**Kelvin principle (***K***).**

*The processes of converting heat to work and work to heat do not run symmetrically. A full conversion of work to heat (internal energy) is possible. However, a full conversion of the heat to the work is not possible in a cyclical process.*

The principle called the Kelvin principle is formulated in various but similar ways. This relates not so much to the original formulation of Kelvin [24,25], but more to a later formulation of Planck (see [26]), called the Kelvin–Planck formulation (see [27]). The above version of the *K* principle reflects, as intended by the author, the essence of all these formulations although this form of the principle comes from Planck or Ostwald ([26]) rather than initially from Kelvin. Nevertheless, traditional formulations identically equivalent to the *K* version of the principle are called the Kelvin principle, not the Kelvin–Planck or Ostwald principle [12–15,28–31]. In the publications [24,25,32], Kelvin expressed his principle as follows: *"It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects"*. Such a formulation, however, seems weaker than the formulation of *K*, as it seems to refer to the performance of work at the expense of the cooler heat, and not more naturally at the expense of the heater heat. Therefore, it is not known which of the following elements Kelvin wanted to emphasize in this formulation: (i) a change in the temperature of the reservoir when taking heat converted into work, (ii) limitations in this conversion of heat to work, (iii) the impossibility of cooling down the coldest body by means of the work performed by this body? Thus, it can be said that Kelvin did not provide his own formulation of the II law of thermodynamics in a very strict and careful manner [2]. Therefore, it is not surprising that we use a version of the rule of the type *K* slightly different from the original formulation (see also [33]). Nevertheless, Kelvin's formulation of the II law of thermodynamics is dated to 1851 in connection with his lectures published in the form of articles in two journals [24,25]. In addition, in the second journal, Kelvin separated a short article in which he repeated the formulation of his principle, and also gave a general interpretation of the II law of thermodynamics [32].

It can be concluded that the Kelvin principle in the *K* version (not original Kelvin version) is the same as the Ostwald principle, which says that there is no *perpetuum mobile* of type II (*πI I*) [26,28]. In other words, there is no heat engine with an efficiency of 100%:

$$
\eta = \frac{W}{Q\_{\rm in}} = \frac{W}{Q\_1} < 1.\tag{5}
$$

It is assumed here that the heat source has a higher temperature than the heat receiver (*Tin* = *T*<sup>1</sup> > *Tout* = *T*2). Formally, however, it is possible to consider taking heat from a reservoir with a lower temperature (conventionally *Q*<sup>2</sup> < 0) and converting it into work *W* > 0:

$$
\eta\_2 = \frac{W}{\left| \ Q\_2 \right|} < 1. \tag{6}
$$

And for the processes going in the opposite direction, the equivalent of this efficiency will have the form *η*˜2 =| *W* | / | *Q*<sup>2</sup> |.

At first glance, the Kelvin principle seems to be very trivial and seems to be significantly weaker than the Carnot principle. Nevertheless, it is allegedly proved that the Kelvin principle in the form of *K* is equivalent to the full formulation of the II law of thermodynamics [12–15]. Criticism of this proof based on combining engine and heat pump processes could only be found in the aforementioned work by a scientist Bhattacharyya [16].
