*3.1. Structure of the Carnot Principle Model*

The main processes mentioned in the Carnot principle are ↓→↓ engine processes with efficiency not higher than *ηC*. This set can be extended to include limit cases in the form of processes: zero efficiency ↓ 0 ↓ and null process 000.

The remainder of the model will be found by eliminating processes, that are inconsistent with the condition of internality of adding processes. Consider the (↓→↓)*η<sup>C</sup>* engine process with a maximum Carnot efficiency of *ηC*. Let us describe the parameters of this non-zero process with three values (*Q*1, *W*, *Q*2). Less efficient processes have the form (*Q*<sup>1</sup> + *q*, *W*, *Q*<sup>2</sup> + *q*), where *q* > 0. Note that such processes are the sum of the Carnot process with the maximum efficiency (↓→↓)*η<sup>C</sup>* and the process with zero efficiency ↓ 0 ↓ (*q*, 0, *q*). On the other hand, adding the ↑ 0 ↑ (−*q*, 0, −*q*) process to the Carnot process would lead to obtaining a virtual (impossible) process with efficiency greater than Carnot efficiency. Thus, processes of type ↑ 0 ↑ should be excluded from the Carnot principle model. Similarly, we will exclude type ↑←↑ (−*Q*<sup>1</sup> − *q*, −*W*, −*Q*<sup>2</sup> − *q*) refrigeration processes, which are the inverse of engine processes with less than maximum efficiency. Well, by adding the Carnot process to this process, we get a process outside the model (*q* > 0):

$$(-Q\_1 - q\_\prime - \mathcal{W}\_\prime - Q\_2 - q) + (Q\_1, \mathcal{W}\_\prime Q\_2) = (-q, 0, -q) \notin \{\mathcal{C}0\}.\tag{12}$$

In addition to the excluded refrigeration processes, there are form ↑←↑ (−*Q*<sup>1</sup> + *p*, −*W*, −*Q*<sup>2</sup> + *p*) refrigeration processes which are not excluded for *p* ≥ 0. It is most convenient to assign the efficiency parameter *η*˜ to such refrigeration processes, which refers to the inverse engine process. Although the inverse engine process is virtual (*η*˜ ≥ *ηC*), the refrigeration process (↑←↑)*η*˜ exists and belongs to the Carnot principle model.

A borderline case of the ineffective refrigeration process of ↑←↑ for *η*˜ → 1 is the process of converting work into heat of the heater ↑← 0. The superposition of the ↑← 0 process with an appropriately selected ↓ 0 ↓ process gives the 0 ←↓ process. And combining the last two processes results in a new type of process ↓←↓. However, as we add a small heat flow in the form ↓ 0 ↓ to the ↑← 0 process, we get ↑←↓.

In summary, the Carnot principle model consists of the following processes:

$$\{\mathbb{C}0\} = \{000, \downarrow 0 \downarrow 0 \preccurlyeq \downarrow \leftarrow \uparrow \downarrow \leftarrow 0, \downarrow \leftarrow \downarrow \star \downarrow \preccurlyeq \downarrow \leftarrow \downarrow \uparrow \} \{\uparrow \rightarrow \downarrow \}\_{0 \le \overline{\eta} \le \overline{\eta} \llcorner \uparrow} \{\uparrow \leftarrow \uparrow \}\_{\overline{\eta} \le \overline{\eta} \le 1}. \tag{13}$$

We can see that the uniform (stable) model of Carnot principle consists of diagrams of 8 types. There are 3 × 3 × 3 = 27 possible all diagram types (make sense or not ). However, some diagrams contradict the principle of conservation of energy: with two zeros 3 × 2 = 6, with one zero 3 × 2 = 6, without zeros 2—which adds up to 6 + 6 + 2 = 14 for diagrams outside areas of consideration. So in the domain of consideration there are 27 − 14 = 13 of diagram types. This means that the Carnot principle model rejects five types of diagrams.

Therefore, it is worth listing the complement to the Carnot model (a set of impossible processes):

$$\overline{\{C0\}} = \{\uparrow 0 \not\uparrow, 0 \to \uparrow, \downarrow \to 0, \uparrow \to \uparrow, \downarrow \to \uparrow, \quad (\downarrow \to \downarrow)\_{\overline{\mathbb{M}} \times \overline{\mathbb{M}} \times \mathbb{T} \vee} (\uparrow \leftarrow \uparrow)\_{0 < \uparrow \mid < \uparrow \!\!/ \sim \uparrow \mid \downarrow}\}.\tag{14}$$

Indeed, five new types of diagrams were created simply by changing the direction of the corresponding arrows. The other two diagrams do not differ in type but have different ranges of efficiency values. Thus, combinatorics is here in line with the resulting model of Carnot's principle.
