Academician Didebulidze’s Thermoelement as a Battery of Closed Thermoelements Connected in Series

L. G. Olert

In the Transactions of the Georgian Industrial Institute (No. 1/15, 1943, p. 273) an article by Academician A. I. Didebulidze was published under the title “Thermoelements of the Author’s System.” We quote in full the description of the experiment he carried out: “One half of a metal wire (constantan) is coated galvanically with copper; if the boundary of deposition is heated, that is, point \(B\), a thermoelectromotive force appears of the same magnitude as at a constantan–copper junction, that is, \(4.1\ mV\) at \(100^\circ\mathrm{C}\). Moreover, it is completely immaterial whether we take the second contact at point \(C\) on the copper or at point \(C_1\), on the constantan.” (Fig. 1). On the basis of the effect he established, the author proposes a new type of thermoelement, “...which, besides great theoretical significance, also has a number of practical design advantages, for example: 1) in the proposed thermoelements there is no thickening at the boundary of two metals; 2) it is possible to make such thermocouples as cannot be made by the ordinary method (by soldering), for example, a pair of thin wires copper–bismuth; and 3) to obtain economy in the manufacture of thermoelements from more valuable metals, if they are deposited on less valuable metals, etc.”

The proposed thermoelement was subjected to investigation, which showed the erroneousness of Academician Didebulidze’s assertion.

Fig. 1

Fig. 2

The phenomenon was studied on a differential diagram of a thermoelement (Fig. 2). A constantan wire \(0.3\ mm\) thick was taken as the basis. The section \(ab\) was coated galvanically with copper, and the ends \(c\) and \(d\) were connected to a galvanometer. When point \(a\) was placed in boiling water and point \(b\) in melting snow, an electromotive force appeared and the needle of the galvanometer was deflected. But at the same time it was found that the magnitude of the e.m.f. in all cases was less than that given by a normal constantan–copper thermoelement, and depended on the thickness of the galvanic coating.

This made it necessary to carry out a quantitative verification of the phenomenon. Five thermoelements were made from the same material, but with different

with a thick copper coating. The section \(ab\) was tested for electrical resistance before and after coating; the resistance of the axial copper layer was calculated by the formula: \(R_m=\dfrac{RR_k}{R_k-R}\), where \(R\) is the resistance of the section after coating, \(R_k\) is the resistance of the same section before coating, and \(R_m\) is the sought resistance of the axial copper.

The experiments showed that the emf of Didebulidze’s thermoelement is approximately proportional to the ratio of the conductivity of the copper to the total conductivity of the section \(ab\), or, what is the same thing, to the ratio of the resistance of the inner core in the section \(ab\) to the sum of the resistances of that same core and of the copper covering it. This dependence can be expressed by the formula:

\[ E=\frac{X\cdot R_k}{R_k+R_m}, \tag{1} \]

where
\[ \begin{aligned} E&\text{ is the emf of the given thermoelement,}\\ X&\text{ is the proportionality coefficient,}\\ R_k&\text{ is the resistance of the inner core in the section }ab,\\ R_m&\text{ is the resistance of the copper layer.} \end{aligned} \]

It is not difficult to see that if \(R_k\) is infinitely large, then \(E\) attains its maximum value and is equal to \(X\). But an infinitely large value of \(R_k\) means that there is no inner core at all and that in the section \(ab\) we have only copper alone, that is, a normal thermoelement with its proper emf. Thus the coefficient \(X\) is the emf of the given thermocouple, and therefore formula (1) may be rewritten in the following general form:

\[ E=\frac{E_{km}R_k}{R_k+R_m}, \tag{2} \]

where
\[ \begin{aligned} E&\text{ is the emf of the given thermoelement,}\\ E_{km}&\text{ is the emf of a normal thermoelement made of the given pair of metals,}\\ R_k&\text{ is the resistance of the inner core in the section }ab,\\ R_m&\text{ is the resistance of the galvanic coating.} \end{aligned} \]

It is therefore clear that the thermoelectromotive force indicated by Academician Didebulidze for constantan coated with copper, at a temperature of \(0—100^\circ=4.1\,mV\), could be obtained only in the case where a thin constantan wire, possessing a very large resistance, was covered with a thick layer of copper, that is, when the ratio

\[ \frac{R_k}{R_k+R_m} \]

was close to 1. With a thinner coating the emf must always be smaller, but then all the advantages of the new thermoelement automatically disappear.

In Fig. 3 a graph is given of the dependence of the emf of Didebulidze’s thermoelement on the ratio

\[ \frac{R_k}{R_k+R_m}; \]

the dependence, as we see, is linear. The circles show the magnitude of the emf found experimentally.

It should be noted that with a sufficiently thick layer of copper coating the emf turned out to be somewhat lower than that calculated by the formula, which can hardly be explained by experimental errors. Rather, one should assume here the influence of secondary causes, namely the different thermal conductivities of copper and constantan, as well as the peculiarity of their spatial arrangement, which must cause a nonuniform distribution of heat. Be that as it may, the deviations are small, and there is no doubt that the dependence we have found plays the principal role.

If we leave aside the design features, it is easy to see that Didebulidze’s thermoelement is an ordinary thermoelement with a conductor of the same material as the leads connected in parallel with it. The role of this conductor is played by

section of the internal conductor between points \(a\) and \(b\). The only difference is that the conductor connected in parallel is, along its entire length, in close contact with the main branch \(ab\) (that is, with the galvanic coating).

To test the assumed analogy, the following experiment was carried out: the same circuit was assembled, except that the section \(ab\), having resistance \(R_k\), was not coated with copper, but copper leads \(e\) and \(f\) were soldered at points \(a\) and \(b\). When the leads were connected through a copper resistance \(R_m\), a thermoelement was obtained that obeyed formula (2). Here the role of the internal conductor of Didebulidze’s thermoelement was played by the section \(R_k\), and the role of the galvanic coating by the copper wire. It was further established that it was entirely immaterial what material the resistance \(R_m\) consisted of, provided only that its junctions with the copper leads \(e\) and \(f\) were at the same temperature. The scheme of the experiment is given in Fig. 4. The graph of the dependence found is shown in Fig. 5.

Fig. 3. \(x=\dfrac{R_k}{R_k+R_m}\); \(y=\) e.m.f. in % relative to a normal thermoelement.

Fig. 4

This experiment showed that the contact of the galvanic coating with the internal conductor in Didebulidze’s thermoelement has almost no effect on the final result.

It was further found that if part of the copper layer is removed along the middle of the branch \(ab\) (Fig. 6), the e.m.f. does not change as a result.

By analogy, the circuit shown in Fig. 7 was assembled. The conductor \(ACB\) connected in parallel (of the same material as the leads) consisted of three series resistances \(A\), \(C\), and \(B\), with \(A=B\). The copper branch \(A_1, B_1, C_1\) likewise consisted of three resistances; \(A_1=B_1\), while \(C_1\) was variable. Points \(c, d, e, f\) were at the same (room) temperature, which was achieved by immersing them in a common vessel with distilled water. When point \(a\) was placed in boiling water and point \(b\) in melting snow, an e.m.f. appeared, and its magnitude depended on the resistances \(ACB\) and \(A_1C_1B_1\), obeying formula (2). But when point \(c\) was soldered to point \(e\), and \(d\) to \(f\), the e.m.f. changed and thereafter was practically constant for different values of \(C_1\). If one starts from the total resistance, the e.m.f. no longer obeyed formula (2), but if the resistances \(C\) and \(C_1\) were discarded, agreement with the formula was again obtained. However, as a result of breaking the branch \(A_1\),

the dependence was again violated. This time it was confirmed only in the case where two thermoelements were considered: \(AA_1\) (with branch \(A\) broken, junction temperatures \(100^\circ\) and \(20^\circ\)) and \(BB_1\) (closed, with junction temperatures \(20^\circ\) and \(0^\circ\)), connected in series by an indifferent conductor.

Thus, after the conductors had been soldered, but before branch \(A\) was broken, agreement with the formula took place only as the result of the pairwise equality

Fig. 5. \(x=\dfrac{R_k}{R_k+R_m}\); \(y=\) t.e.m.f. in % relative to a normal thermoelement.

of the resistances, and when this condition was violated there was no agreement either. Meanwhile, each thermoelement taken separately still

Fig. 6

Fig. 7

obeyed formula (2). Consequently, it can be applied only to a single element, taking its limits from junction to junction.

But in Didebulidze’s thermoelement we have close contact between the main branch and the conductor connected in parallel along their entire length. There is an infinite number of “junctions” there. Therefore, we may consider

regard it as a battery of an infinite number of infinitely small closed thermoelements connected in series. But this is valid only for a zone of temperature gradients. If the branch with the galvanic coating is sufficiently long, then its middle part over some extent will be at the same temperature, and therefore there are no longer any thermoelements there, but only an indifferent conductor connecting in series two batteries; and its composition is of no significance. Hence, naturally, follows the invariance noted above of the e.m.f. when part of the galvanic coating is removed in the middle zone of the branch.

Consequently, formula (2), as applied to Didebulidze’s thermoelement, is valid (and even then only approximately) only insofar as we have a uniform galvanic coating along the entire length of the conductor. If, however, it is nonuniform, then there will be no dependence either; it will then have to be sought in the sum of the e.m.f.’s of an infinite number of infinitely small sections of the branch.

Let us now examine the general case of a thermoelement with a conductor connected in parallel, independently of the material.

Fig. 8

If to a thermoelement consisting of the metals \(n\) and \(m\) we connect in parallel an \(m\)-branch of the metal \(q\) (Fig. 8), then in fact we shall have two summed thermoelements with a conductor connected in parallel. The first of them is \(nm\), and to it is connected the conductor \(q\); the second is \(nq\), and to it—the conductor \(m\). The sum of the equations of these two thermoelements will give us the general formula for a thermoelement with a conductor connected in parallel. We have

\[ E_1=\frac{E_{nm}R_q}{R_m+R_q}, \qquad E_2=\frac{E_{nq}R_m}{R_m+R_q}, \]

\[ E=E_1+E_2=\frac{E_{nm}R_q+E_{nq}R_m}{R_m+R_q}. \tag{3} \]

The e.m.f.’s of different pairs may have either the same or different signs; in the latter case we shall have not a sum, but a difference; if in this case \(E_{nm}R_q\) turns out to be numerically equal to \(E_{nq}R_m\), then as a result we shall obtain 0; such a thermoelement will give no current at all to the external circuit. The value of \(E\) may range from \(E_{nm}\) to \(E_{nq}\), depending on the resistances; it cannot go beyond these limits.

The experiments performed confirmed the correctness of the formula derived. In Figs. (9) and (10) are given graphs of the dependence found for two variants subjected to verification (Fig. 9—constantan–copper; Fig. 10—wire of unknown composition–copper).

It is necessary to note that formula (3) is general for a thermoelement with a conductor connected in parallel, or even without one. Indeed, if we connect the same material from which the leads are made, that is, not \(q\), but \(n\), then the term \(E_{nq}R_m\) becomes \(E_{nn}R_m=0\) (since \(E_{nn}=0\)); as a result formula (2) remains. If instead of \(q\) we put \(m\), then all resistances cancel and we obtain only \(E_{nm}\), that is, the e.m.f. of a normal thermoelement. The same result is obtained if we approach the question differently, taking \(R_q=\infty\).

With three different metals \(m, q, p\) and leads \(n\), we must consider three thermoelements, each of which has two conductors connected in parallel. The sum of the e.m.f.’s of these elements will give the result-

generating e.m.f. By simple algebraic transformations we obtain:

\[ E=\frac{E_{mq} R_q R_p+E_{mp} R_m R_p+E_{np} R_m R_q}{R_m R_q+R_m R_p+R_q R_p}, \tag{4} \]

Fig. 9. \(x=\dfrac{R_k}{R_m+R_k};\quad z=\dfrac{R_m}{R_m+R_k};\quad y\)—thermo-e.m.f. in mV.

The metal \(q\) is connected to the galvanometer.
For \(x=0\) (i.e., \(z=1\)) we have the usual thermocouple \(q\)—constantan.
For \(z=0\) (i.e., \(x=1\)) we have the thermocouple \(q\)—copper.

Fig. 10. \(x=\dfrac{R_q}{R_m+R_q};\quad z=\dfrac{R_m}{R_m+R_q};\quad y\)—thermo-e.m.f. in mV.

Constantan \(k\) is connected to the galvanometer.
For \(x=0\) (i.e., \(z=1\)) we have the thermocouple \(k\)—\(q\).
For \(z=0\) (i.e., \(x=1\)) we have the thermocouple \(k\)—\(m\) (constantan—copper).

It may be expected that a certain dependence arising from formula (3) should hold for binary eutectic alloys, and from formula (4) for ternary eutectic alloys, since each eutectic inclusion may be regarded as a conductor connected in parallel or, conversely, as part of the main branch of the thermoelement. On this basis one may establish a quantitative thermoelectric analysis of alloys that yield a eutectic, although ternary alloys will give an indeterminate equation; a way out of this indeterminacy may be sought in determining the T.E.M.F. at different temperatures, if the inversion point of one of the components differs sharply from the inversion points of the other two components. However, application of the indicated dependence to alloys may be complicated by secondary influences, for example, by the differing thermal conductivities of the constituent parts.