notation has been introduced
\[ m^{2}=\frac{\alpha P}{\lambda f}, \]
where \(\alpha\) — heat-transfer coefficient,
\(P\) — perimeter,
\(f\) — cross-sectional area.
The shunt terminals are massive parts made of a material of high thermal and electrical conductivity (usually copper).
Fig. 1. 1—plate; 2—terminal; 3—current-carrying bus. In the upper drawing, curves are shown of the approximate distribution of temperatures along the plate at different moments of time during heating.
Under the action of short-duration overload currents, it is permissible to regard their temperature as constant.
Then the boundary conditions of the problem are written as follows:
\[ \vartheta\left(\tau,\frac{l}{2}\right)=\vartheta_{\mathrm{n}}, \tag{2} \]
\[ \vartheta'_x(\tau,0)=0, \tag{3} \]
where \(l\) is the length of the plates,
\(\vartheta_{\mathrm{n}}\) is the excess temperature of the terminal.
Condition (3) is the mathematical expression of the symmetry of the temperature field of the plate.
The choice of the initial condition depends on the type of initial temperature distribution along the shunt. In the general case
\[ f(x)=\vartheta(0,x). \tag{4} \]
If heating begins from the ambient temperature, then, to simplify the solution, we take
\[ f(x)=0,\qquad \vartheta_{\mathrm{n}}=0. \]
A more complicated case will be the sudden heating of a shunt which, before this, had been heated for a long time under normal load. Then by \(f(x)\) we shall understand the initial temperature distribution along the shunt. It is clear that, under our conditions, \(f(x)\) will be a continuous, symmetric, and even function of \(x\).
To solve equation (1), we assume that the temperature is represented as the sum of two as yet unknown functions of the form
\[ \vartheta(\tau,x)=Z(\tau,x)+w(x). \tag{5} \]
Since \(w(x)\) depends only on the coordinate, substitution into (1) gives
\[ \frac{1}{a}Z'_\tau=Z''_{xx}-m^2Z+\left\{w''-m^2w+\frac{q_e'}{\lambda}\right\}. \tag{6} \]
We define the function \(w\) so that the expression in braces is equal to zero. In doing so, for \(w\) we require the boundary conditions to be satisfied in the form
\[ w\left(\frac{l}{2}\right)=\vartheta_{\mathrm{n}}, \tag{7} \]
\[ w'(0)=0. \tag{8} \]
Then, after solving the differential equation, \(w\) takes the form
\[ w=\frac{q_v}{\lambda m^2}-\left(\frac{q_v}{\lambda m^2}-\vartheta_{\mathrm{n}}\right) \frac{\operatorname{ch}mx}{\operatorname{ch}\frac{ml}{2}}. \tag{9} \]
To find the second function, from expression (6) we obtain
\[ \frac{1}{a}Z'_\tau=Z''_{xx}-m^2Z, \tag{10} \]
with the initial and boundary conditions
\[ Z(0,x)=f(x)-w(x)=\varphi(x), \tag{11} \]
\[ Z'(\tau,0)=0, \tag{12} \]
\[ Z\left(\tau,\frac{l}{2}\right)=0. \tag{13} \]
Applying the usual method of separation of variables, we have
\[ Z(\tau,x)=\sum_{1}^{\infty} C_n \exp\left[-a\left(\beta_n^2+m^2\right)\tau\right]\cos \beta_n x, \tag{14} \]
where the eigenvalues are found from
\[ \beta_n=(2n-1)\frac{\pi}{l}; \qquad n=1,2,3,\ldots \tag{15} \]
The constants \(C_n\) are determined from the initial condition as the coefficients of the expansion of the function \(\varphi(x)\) in a Fourier series
\[ C_n=\frac{4}{l}\int_{0}^{l/2} \varphi(x)\cos \beta_n x\,dx. \tag{16} \]
The final form of the integral is established for a specified initial temperature distribution. In practice, one may encounter sudden heating from ambient temperature, when
\[ f(x)=0, \tag{17} \]
and the case when the shunt has been heated to some stationary state under internal heat generation \(q_v\).
For the latter, as shown in [1],
\[ f(x)=\frac{q_v}{\lambda m^2}-\left(\frac{q_v}{\lambda m^2}-\vartheta_{\mathrm{н}}\right) \frac{\operatorname{ch} mx}{\operatorname{ch}\dfrac{ml}{2}}. \tag{18} \]
Let us note that for \(q_v=0\) and \(\vartheta_{\mathrm{н}}=0\), expression (18) identically becomes equality (17). Therefore, taking as the basis the temperature distribution described by expression (18), after transformations we obtain
\[ \vartheta(\tau,x)= \frac{q'_v}{\lambda m^2} -\left(\frac{q'_v}{\lambda m^2}-\vartheta_{\mathrm{н}}\right) \frac{\operatorname{ch} mx}{\operatorname{ch}\dfrac{ml}{2}} \]
\[ -\frac{4(q'_v-q_v)}{\pi l} \sum_{1}^{\infty} \frac{(-1)^{n-1}\exp\left[-a\left(\beta_n^2+m^2\right)\tau\right]\cos \beta_n x} {(2n-1)\left(\beta_n^2+m^2\right)}, \tag{19} \]
The most heated point will be at the middle of the plate length, at \(x=0\)
\[ \vartheta(\tau,0)= \frac{q'_v}{\lambda m^2} -\left(\frac{q'_v}{\lambda m^2}-\vartheta_{\mathrm{н}}\right) \frac{1}{\operatorname{ch}\dfrac{ml}{2}} - \]
\[ -\frac{4(q'_v-q_v)}{\pi l} \sum_{1}^{\infty} \frac{(-1)^{n-1}\exp\left[-a\left(\beta_n^2+m^2\right)\tau\right]} {(2n-1)\left(\beta_n^2+m^2\right)}. \tag{20} \]
In most cases of applying formulas (19) and (20), one may restrict oneself to the first term of the infinite sum, since the subsequent terms decrease very rapidly.
Calculations of the maximum temperature during heating were carried out. For one of the variants, for example, the following data were obtained:
overload. When an overload current three times as large acts on the shunt for 5.55 sec, the calculation performed by the method of work [1] gives a maximum overheating of 150°C. Applying formula (20) and reckoning the heating from the ambient temperature, a value of the maximum overheating temperature of 146.5°C is obtained. The same formula (20), used to calculate the heating of a shunt preheated by the rated current, gives an overheating temperature of 255°C.
Conclusions
For an approximate determination of the maximum overheating temperature of shunt elements under the action of overload currents, one may use the dependence published in work [1]. If it is necessary to refine the calculation, one should use the more accurate expressions (19) and (20) of the present work.
In particular, checking the temperatures by the refined dependences is recommended in those cases where there is a possibility of overload currents acting on a preheated shunt.
References
- L. G. Fuks. Thermal calculation of plate shunts for large direct currents. Priborostroenie, No. 3, 1960.
- A. V. Lykov. Theory of Heat Conduction. GITTL, Moscow, 1952.
- V. I. Smirnov. A Course of Higher Mathematics. Vol. 3, Part 2, 7th ed., GITTL, Moscow, 1956.
- H. S. Carslaw. Theory of Heat Conduction. OGIZ, Moscow, 1947.