**Carnot principle (***C***0)**

*The efficiency of the heat conversion process Qin* > 0 *to work W in the device operating in the range between the temperature of the heat source Tin and the temperature of the heat receiver Tout, cannot be greater than the ratio of the difference of these temperatures to the temperature of the heat source:*

$$\frac{\mathcal{W}}{Q\_{in}} \le \frac{T\_{in} - T\_{out}}{T\_{in}}.\tag{1}$$

Such a formulation does not explicitly contain the information that the temperature of the heat source must be higher than that of the heat receiver. However, the implicit condition that the engine performs real positive work (*W* > 0) entails the aforementioned temperatures relationship *Tin* = *T*<sup>1</sup> > *Tout* = *T*2. In this case, the condition of the Carnot principle takes the form:

$$\eta := \frac{\mathcal{W}}{Q\_1} \le \frac{T\_1 - T\_2}{T\_1} =: \eta\_{\mathcal{C}}.\tag{2}$$

Mathematically speaking, we can also consider the process of work performed by an external force (*W* < 0), then *Tin* = *T*<sup>2</sup> < *Tout* = *T*<sup>1</sup> and:

$$\frac{W}{|\,|\,Q\_2|} \le \frac{T\_2 - T\_1}{T\_2} < 0.\tag{3}$$

This correct condition applies to refrigeration processes in which, thanks to external work, it is possible to pump heat from a lower to a higher temperature. Such processes exhibit high coefficients of cooling efficiency | *Q*<sup>2</sup> | / | *W* | or the efficiency of the heat pump | *Q*<sup>1</sup> | / | *W* |≥ 1. However, it is convenient to introduce an efficiency parameter *η*˜ ≤ 1 for the refrigeration processes, which corresponds to an engine process running in the opposite direction. Thus, such a parameter is determined analogously to (2), with the sense that the

minuses are shortened *η*˜ =| *W* | / | *Q*<sup>1</sup> |= (−*W*)/(−*Q*1) = *W*/*Q*1. To find the condition for *η*˜, let us transform (3) by virtue of *W* =| *Q*<sup>2</sup> |−| *Q*<sup>1</sup> |< 0:

$$1 - \frac{\left|\left|Q\_1\right|\right|}{\left|Q\_2\right|} \le 1 - \frac{T\_1}{T\_2} \implies \frac{\left|\left|Q\_2\right|\right|}{\left|\left|Q\_1\right|\right|} \le \frac{T\_2}{T\_1} \implies \eta \ge \eta\_C. \tag{4}$$

Thus, the efficiency parameter of *η*˜ cooling processes is not lower than the Carnot efficiency (unlike the engine processes which have the efficiency not greater than the Carnot efficiency). This result was obtained here on the basis of the application of the Carnot principle for negative work *W* < 0. This result will be later deduced through independent considerations of a different kind.

Historically speaking, Carnot in 1824 [1] did not yet know the concept of absolute temperature and only defined the dependence of efficiency on a temperature difference [2]. Nevertheless, his considerations on the isothermal-adiabatic cycle implied the existence of maximum efficiency and minimal temperature. This minimum temperature along with the absolute temperature scale were formally introduced by Thomson (Lord Kelvin) [3]. Such absolute temperature scale is adopted in this work. This temperature is consistent with the zeroth law of thermodynamics, and its changes are consistent with the operation of a thermometer—so there is no need to give its elementary definition. The expression for the efficiency *ηC*(*T*1, *T*2) of the Carnot cycle was known as the Carnot function, which was not originally fully defined by Carnot. This function was intensely sought after by Clausius and Kelvin. Thus, the above-mentioned scholars, who had always referred to Carnot, also made an important contribution to the contemporary form of the *C*0 principle [4].

Carnot, as a pioneer of thermodynamics [3,5,6], did not know yet either the law of conservation of energy or the I law of thermodynamics [7], so he used the concept of the flow of the indestructible fluid of heat (caloric) [2]. Nevertheless, the Carnot principle can be considered in the context of the II law of thermodynamics [8–10], putting aside the fact that Carnot did not know the I law of thermodynamics. In the context of the [11] research of the finite-time Carnot cycle, it can be concluded that the idea of a Carnot engine, in a sense, is still not a closed concept in terms of engineering and experience.
