We introduce a natural way of visualizing the entropy production in heat transfer processes between a system and a thermal reservoir. This representation highlights the asymmetric character of the heating and cooling processes when they are analyzed from the second-law perspective.

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 PV and TS 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.

To illustrate the method, let us consider the thermalization of 1 kg of an incompressible solid (system A), with constant heat capacity C and initial temperature T1, which is placed in contact with a thermal reservoir (system B) at temperature T 2 > T 1. Since entropy is a state function, the entropy change of the solid can be found by integrating the Gibbs relation TdS = d U + PdV along a reversible path linking the initial and final states. For an incompressible solid, dU = CdT, we obtain
Δ S 1 2 A = T 1 T 2 C T d T .
(1)
Since no work is involved, the energy absorbed by the solid is Q A = C ( T 2 T 1 ), and the entropy change of the reservoir is
Δ S 1 2 B = Q B T B = Q A T B = C ( T 2 T 1 ) T 2 .
(2)
Combining Eqs. (1) and (2), we conclude that the global entropy increase in reaching equilibrium is
Δ S 1 2 Univ = T 1 T 2 C T d T C T 2 ( T 2 T 1 ) .
(3)

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.

Fig. 1.

Entropy production associated with the cooling (blue, upper region) and heating (red, lower region) processes undergone by an incompressible solid. Both processes are performed by placing the solid in contact with a thermal reservoir. From the diagram, we conclude that cooling the system from T2 to T1 always generates more entropy than heating it from T1 to T2.

Fig. 1.

Entropy production associated with the cooling (blue, upper region) and heating (red, lower region) processes undergone by an incompressible solid. Both processes are performed by placing the solid in contact with a thermal reservoir. From the diagram, we conclude that cooling the system from T2 to T1 always generates more entropy than heating it from T1 to T2.

Close modal
Following the same procedure, for a cooling process due to energy exchange between a system at initial temperature T2 and a reservoir at temperature T1, we obtain
Δ S 2 1 Univ = C T 1 ( T 2 T 1 ) T 1 T 2 C T d T ,
(4)
which is represented by the blue (upper) area in Fig. 1.

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.

This diagrammatic approach can be generalized to any thermalization process with a heat reservoir B, provided that some property Y of system A adopts the same value in the initial and final states. It is not necessary that Y is constant during the process: in fact, Y might not even be well defined during the whole process. Under this condition, it is possible to show that if the system performs W A units of work during the thermalization process (considering the work done by the system as positive), the entropy production is
Δ S 1 2 Univ = X 1 X 2 [ S A X | Y = Y 1 1 T B U A X | Y = Y 1 ] d X W A T B ,
(5)
where X is another, conveniently chosen, independent property of A (in the previous example, X is the temperature and Y is the specific volume). Interpreting the integrals as areas, Eq. (5) can be exploited to generate entropy production diagrams for a great variety of physical situations.

For instance, the analysis of the quasi-static isobaric expansion of a perfect gas (CP, CV 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.

Fig. 2.

Diagrammatic representation of the entropy production (per unit of mass flowing) in the cooler (blue, upper region) and in the heater (red, lower region) for the Brayton cycle. The turbine and the compressor are isentropic, but the diagram can be generalized to include irreversibility in those elements.

Fig. 2.

Diagrammatic representation of the entropy production (per unit of mass flowing) in the cooler (blue, upper region) and in the heater (red, lower region) for the Brayton cycle. The turbine and the compressor are isentropic, but the diagram can be generalized to include irreversibility in those elements.

Close modal
The following expressions are obtained for the entropy generations in the heater and the cooler (per unit of mass flowing m ̇ = δ m / δ t):
{ s gen Heater = S ̇ gen Heater m ̇ = T 1 T 2 C P T d T C P ( T 2 T 1 ) T H , s gen Cooler = S ̇ gen Cooler m ̇ = C P ( T 3 T 4 ) T L T 4 T 3 C P T d T
(6)
where TH and TL are the temperatures of the reservoirs. The diagrammatic representation of these quantities in Fig. 2 allows a quick comparison between the different contributions to entropy production in the cycle and could be a useful tool for its thermodynamic optimization, which is based on the minimization of entropy production.3 

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).

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