Heating of Resistance Elements of a Rod Shunt under Direct Current
Abstract Generated abstract
This paper analyzes transient heating of the resistance rods in a direct current shunt during short switching current surges, which may exceed rated current and therefore govern allowable rod dimensions. The shunt rod is modeled as a thin cylinder with uniform internal heat generation, insulated lateral surface, and ends held at the initial terminal temperature, and the nonstationary heat conduction problem is solved by Laplace transformation. The resulting series expression, together with a practical one-term approximation for the hottest point, relates temperature rise to current density and shunt parameters. Calculations for class 0.5 shunts suggest that permissible current densities can be raised to about 4 A/mm² without exceeding the specified temperature limit, allowing substantial reductions in resistance-element volume.
Full Text
HEATING OF THE RESISTANCE ELEMENTS OF A ROD SHUNT UNDER THE ACTION OF DIRECT CURRENT
Yu. A. Korolenko
(Presented by Prof. G. I. Fuks, Doctor of Technical Sciences)
In the communication “On the question of the thermal calculation of rod shunts” [6], relations were given that make it possible to calculate the temperature field of a rod shunt under steady-state conditions when direct current passes through the shunt. On the basis of these relations it can be shown that very high volumetric heat generations may be permitted in shunts without exceeding the maximum temperatures of the system allowable under GOST 8042-56 [4].
At the same time, it is known that at the moment of switching into circuits, short-time surges of current are produced, exceeding the rated current by several (up to 10–12) times. It is obvious that precisely these short-time, but very considerable, heat generations must determine the limit to which the dimensions of the resistance elements (rods) of the shunt can be reduced.
The investigation of the temperature field of a shunt during the period of flow of the extracurrent is the subject of the present work.
In the consideration we assume:
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The resistance of the material of the rods (manganin) is higher than the resistance of the terminals and buses (copper) by 25–35 times. Therefore we assume that during the period of flow of the extracurrent, heat is liberated only in the rods, disregarding heat generation in the buses and terminals. Owing to the massiveness and high thermal conductivity of the terminals, their temperature over the short period of action of the extracurrents does not have time to change appreciably and may be taken as constant, equal to the initial temperature \(t_0\).
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Heat transfer from the shunt rods to the surrounding medium—by convection and radiation—is determined by the temperature difference between the shunt rods and the surrounding medium. During the short period of heating under the action of the extracurrent, this heat transfer cannot be significant and is not taken into account in the consideration.
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The character of the action of the extracurrent depends on the electrical parameters of the shunt circuit and, for series shunts, as a rule, is unknown. Therefore GOST 8042-56 proposes that the extracurrent be considered constant in magnitude and acting during the time \(\tau = \dfrac{50}{N^2}\) seconds [4].
The value \(N=\dfrac{I_{\max}}{I_{\mathrm{n}}}\) is given by the State Standard as a function of \(I_{\mathrm{n}}\), the rated current for the shunt.
On the basis of the foregoing, the problem may be formulated as follows.
A thin cylinder of length \(2L\) and cross-sectional area \(F\) is given. The initial temperature of the rod is equal to \(t_0\). Inside the rod there acts
Fig. 1.
a constant heat source with specific power
\[ q_v=\frac{0.86N^2 I_{\mathrm{n}} V_{\mathrm{n}}}{2L\cdot F} \]
\(\mathrm{kcal}/\mathrm{m}^3\cdot\mathrm{h}\). Heat is removed from the rod only by thermal conduction through the ends, which have the constant temperature \(t_0\). The lateral surface of the rod is assumed insulated. It is required to find the temperature distribution along the length of the rod at any instant of time. Mathematically, the problem is described by the following system of equations:
\[ \frac{\partial t}{\partial \tau} = a\frac{\partial^2 t}{\partial x^2} + \frac{q_v}{c\gamma}, \tag{1} \]
\[ t_{x0}=t_0, \tag{2} \]
\[ t_{L\tau}=t_0, \tag{3} \]
\[ \frac{\partial t_{x\tau}}{\partial x_{x=0}}=0. \tag{4} \]
Applying a Laplace transform of the form
\[ F(s)=\int_0^\infty T(x\tau)e^{-s\tau}\,d\tau \qquad (\text{see [2], [3]}) \]
to the system of equations 1—4, we obtain:
\[ T''_{xs} - \frac{s}{a} \left( T_{xs} - \frac{t_0}{s} - \frac{q_v}{s^2 c\gamma} \right) =0, \tag{5} \]
\[ T_{Ls}=\frac{t_0}{s}, \tag{6} \]
\[ \frac{\partial T_{xs}}{\partial x_{x=0}}=0. \tag{7} \]
The solution of equation (5) will be
\[ T_{xs}=Ae^{\sqrt{\frac{s}{a}}\,x}+Be^{-\sqrt{\frac{s}{a}}\,x}+\frac{t_o}{s}+\frac{q_v}{s^2 c\gamma}. \tag{8} \]
From conditions (6) and (7) it follows that
\[ A=B=-\frac{q_v}{s^2 c\gamma\left(e^{\sqrt{\frac{s}{a}}\,L}+e^{-\sqrt{\frac{s}{a}}\,L}\right)} \tag{9} \]
and the solution of equation (5) takes the form:
\[ T_{xs}=\frac{q_v}{s^2 c\gamma}+\frac{t_o}{s} -\frac{q_v\,\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s^2 c\gamma\,\operatorname{ch}\sqrt{\frac{s}{a}}\,L}. \tag{10} \]
Applying the inverse Laplace transform to the first two terms of equation (10), we find their originals
\[ L^{-1}\left(\frac{q_v}{s^2 c\gamma}\right)=\frac{q_v}{c\gamma}\tau, \tag{11} \]
\[ L^{-1}\left(\frac{t_o}{s}\right)=t_o. \tag{12} \]
To find the original of the third term of equation (10), we rewrite it in the form
\[ \frac{q_v\,\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {sc\gamma\,\operatorname{ch}\sqrt{\frac{s}{a}}\,L}\cdot\frac{1}{s}. \]
The first part of the complex is a ratio of generalized polynomials. Applying the expansion theorem, we find:
\[ L^{-1}\left( \frac{q_v\,\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {sc\gamma\,\operatorname{ch}\sqrt{\frac{s}{a}}\,L}\cdot\frac{1}{s} \right) = \frac{q_v}{c\gamma}\tau -2\frac{q_v L^2}{\lambda} \sum_{1}^{\infty} \frac{(-1)^{n+1}}{\mu_n^3} \left(1-e^{-\mu_n^2\frac{a\tau}{L^2}}\right) \cos\mu_n\frac{x}{L}, \tag{13} \]
where
\[ \mu_n=(2n-1)\frac{\pi}{2}. \]
The equation of the temperature field of the shunt rod during heating will now be written as:
\[ t_{x\tau}-t_o=2\frac{q_v L^2}{\lambda}\sum_1^\infty \frac{(-1)^{n+1}}{\psi_n^3} \left(1-e^{-\psi_n^2 \frac{a\tau}{L^2}}\right) \cos\psi_n\frac{x}{L}. \tag{14} \]
Substitution of the boundary conditions into (14) gives:
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For \(\tau=0\)
\[ t_{xo}=t_o. \] -
For \(x=L\)
\[ t_L=t_o. \] -
\[ \left.\frac{\partial t}{\partial x}\right|_{x=0}=0. \]
It can be shown that for \(x=0\) and \(\tau=\infty\)
\[ t_{o\tau}-t_o=0.5\,\frac{q_v L^2}{\lambda}. \]
The latter coincides with the known formula for the temperature difference in a plane wall with internal heat generation [4,5].
Equation (14) is a rapidly convergent series. Calculations show that even for \(x=0\), neglecting all terms of the series except the first introduces an error not exceeding 2% in the direction of increasing the result. With this simplification, equation (14) takes the form
\[ t_{o\tau}-t_o=0.506\,\frac{q_v L^2}{\lambda} \left(1-e^{-2.47\frac{a\tau}{L^2}}\right). \tag{15} \]
Substituting in (14) and (15)
\[ q_v=\frac{0.86\,N^2 I_n V_n}{F\cdot 2L} \left[\frac{\text{kcal}}{m^3\,\text{h}}\right] \quad \text{and} \quad L=\frac{V_n}{2\rho i_n}, \]
where
\[ i_n=\frac{I_n}{F}\left[\frac{A}{\text{mm}^2}\right], \]
the equations take the form:
\[ t_{x\tau}-t_o= 215\cdot 10^3 N^2\,\frac{V_n^2}{\lambda\rho} \sum_1^\infty \frac{(-1)^{n+1}}{\psi_n^3} \left(1-e^{-\psi_n^2\frac{a\rho^2}{V_n^2}i_n^2\tau}\right) \cos\psi_n\frac{x}{L}, \tag{14a} \]
\[ t_{o\tau}-t_o= 110\cdot 10^3 N^2\,\frac{V_n^2}{\lambda\rho} \left(1-e^{-9.88\frac{a\rho^2}{V_n^2}i_n^2\tau}\right). \tag{15a} \]
From expressions (14a) and (15a) it is seen that the temperature rise of each point of the rod during the period of action of the switching-on current depends only on the specific current density
\[ i_n=\frac{I_n}{F},\quad \left[\frac{A}{\text{mm}^2}\right] \]
in the shunt rods.
According to GOST 8042-56, the temperature of the hottest part of the resistance of a class 0.5 shunt must not exceed the ambient temperature by more than \(120^\circ\text{C}\).
The maximum current-density values for shunts of various types of class 0.5, determined from this condition by equation (15a), are given in Table 1.
In the calculation it was assumed:
temperature-conductivity coefficient of manganin \(a = 20 \cdot 10^{-3}\ \text{m}^2/\text{hour};\)
thermal-conductivity coefficient of manganin \(\lambda = 20\ \text{kcal}/\text{m}\cdot\text{hour}\cdot\text{deg};\)
specific resistance of manganin \(\rho = 0.5\ \Omega\cdot\text{mm}^2/\text{m}.\)
Table 1
| \(J_n\) | \(N=\dfrac{J_{\max}}{J_n}\) | \(\tau=\dfrac{50}{N^2}\) | \multicolumn{4}{c}{\(i\) in \(\text{A}\cdot\text{mm}^{-2}\) for shunts with \(\gamma_n\)} |
|---:|---:|---:|---:|---:|---:|---:|
| amperes | | seconds | \(45\ \mu\Omega\) | \(75\mu\Omega\) | \(100\mu\Omega\) | \(150\mu\Omega\) |
| 0–500 | 10 | 0.5 | 4.05 | 93.91 | 4.09 | 3.90 |
| 500–2000 | 3 | 6 | 4.73 | 4.51 | 3.94 | 3.90 |
| 2000 | 1.5 | 22.3 | -- | 6.02 | 4.59 | 4.24 |
It is evident from the table that for shunts of class 0.5 the current density may be increased, without exceeding the permissible temperature values, from \(1\text{–}2\ \dfrac{\text{A}}{\text{mm}^2}\) to \(4\ \dfrac{\text{A}}{\text{mm}^2}\), which makes it possible to reduce the volume of the shunt resistance elements by \(4 \div 16\) times.
Conclusion
The solution presented makes it possible to determine the possible overheating of the resistance elements of shunts when they are heated by the making current, on the basis of which one can take a more substantiated approach to the design of shunts.
LITERATURE
- GOST 8042-56 (Calibrated shunts), Moscow, 1956.
- Lykov A. V. Thermal Conductivity of Nonstationary Processes, GITTL, M.-L., 1948.
- Lykov A. V. Theory of Heat Conduction, Moscow, GITTL, 1952.
- Mikheev M. A. Fundamentals of Heat Transfer, Moscow, GEI, 1956.
- Gröber and Erk. Fundamentals of the Theory of Heat Exchange, GITTL, 1936.
- Korolenko Yu. A. On the thermal calculation of rod shunts. Fizika, No. 3, 1959.
- Buchgolz. Besondere Probleme der erwarmung elektrischer Leiter. Zeitschrift für angew. Math. u. Mech. B. 9. H 4, 1929.