*3.3. Structure of Models for Kelvin Principle*

The structure of the Kelvin principle models is the most complex of all the principles considered in this paper. It turns out that this principle allows scenarios of models with both "top-down" ↓ and "bottom-up" ↑ heat flows. The zero-efficiency scenario model allows for both directions simultaneously. Thus, the structure of the Kelvin models can be represented as follows:

$$\{K\} = \{K\}\_{0'}^{\uparrow} \; \{K\}\_{0 < \eta\_m < 1'}^{\downarrow} \; \{K\}\_{1-'}^{\downarrow} \; \{K\}\_{0 < \eta\_m < 1'}^{\uparrow} \; \{K\}\_{1-}^{\uparrow}. \tag{19}$$

Models containing processes of arbitrarily high efficiency but less than one (1−), which should not be confused with models with a specified maximum efficiency of *ηm*, have been distinguished here. The zero-efficiency model is as follows:

$$\{K\}\_0^\downarrow = \{000, \downarrow 0 \downarrow, 0 \leftarrow \downarrow, \uparrow \leftarrow 0, \downarrow \leftarrow \downarrow, \uparrow \leftarrow \downarrow, \text{ no regimes, } \uparrow \leftarrow \uparrow, \uparrow \ 0 \uparrow\}.\tag{20}$$

Models with a natural direction of heat flow ("downwards") and with an intermediate maximum efficiency coincide with the analogous models for the Clausius I principle:

$$\{K\}\_{0\le\eta\_m<1}^\downarrow = \{000, \downarrow 0 \downarrow, 0 \gets \downarrow, \uparrow \gets 0, \uparrow \gets \downarrow, \uparrow \gets \downarrow, \uparrow \gets \downarrow\} \cup \iota\_{\ell} \leq\_{\eta\_m \leq \eta\_m, \prime} (\uparrow \leftarrow \uparrow)\_{\eta\_m \leq \tilde{\eta} \leq 1} \}. \tag{21}$$

On the other hand, the model containing the efficiencies arbitrarily close to one, differs from the analogous model of the *C I* principle only by the lack of processes with the efficiency of one:

$$\{K\}\_{1^{-}}^{\downarrow} = \{000, \downarrow 0 \downarrow, 0 \leftarrow \downarrow, \uparrow \leftarrow 0, \downarrow \leftarrow \downarrow, \uparrow \leftarrow \downarrow, \text{ no } \text{refrigerators}, \downarrow \rightarrow \downarrow\}.\tag{22}$$

In models with the opposite direction of heat flow, it is enough to reverse the directions of the corresponding arrows:

$$\{K\}\_{0\leqslant\eta\_{\mathfrak{n}}<1}^{\uparrow} = \{000, \uparrow 0 \not\downarrow, 0 \gets \downarrow, \uparrow \gets 0, \uparrow \gets \uparrow, \downarrow \gets \uparrow, \ (\uparrow \to \uparrow)\_{0\leqslant\eta\_{2}\leq\eta\_{\mathfrak{n}}, \prime}(\downarrow \gets \downarrow)\_{\eta\_{\mathfrak{n}}\leq\eta\_{2}<1}\},\tag{23}$$

$$\{K\}\_{1-}^{\top} = \{000, \uparrow 0 \uparrow, 0 \leftarrow \downarrow, \uparrow \leftarrow 0, \uparrow \leftarrow \uparrow, \downarrow \leftarrow \uparrow, \text{ no \textquotedblleft referators\textquotedblright}, \uparrow \rightarrow \uparrow\}.\tag{24}$$

The term "*refrigerators*" is given here in quotation marks, because the heat in such a device would be pumped from a higher temperature to a lower temperature, in these models. It is worth noting that these models do not have the usual engines (*no engines*), i.e., processes in which the work is done at the expense of the heat of the heater, not the cooler. The lack of typical engines ensures the consistency of these models. Such a class of models obviously contradicts the tendency towards thermodynamic equilibrium, but the Kelvin principle does not seem to determine this condition.

Above all, however, both classes of heat flow direction, similarly to the Clausius I principle, have the efficiency values *ηm*, which may exceed the Carnot efficiency *η<sup>C</sup>* (*η<sup>m</sup>* > *ηC*). The value of *η<sup>m</sup>* < 1 is not related in any way to the value of *ηC*. Hence, we have whole classes (sets families) of Kelvin (and Clausius I) models. Models for which *η<sup>m</sup>* > *η<sup>C</sup>* contain more engine diagrams than the uniform (stable) and common model for the Carnot principle and Claussius II principle.
