A diagrammatic representation of entropy production

The entropy produced during a physical process is a measure of the energy that can no longer be employed to generate work. For this reason, it is a crucial quantity in thermodynamic analysis.

Despite the difficulties that entropy analysis presents for students of thermodynamics, multiple representations of entropy production are rarely found in textbooks. Unlike work and heat, which in some cases can be represented as areas in the P–V and T–S diagrams, a visual representation of entropy production in the classical framework is, to the best of our knowledge, still lacking. In this short note, we show how such a graphical analysis can be included in thermodynamics courses at the undergraduate level.

This expression can be interpreted in terms of the areas plotted in Fig. 1. The first term is the area under the curve $f ( T ) = C / T$ over the range from T₁ to T₂. The second term is the area below the line $C / T 2$ over the same range. The difference between these is the red area shown in Fig. 1 and represents the global entropy production, $Δ S 1 → 2 Univ$⁠.

An advantage of the graphical representation is that it clearly shows the asymmetry between the heating and cooling processes in terms of the degree of irreversibility involved. Figure 1 shows that, due to the convex character of f(T), cooling a solid from T₂ to T₁ by energy exchange with a reservoir is always further from the reversible limit than heating from T₁ to T₂ by the same procedure. Moreover, the diagrammatic approach shows that this asymmetry grows with increasing temperature differences. This is due to the different rates at which the corresponding areas grow, both as T₁ approaches zero and T₂ tends to infinity. In fact, analysis shows that, for $T 2 ≫ T 1$⁠, $Δ S 1 → 2 Univ ≃ C log T 2 / T 1$⁠, showing a logarithmic dependence, while $Δ S 2 → 1 Univ ≃ C T 2 / T 1$⁠, exhibiting a linear trend with $T 2 / T 1$⁠.

For instance, the analysis of the quasi-static isobaric expansion of a perfect gas (C_P, C_V constant) due to energy exchange with a large reservoir gives rise to a similar representation, but in a $C P / T − T$ diagram. If the gas is in equilibrium with a reservoir and the expansion is performed by abruptly diminishing the external pressure to another constant value, the entropy production until the new equilibrium is reached can be obtained by plotting R/P as a function of P, where R is the gas constant.² We invite the interested reader to construct these diagrams and carry out the analysis.

The example of an abrupt change in pressure shows that the method can be applied to processes far away from the quasi-static limit. In that sense, it is important to note that the function f that generates the diagram (for example f(T) in Fig. 1) is not related to the real trajectory followed by the system. It represents the auxiliary internally reversible process considered to find the entropy variation of A. This is a strong point in favor of the diagrammatic approach, since the diagram can be constructed even in situations in which the real process cannot be represented.¹ 

The diagrammatic approach can be applied to analyze more complex processes, such as power cycles. One example is the Brayton cycle operating with a perfect gas (see Fig. (2)). In this model, the high-pressure gas coming from the compressor is directed into a combustion chamber, where it is mixed with fuel and ignited at constant pressure. For simplicity, we model this stage as an isobaric heat exchange with a thermal reservoir. The high enthalpy gas expands adiabatically in the turbine, producing power. The exit gas is cooled at constant pressure and sent to the compressor. We assume that the compressor and the turbine are isentropic, so the only irreversibilities in the cycle are associated with the heat transfer between the gas and the reservoirs.

In conclusion, we believe that this diagrammatic approach to entropy production is appropriate for teaching thermodynamics at the undergraduate level. Knowledge of calculus in one variable is the only mathematical prerequisite. Likewise, since it provides a complementary perspective to analytic or numerical calculations, we believe that this tool could be valuable for thermodynamic analysis in many different contexts.

This work was partially supported by Agencia Nacional de Investigación e Innovación and Programa de Desarrollo de las Ciencias Básicas (Uruguay).