Entropy as non-workable energy: an easy engineering interpretation for teaching thermodynamics in engineering degree courses

14CNIT 14th National and 5th International Conference in Engineering Thermodynamics 1 14th National and 5th International Conference in Engineering Thermodynamics. Universidad de Zaragoza, Zaragoza (Spain), June 4-6, 2025. 14CNIT-262 Entropy as non-workable energy: an easy engineering interpretation for teaching thermodynamics in engineering degree courses Fernando Varela Díez1,*, Susana Sánchez-Orgaz2, Mercedes Ibarra Mollá1, José Daniel Marcos del Cano1, Alicia Mayoral Esteban1 1 Dept. of Energy Engineering, Universidad Nacional de Educación a Distancia (UNED), C/ Juan del Rosal 12, 28040 Madrid, Spain, Phone: +34-91-3986468 2 Dept. of Energy Engineering Universidad Politécnica de Madrid, José Gutiérrez Abascal, 28006, Madrid, Spain, Phone: +3491-0677191 * [email protected] 1. Introduction The introduction of the second law of thermodynamics necessitates the discussion of entropy, a concept that often requires clarification for students. In pedagogical practice, entropy is frequently described as "disorder" to facilitate comprehension. However, this analogy originates from statistical thermodynamics, a framework that is typically beyond the scope of undergraduate curricula and of limited utility in engineering applications. Consequently, students often perceive entropy as an abstract quantity used primarily for verifying the second law in engineering system analyses rather than as a physically meaningful property. This study proposes an alternative pedagogical approach to enhance students’ conceptual understanding of entropy, particularly in the context of engineering applications. Although the interpretation proposed here has been briefly mentioned in some texts (see [1], [2], [3]), it has neither been justified nor developed in any of them, thus failing to take advantage of its pedagogical approach. Zemansky [4] does a similar work but considering the variation of entropy in the universe instead of the system, which is our purpose. 2. The concept of “workable energy” Consider the maximum total work that can be extracted from a closed system under given pressure and temperature (𝑃𝑃,𝑇𝑇) conditions in a process that brings it to the dead state (𝑃𝑃0,𝑇𝑇0), including work done against the atmosphere. The energy balance of the system is given by: 𝑈𝑈0−𝐸𝐸 =𝑄𝑄+𝑊𝑊 [1] The entropy balance for the surroundings is: Δ𝑆𝑆𝑎𝑎𝑎𝑎𝑎𝑎 = −𝑄𝑄 𝑇𝑇0 [2] The entropy balance for the universe is: Δ𝑆𝑆𝑢𝑢=Δ𝑆𝑆+Δ𝑆𝑆𝑎𝑎𝑎𝑎𝑎𝑎 =(𝑆𝑆0−𝑆𝑆)+ −𝑄𝑄 𝑇𝑇0 =𝜎𝜎+𝜎𝜎𝑄𝑄 [3] By combining these equations, the energy balance can be expressed as: 𝑈𝑈 0 −𝐸𝐸 =𝑇𝑇 0 (𝑆𝑆 0 −𝑆𝑆)−𝑇𝑇 0 �𝜎𝜎+𝜎𝜎 𝑄𝑄 �−𝑊𝑊 [4] Thus, the work produced in the process is: −𝑊𝑊 =(𝐸𝐸−𝑈𝑈0)−𝑇𝑇0(𝑆𝑆−𝑆𝑆0)−𝑇𝑇0�𝜎𝜎+𝜎𝜎𝑄𝑄� [5] or equivalently: −𝑊𝑊 = ( 𝐸𝐸−𝑈𝑈 0) −𝑇𝑇 0( 𝑆𝑆−𝑆𝑆 0) −𝐼𝐼 [6] Since the total irreversibility 𝐼𝐼 is always positive, the maximum extractable work is: 14CNIT 14th National and 5th International Conference in Engineering Thermodynamics 2 14th National and 5th International Conference in Engineering Thermodynamics. Universidad de Zaragoza, Zaragoza (Spain), June 4-6, 2025. −𝑊𝑊 𝑎𝑎á𝑥𝑥 =(𝐸𝐸−𝑈𝑈 0 )−𝑇𝑇 0 (𝑆𝑆−𝑆𝑆 0 ) [7] This expression differs slightly from the definition of exergy, as it explicitly accounts for the work done against the atmosphere. Although this work cannot be utilized, it still is work extracted from the energy of the system and therefore corresponds to highquality energy. Similar to exergy, this quantity is a state function, which we define as workable energy 𝐸𝐸𝐸𝐸 - energy convertible into work: 𝐸𝐸𝐸𝐸 = ( 𝐸𝐸−𝑈𝑈 0) −𝑇𝑇 0( 𝑆𝑆−𝑆𝑆 0) [8] This function represents the portion of a system's energy that is convertible into work and therefore can be interpreted as high quality energy. The corresponding specific function is 𝑒𝑒𝐸𝐸 = (𝑒𝑒−𝑢𝑢 0 ) + 𝑇𝑇 0 (𝑠𝑠−𝑠𝑠 0 ) [9] By analogy with anergy, we define non-workable energy 𝐸𝐸𝐸𝐸𝐸𝐸 as the portion of a system’s energy that cannot be converted into work under any circumstances (low quality energy): 𝐸𝐸𝐸𝐸𝐸𝐸 =𝐸𝐸−𝐸𝐸𝐸𝐸 =𝑈𝑈 0 +𝑇𝑇 0 (𝑆𝑆−𝑆𝑆 0 ) 𝑒𝑒𝐸𝐸𝐸𝐸 =𝑒𝑒−𝑒𝑒𝐸𝐸 =𝑢𝑢0+𝑇𝑇0(𝑠𝑠−𝑠𝑠0) [10] Applying this concept to any process in a closed system, we obtain: Δ𝐸𝐸𝐸𝐸𝐸𝐸 =Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸 =𝑇𝑇 0 ·Δ𝑆𝑆 [11] Δ𝑆𝑆 = Δ𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇0 Δ𝑠𝑠= Δ𝑒𝑒𝐸𝐸𝐸𝐸 𝑇𝑇0 𝑑𝑑𝑒𝑒𝐸𝐸𝐸𝐸 =𝑇𝑇0·𝑑𝑑𝑠𝑠 [12] In open systems, where the system's mass is variable, applying the same concept, 𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑈𝑈 0 𝑑𝑑𝑑𝑑 +𝑇𝑇0 𝑑𝑑(𝑆𝑆−𝑆𝑆 0 ) 𝑑𝑑𝑑𝑑 =𝑢𝑢0 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 +𝑇𝑇0�𝑑𝑑 𝑑𝑑𝑠𝑠 𝑑𝑑𝑑𝑑 +𝑠𝑠 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 −𝑠𝑠0 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 �=𝑒𝑒𝐸𝐸𝐸𝐸· 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 +𝑇𝑇0·𝑑𝑑 𝑑𝑑𝑠𝑠 𝑑𝑑𝑑𝑑 [13] As 𝑑𝑑𝐸𝐸𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑 =𝑒𝑒𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 +𝑑𝑑 𝑑𝑑𝑒𝑒𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑 [14] We obtain 𝑑𝑑 𝑑𝑑𝑒𝑒𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑 =𝑇𝑇0·𝑑𝑑 𝑑𝑑𝑠𝑠 𝑑𝑑𝑑𝑑 𝑑𝑑𝑒𝑒𝐸𝐸𝐸𝐸 𝑑𝑑𝑑𝑑 =𝑇𝑇0· 𝑑𝑑𝑠𝑠 𝑑𝑑𝑑𝑑 𝑑𝑑𝑒𝑒𝐸𝐸𝐸𝐸 =𝑇𝑇0·𝑑𝑑𝑠𝑠 [15] Thus, an increase in entropy corresponds to an increase in the system’s non-workable energy per unit of ambient temperature. Consequently, the increase in entropy in a process is directly proportional to an increase in non-workable energy within the system. The behaviour of entropy is tied to the behaviour of low-quality energy, but not directly to that of the high-quality energy. In other words, a rise in entropy means low-quality energy increase, but high quality energy can decrease (or increase, or remain constant depend on the case). In the next section we will analyse this behaviour to illustrate this fact. 3. Understanding the entropy variation in closed systems Let us now analyse the entropy variation of a closed system from the workable energy point of view. Case a) In Isolated systems, energy is preserved, and entropy always increases due to the second principle: Δ𝐸𝐸 = 0 Δ𝑆𝑆=𝜎𝜎 ≥ 0 [16] So 14CNIT 14th National and 5th International Conference in Engineering Thermodynamics 3 14th National and 5th International Conference in Engineering Thermodynamics. Universidad de Zaragoza, Zaragoza (Spain), June 4-6, 2025. Δ𝐸𝐸𝐸𝐸𝐸𝐸=𝑇𝑇 0 Δ𝑆𝑆 ≥ 0 Δ𝐸𝐸𝐸𝐸=Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸𝐸𝐸 ≤0 [17] In this case, the increase in entropy in the system reflects directly the destruction of workable energy, which is converted directly into non-workable energy, representing a degradation of the system's energy. Fig. 1. Entropy increase in an isolated system Case b). If the system is not isolated, the following scenarios may occur: Δ𝐸𝐸 > 0; Δ𝑆𝑆 > 0. Isolated volumetric compressor with friction. Δ𝐸𝐸𝐸𝐸𝐸𝐸 =𝑇𝑇0Δ𝑆𝑆≥0 Δ𝐸𝐸𝑤𝑤=Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸𝐸𝐸 𝑢𝑢𝐸𝐸𝑑𝑑𝑒𝑒𝑢𝑢𝑢𝑢𝐸𝐸𝑒𝑒𝑑𝑑 Δ𝐸𝐸 > 0; Δ𝑆𝑆 = 0 Isolated volumetric compressor without friction. Δ𝐸𝐸𝐸𝐸𝐸𝐸 =𝑇𝑇0Δ𝑆𝑆= 0 Δ𝐸𝐸𝑤𝑤=Δ𝐸𝐸 > 0 Δ𝐸𝐸 > 0; Δ𝑆𝑆 < 0. Non-isolated volumetric compressor with heat rejection. (𝑊𝑊+𝑄𝑄> 0) Δ𝐸𝐸𝐸𝐸𝐸𝐸 =𝑇𝑇0Δ𝑆𝑆< 0 Δ𝐸𝐸𝑤𝑤=Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸𝐸𝐸 > 0 Δ𝐸𝐸 < 0; Δ𝑆𝑆 < 0. Uninsulated DHW tank: Heat rejection. Δ𝐸𝐸𝐸𝐸𝐸𝐸 =𝑇𝑇0Δ𝑆𝑆< 0 Δ𝐸𝐸𝑤𝑤=Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸𝐸𝐸 𝑢𝑢𝐸𝐸𝑑𝑑𝑒𝑒𝑢𝑢𝑢𝑢𝐸𝐸𝑒𝑒𝑑𝑑 𝐸𝐸𝑛𝑛𝑤𝑤 𝐸𝐸𝑤𝑤 𝐸𝐸𝑛𝑛𝑤𝑤 𝐸𝐸𝑤𝑤 𝑇𝑇0Δ𝑆𝑆 Δ𝐸𝐸 𝐸𝐸𝑛𝑛𝑤𝑤 𝐸𝐸𝑤𝑤 𝐸𝐸𝑛𝑛𝑤𝑤 𝐸𝐸𝑤𝑤 Δ𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑇𝑇0Δ𝑆𝑆 Δ𝐸𝐸 𝑇𝑇0Δ𝑆𝑆 Δ𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝐸𝐸𝑛𝑛𝑤𝑤 𝐸𝐸𝑤𝑤 𝐸𝐸𝑛𝑛𝑤𝑤 𝐸𝐸𝑤𝑤 𝑇𝑇0Δ𝑆𝑆 14CNIT 14th National and 5th International Conference in Engineering Thermodynamics 4 14th National and 5th International Conference in Engineering Thermodynamics. Universidad de Zaragoza, Zaragoza (Spain), June 4-6, 2025. Δ𝐸𝐸 < 0; Δ𝑆𝑆 > 0. Expansion in an isolated piston with friction. Δ𝐸𝐸𝐸𝐸𝐸𝐸 =𝑇𝑇0Δ𝑆𝑆> 0 Δ𝐸𝐸𝐸𝐸 =Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸𝐸𝐸 < 0 Δ𝐸𝐸 < 0; Δ𝑆𝑆 = 0. Expansion in an isolated piston without friction. Δ𝐸𝐸𝐸𝐸𝐸𝐸 =𝑇𝑇0Δ𝑆𝑆= 0 Δ𝐸𝐸𝐸𝐸 =Δ𝐸𝐸−Δ𝐸𝐸𝐸𝐸𝐸𝐸 < 0 4. Assessing the effectiveness of the proposed pedagogical approach We would perform an experimental study using a Randomized Controlled Trial (RCT) design. 4.1 Procedure Students will be randomly divided into two groups: • Group A (Control): Will receive the traditional explanation of entropy (e.g., based on formal definitions from the second law of thermodynamics). • Group B (Experimental): Will receive the new explanation (e.g., using visual analogies, interactive simulations, or models inspired by information theory). 4.2 Assessment Instruments • Pre-test: A test covering basic thermodynamics concepts administered before the intervention to ensure similar baseline knowledge across groups. • Intervention: Delivery of the two different teaching methods. • Post-test: A specific test on entropy, including theoretical questions and application problems. • Perception Survey: A questionnaire assessing students' perceived clarity, motivation, and understanding. 4.3 Data Analysis • Post-test scores will be compared between groups using an independent samples t-test. • Survey responses will be analyzed to evaluate subjective perceptions of the teaching methods. 4.4 Success Criteria • The experimental group should demonstrate a statistically significant improvement in understanding and applying the concept of entropy. • Higher levels of perceived clarity and satisfaction as indicated in the survey responses. 5. Conclusions Through the development of the concept of non-workable energy, a macroscopic interpretation of the concept of entropy (specifically, its variation) has been identified, with potential applications in engineering education. In recent years, it has been observed that the use of this interpretation help undergraduate engineering students understand the concept of entropy, providing an engineering-based interpretation that is relatively easy to comprehend. 6. References [1] Klein, J.F., Physical significance of entropy or of the second law, The Scientific Press, 1910. [2] Kostic, M. J., The Elusive Nature of Entropy and Its Physical Meaning. Entropy, 2014. 16: p. 953-967. [3] Wu, J. Understanding entropy through a discussion of exergy. European Journal of Physics, 2020. 41: 19p [4] Zemansky, M. W., Dittman, R. H., Heat and Thermodynamics. McGraw Hill, 1981. 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑇𝑇0Δ𝑆𝑆 Δ𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 Δ𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸