HEATING OF RESISTANCE ELEMENTS OF ROD SHUNTS UNDER THE ACTION OF SWITCH-ON CURRENT

Yu. A. Korolenko

(Presented by Prof. G. I. Fuks, Doctor of Technical Sciences)

In [8] a method is considered for calculating the temperature of the resistance elements of rod shunts at the moment the current is switched on.

The switch-on current \(I_{\max}\) is assumed, according to GOST 8042-56 [4], to be constant in magnitude, exceeding by \(N\) times the rated current of the shunt \(I_n\), and acting for

\[ \tau = \frac{50}{N^2} \]

seconds. The article shows that the excess temperature of the shunt rods \(t_x\) over their initial temperature \(t_0\) depends mainly on the current density in the shunt rods

\[ i_n = \frac{I_n}{F}. \]

By specifying the permissible overheating of the centers of the shunt rods, one can determine the current density \(i_n\) permissible for it, according to which the entire further calculation of the shunt can be carried out.

The principal shortcoming of the method set forth in [8] is that the switch-on current is assumed to be constant in magnitude. In reality, the current varies according to an exponential law \(I = I_n(1 + me^{-kt})\), and the volumetric heat generation in the shunt rods—\(q_v\)—correspondingly follows the law

\[ q_v = \frac{0.86 I^2 R}{2L \cdot F} = q_n(1 + 2me^{-kt} + m^2 e^{-2kt}), \]

where \(q_n\) is the volumetric heat generation from \(I_n\), and the coefficients \(m\) and \(k\) depend on the electrical characteristics of the circuit in which the shunt is installed.

For production shunts for relatively small currents (up to 2000–5000 amperes), as a rule, the electrical parameters of the circuits in which they will be installed are unknown. Therefore, the calculation method based on averaged characteristics, set forth in [8], is the only possible one for such shunts.

When calculating shunts for large currents (more than 5000–7000 amperes), in order to reduce their dimensions and weight, it is necessary to take fuller account of the actual conditions of their operation and, above all, the actual character of the change of current in the shunt circuit. In addition, the initial temperature of the shunt rods may be not

constant along their length, \(t_0\), and \(t_{0x}=f(x)\) (for example, in the case of switching on a circuit after it has been opened for a short time, during which the shunt has not had time to cool).

The investigation of the temperature field of the shunt rods during the period of current switching, taking into account the features noted above, is the subject of the present work.

In considering the problem we assume:

1. The resistance of the material of the rods (manganin) is 20–30 times higher than the resistance of the busbars and lugs (copper). Therefore it may be assumed that, during the flow of the extra-current, heat is released only in the rods, and heat generation in the busbars and lugs may be neglected.

2. Owing to the massiveness and high thermal conductivity of the lugs, their temperature during the short period of action of the extra-current does not have time to change appreciably and may be taken as constant, equal to \(t_n\).

3. The heat transfer from the shunt rods to the surrounding medium during the action of the extra-current is negligibly small in comparison with the amount of heat conducted away into the lugs and busbars, and is not taken into account.

4. The character of the change in direct current when the circuit is closed is described by the equation \(I=I_n(1-me^{-k\tau})\), where \(m\) and \(k\) depend on the electrical parameters of the circuit [10].

These quantities are considered known.

1. Owing to the small diameter of the shunt rods in comparison with their length

\[ \left(\frac{2L}{d}=12\text{--}16\right) \]

and the relatively high thermal conductivity of their material, we neglect the variation of temperature over the radius of the rods.

1. The initial temperature of all elements of the shunt (rods, lugs, busbars) is either identical and equal to \(t_0\) (initial switching), or \(t_{0x}=f(x)\) (repeated switching).

On the basis of the foregoing, the problem for repeated switching may be formulated as follows: a thin rod of length \(2L\) with initial temperature \(t_{0x}=f(x)\) is given. The lateral surface of the rod is insulated. The noninsulated ends of the rod have the constant temperature \(t_n\). Inside the rod there acts a heat source with specific power

\[ q_\tau=\frac{0.86 I^2 R}{2L\cdot F} = q_n\left(1-2me^{-k\tau}+m^2e^{-2k\tau}\right). \]

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:

\[ \frac{\partial t}{\partial \tau} = a\frac{\partial^2 t}{\partial x^2} + \frac{q_n\left(1-2me^{-k\tau}+m^2e^{-2k\tau}\right)}{c\gamma}; \tag{1} \]

\[ t_{x0}=f(x)\quad (\text{see }[7]); \tag{2} \]

\[ t_{L\tau}=t_n \ \text{and}\ t_{L0}=t_n; \tag{3} \]

\[ t'_{0\tau}=0. \tag{4} \]

Applying to equations 1--4 a Laplace transform of the form

\[ F_{(s)}=\int_{0}^{\infty} T(x\tau)e^{-s\tau}\,d\tau, \]

we obtain:

\[ T''_{xs}-\frac{s}{a}T_{xs}+\frac{t_{x0}}{s}+\frac{q_H}{s^2c\gamma} +\frac{2mq_H}{sc\gamma}\frac{1}{k+s} +\frac{m^2q_H}{sc\gamma}\frac{1}{2k+s}=0; \tag{5} \]

\[ T_{Ls}=-\frac{t_H}{s}; \tag{6} \]

\[ T'_{0s}=0. \tag{7} \]

The solution of equation (5) is

\[ T_{xs}=Ae^{\sqrt{\frac{s}{a}}\,x} +Be^{-\sqrt{\frac{s}{a}}\,x} -\frac{t_{x0}}{s} +\frac{q_H}{s^2c\gamma} +\frac{2mq_H}{sc\gamma}\frac{1}{k+s} +\frac{m^2q_H}{sc\gamma}\frac{1}{2k+s}. \tag{8} \]

Taking into account (6) and (7), we have

\[ A=B=- \frac{ \frac{q_H}{s^2c\gamma} +\frac{2mq_H}{sc\gamma}\frac{1}{k+s} +\frac{m^2q_H}{sc\gamma}\frac{1}{2k+s} }{ e^{\sqrt{\frac{s}{a}}\,L}+e^{-\sqrt{\frac{s}{a}}\,L} }. \tag{9} \]

Now

\[ \begin{aligned} T_{xs}={}&-\frac{t_{x0}}{s} +\frac{q_H}{s^2c\gamma} +\frac{2mq_H}{sc\gamma}\frac{1}{k+s} +\frac{m^2q_H}{sc\gamma}\frac{1}{2k+s} \\ &-\frac{q_H}{s^2c\gamma} \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} -\frac{2mq_H}{sc\gamma} \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} \\ &-\frac{m^2q_H}{sc\gamma} \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L}. \end{aligned} \tag{10} \]

Applying the inverse Laplace transform, we find the originals of the first four terms of equation (10):

\[ L^{-1}\left(-\frac{t_{x0}}{s}\right)=-t_{x0}; \tag{11} \]

\[ L^{-1}\left(\frac{q_H}{s^2c\gamma}\right)=\frac{q_H}{c\gamma}\tau; \tag{12} \]

\[ L^{-1}\left(\frac{2mq_H}{sc\gamma}\frac{1}{k+s}\right) =\frac{2mq_H}{c\gamma}\left(1-e^{-k\tau}\right); \tag{13} \]

\[ L^{-1}\left(\frac{m^2q_H}{sc\gamma}\frac{1}{2k+s}\right) =\frac{m^2q_H}{c\gamma}\left(1-e^{-2k\tau}\right). \tag{14} \]

Using the expansion theorem, we find the original of the fifth term of the equation

\[ L^{-1}\left( \frac{q_H}{s^2 c\gamma_i}\, \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} \right) = \frac{q_H}{c\gamma_i}\tau - q_H\,\frac{2L^2}{a c\gamma_i} \sum_{1}^{\infty} \frac{(-1)^{n+1}}{\mu_n^3} \cos \mu_n \frac{x}{L} \left(1-e^{-\mu_n^2\frac{a}{L^2}\tau}\right), \tag{15} \]

where

\[ \mu_n=(2n-1)\frac{\pi}{2}. \]

To find the originals of the sixth and seventh terms of the equation, we rewrite them in the form:

\[ \frac{2q_H}{s c\gamma_i}\, \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} \cdot \frac{m}{\kappa+s} \quad \text{and} \quad \frac{q_H}{s c\gamma_i}\, \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} \cdot \frac{m^2}{2\kappa+s}. \]

Applying successively Borel’s theorem and the expansion theorem (see [1], [2], [3]), we find the originals of these complexes:

\[ L^{-1}\left( \frac{2q_H}{s c\gamma_i}\, \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} \frac{m}{\kappa+s} \right) = \]

\[ = \frac{2mq_H}{\kappa c\gamma_i} \left(1-e^{-\kappa\tau}\right) + e^{-\kappa\tau}\cdot \frac{4mq_H}{c\gamma_i} \sum_{1}^{\infty} \frac{(-1)^{n+1}}{\mu_n} \cdot \]

\[ \cdot \frac{\cos \mu_n \frac{x}{L}} {\kappa-\mu_n^2\frac{a}{L^2}} \left[ 1-e^{-\left(\kappa-\mu_n^2\frac{a}{L^2}\right)\tau} \right]; \tag{16} \]

\[ L^{-1}\left( \frac{m^2}{2\kappa+s}\, \frac{q_H}{s c\gamma_i}\, \frac{\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {\operatorname{ch}\sqrt{\frac{s}{a}}\,L} \right) = \]

\[ = \frac{m^2q_H}{2\kappa c\gamma_i} \left(1-e^{-2\kappa\tau}\right) - e^{-2\kappa\tau}\cdot 2\frac{m^2q_H}{c\gamma_i} \sum_{1}^{\infty} \frac{(-1)^{n+1}}{\mu_n} \cdot \]

\[ \cdot \frac{\cos \mu_n \frac{x}{L}} {2\kappa-\mu_n^2\frac{a}{L^2}} \left[ e^{\left(2\kappa-\mu_n^2\frac{a}{L^2}\right)\tau}-1 \right]. \tag{17} \]

The equation of the temperature field of the shunt rod during heating will now be written as follows:

\[ \begin{aligned} t_{x\tau}-t_{x0} &= \frac{q_H\cdot 2L^2}{ac\gamma} \sum_{1}^{S}\frac{(-1)^{n+1}}{\psi_n^3} \cos \psi_n\frac{x}{L} \left(1-e^{-\psi_n^2\frac{a}{L^2}\tau}\right) \\ &\quad +e^{-k\tau}\cdot 4\frac{mq_H}{c\gamma} \sum_{1}^{S}\frac{(-1)^{n+1}}{\psi_n} \frac{\cos \psi_n\frac{x}{L}}{k-\psi_n^2\frac{a}{L^2}} \left[e^{\tau\left(k-\psi_n^2\frac{a}{L^2}\right)}-1\right] \\ &\quad +e^{-2k\tau}\cdot 2\frac{m^2q_H}{c\gamma} \sum_{1}^{S}\frac{(-1)^{n+1}}{\psi_n} \frac{\cos \psi_n\frac{x}{L}}{2k-\psi_n^2\frac{a}{L^2}} \left[e^{\tau\left(2k-\psi_n^2\frac{a}{L^2}\right)}-1\right]. \end{aligned} \tag{18} \]

Substitution of the boundary conditions into (18) gives:

1. For \(\tau=0\), \(t_{0x}=t_{x0}\).
2. For \(x=L\), \(t_{L\tau}=t_{0L}=t_H\).
3. \[ \left.\frac{\partial t}{\partial x}\right|_{x=0}=0. \]

Expression (18) consists of three terms. The first term reflects heating under the action of the constant component of the switch-on current and is completely identical to formula (14) given in [8].

The second and third terms of expression (18) reflect the influence on the change in the rod temperature of the variable component of the switch-on current.

It can be shown that for \(x=0\) and \(\tau=\infty\), \(t_{0\tau}-t_0=0.5\,q_H\frac{L^2}{\lambda}\).

The latter coincides with the known formula giving the temperature difference in a plane wall with internal heat generation [5], [6].

To find the temperature field of the shunt rod when it is heated from a completely cooled state (initial switching-on), it is sufficient in equation (18) to set \(t_{x0}=t_0=\mathrm{const}\).

Analysis of expression (18) shows that it is a sum of rapidly converging series.

For practical use, one may restrict oneself to taking into account only the first terms of each of the series, which gives an error on the high side not exceeding \(4\%\) even for \(x=0\). In this case expression (18) takes the form:

\[ \begin{aligned} t_{0\tau}-t_0 &= 0.516\,q_H\frac{L^2}{\lambda} \left(1-e^{-2.47\frac{a}{L^2}\tau}\right) \\ &\quad +2.55\,q_H \frac{e^{-k\tau}m}{c\gamma\left(k-2.47\frac{a}{L^2}\right)} \left[e^{\tau\left(k-2.47\frac{a}{L^2}\right)}-1\right] \\ &\quad +0.637\,q_H \frac{m^2\cdot e^{-2k\tau}}{c\gamma\left(k-1.23\frac{a}{L^2}\right)} \left[e^{\tau\left(2k-2.47\frac{a}{L^2}\right)}-1\right]. \end{aligned} \tag{18a} \]

Replacing

\[ q_{н}=\frac{0{,}86 J_{н}V_{н}}{2L\cdot F} \quad \text{and} \quad L=\frac{V_{н}}{2\rho i_{н}}, \]

equation (18) and (18a) can be put into a form similar to equations (15) and (15a) of work [8], which makes it possible, for a specified overheating of the center of the shunt, to determine the value of the current density permissible for it.

Conclusion

The solution presented makes it possible to determine the possible overheating of the resistance elements of a shunt when they are heated by the switch-on current.

Special attention should be paid to the fact that, for calculating shunts for large currents, when it is desirable to give them the smallest possible dimensions, it is necessary to know the electrical parameters (capacitance, inductance, active resistance) of the circuit for installation in which the shunt is intended.

References

1. Lykov A. V. Theory of Heat Conduction, GITTL, Moscow, 1952.

2. Lykov A. V. Heat Conduction of Nonstationary Processes, GITTL, Moscow–Leningrad, 1948.

3. Carslaw H., Jaeger J. Operational Methods in Applied Mathematics, GITTL, Moscow, 1948.

4. GOST 8042-56, Moscow, 1956.

5. Mikheev M. A. Fundamentals of Heat Transfer, Moscow, GEI, 1956.

6. Gröber and Erk. Fundamentals of the Theory of Heat Exchange, GITTL, Moscow–Leningrad, 1936.

7. Korolenko Yu. A. On the thermal calculation of rod shunts. Fizika, No. 3, 1959.

8. Korolenko Yu. A. Heating of the resistance elements of rod shunts under the action of direct current (the article is included in this collection).

9. Buschgolz. Besondere Probleme der erwärmung elektrischer Leiter. Zeitschrift für angew. Math. u. Mech. H. 4. B. 9. 1929.

10. Netushil A. V. and Strakhov S. V. Fundamentals of Electrical Engineering, Part II, GEI, 1955.