Comparative Performance Analysis of a Simplified Curzon-Ahlborn Engine

This paper presents a finite-time thermodynamic optimization based on three different optimization criteria: Maximum Power Output (MP), Maximum Efficient Power (MEP), and Maximum Power Density (MPD), for a simplified Curzon-Ahlborn engine that was first proposed by Agrawal. The results obtained for the MP are compared with those obtained using MEP and MPD criteria. The results show that when a Newton heat transfer law is used, the efficiency values of the engine working in the MP regime are lower than the efficiency values (τ) obtained with the MEP and MPD regimes for all values of the parameter τ=T2/T1, where T1 and T2 are the hot and cold temperatures of the engine reservoirs (T2<T1), respectively. However, when a Dulong-Petit heat transfer law is used, the efficiency values of the engine working at MEP are larger than those obtained with the MP and the MPD regimes for all values of τ. Notably, when 0<τ<0.68, the efficiency values for the MP regime are larger than those obtained with the MPD regime. Also, when 0.68<τ<1, the efficiency values for the aforementioned regimes are similar. Importantly, the parameter τ plays a crucial role in the engine performance, providing guidance during the design of real power plants.


Introduction
The concept of Carnot's efficiency is one of the cornerstone of thermodynamics. It serves as the upper bound for the heat engine efficiency between two heat reservoirs; however, when the engines are operating infinitely slower, this is obviously unrealistic. From the second half of the 20th century, research has focused on identifying performance limits of thermodynamic processes and optimizing thermodynamic cycles. Novikov [1], Chambdal [2], and Curzon-Ahlborn [3] were the first to extend the Carnot cycle, considering the irreversibilities of finite time, to show that a Carnot engine with heat resistance in its reservoirs has a maximal power production, and this maximum thermal efficiency can be described by η CA = 1 − √ T 2 /T 1 = 1 − τ. From the pioneer work of Curzon-Ahlborn, a new branch of irreversible thermodynamics appeared called Finite Time Thermodynamics (FTT), which have inspired many articles that focused on power optimization or minimization of fixed costs for heat engines, endoreversibles, non-endoreversibles, and finite size constrains under various heat transfer laws, including linear and non-linear, among others [4][5][6][7][8][9][10]. Extensive details about the FTT background can be found in Bejan [11] and Cheng et al. [12]. However, the above mentioned works did not consider the effect of engine size related to investment cost. To incorporate the effects of size in performance

Agrawal's Model
The temperatures of hot and cold reservoirs and the temperatures of the working fluid substance for a Curzon-Ahlborn engine are related by: where T 1 is the temperature of the hot reservoir, T 2 is the temperature of the cold reservoir, and T 1W and T 2W are the working fluid temperatures of the heat engine at the hot and cold isotherms, respectively, as depicted in Figure 1. Also, in their famous paper, Curzon and Ahlborn [3] defined x = T 1 − T 1W and y = T 2W − T 2 as the temperature difference between thermal reservoirs and the isothermal branches of the internal cycle. Moreover, by using an algebraic method, Agrawal [20] proposed a simplified version of the Curzon-Ahlborn engine to help undergraduate students more easily understand the theory, in which, by assigning the same thermal resistance to the same temperature differences at the upper and the lower cycle isotherm, he obtained similar efficiency values to those obtained by Curzon-Ahlborn for real power plants. Furthermore, this model has remarkable similarities with other results obtained from finite time thermodynamics [21,22]. Now, by considering a Newton heat transfer law from the hot reservoir to the working fluid (Q 1 ) and from the working fluid to the cold reservoir (Q 2 ), we obtain: where i = 1 at the hot isotherm and i = 2 at the cold isotherm of the cycle. For simplicity, an equal thermal conductance factor is considered in both heat transfer processes, α.
In accordance with the procedure followed by Agrawal [20] and repeated by Páez-Hernández et al. [21], the power output can be written as:

Maximum Power Output
In order to investigate the efficiency when aheat engine is working on aMaximum Power Output regime, let us consider that the temperatures of the engine working fluid, 1 and 2 , work as a Carnot engine, so its efficiency is = 1 − , where isthe ratio of working temperatures, = 2 1 ⁄ . So, we obtain: where we are using the Agrawal assumption, in which the difference intemperatures is = . Now, from the last equation, we can solve obtaining: which, substituted in Equation (2), leads to: From the condition ̇⁄ = 0, the value of , where ̇ has a maximum value, is: Then, the efficiency at maximum power output regime yield is: and the maximum value of power output provides:

Maximum Power Density
Instead of using the Maximum Power Output and the efficiency in order to analyze the performance of thermodynamic cycles, recently Sahinet al. [13] introduced the Maximum Power Density criterion, which involves maximizing the ratio of the power to the maximum specific volume in the cycle. For the system that we are considering, the MPD can be defined as:

Maximum Power Output
In order to investigate the efficiency when a heat engine is working on a Maximum Power Output regime, let us consider that the temperatures of the engine working fluid, T 1W and T 2W , work as a Carnot engine, so its efficiency is η = 1 − θ, where θ is the ratio of working temperatures, θ = T 2W /T 1W . So, we obtain: where we are using the Agrawal assumption, in which the difference in temperatures is x = y. Now, from the last equation, we can solve x obtaining: which, substituted in Equation (2), leads to: From the condition d W has a maximum value, is: Then, the efficiency at maximum power output regime yield is: and the maximum value of power output provides: .

Maximum Power Density
Instead of using the Maximum Power Output and the efficiency in order to analyze the performance of thermodynamic cycles, recently Sahin et al. [13] introduced the Maximum Power Density criterion, which involves maximizing the ratio of the power to the maximum specific volume in the cycle. For the system that we are considering, the MPD can be defined as: where . W PD is the Power Density, . W MPV is the Maximum Power Output, and V is the Maximum Volume in the cycle.
Our next objective was to analyze in more detail the Power Density and establish a set of equations similar to Equations (7) and (8). Thus, following the process used in the previous section, we proceeded to maximize the Power Density.
where we assume that the maximum volume in the cycle is an ideal gas, which can be written as: where m is the mass of the working fluid, and R is the ideal gas constant. In this analysis, the minimum pressure P min in the cycle is taken to be constant. It is important to note that Equations (10) and (11) contain the mR constant parameter instead of nR, where n is the quantity of moles, as is common in classical equilibrium thermodynamics. This does not influence the analysis that is being performed because, during the processes of power derivation and normalization, this parameter disappears. In addition, as the mass and the number of molecules in a substance are proportional, this allows using these expressions without any conceptual problems. Now, the condition d W MPD has a maximum value: Then, the efficiency of engine at maximum power regime yields:

Maximum Efficient Power
Let us analyze the Maximum Efficient Power regime. Yilmaz et al. [17] introduced the Maximum Efficient Power criterion as the multiplication of power output by cycle efficiency, obtaining: As above, proceed to maximize the Efficient Power. Then, from Equation (13), we obtain: .
Then, the efficiency of the engine at MEP regime results in: It is important to notice that in all the regimes considered above, we use a Newton heat transfer law for the interchange of energy between the heat reservoirs.

Performance Using Different Criteria for the Dulong-Petit Heat Transfer Law
Following the same procedure performed by Páez-Hernández et al. [21], we analyzed the engine shown in Figure 1, but considering a Dulong-Petit heat transfer law, whichis .
where α is the thermal conductance and k = 5/4 is the exponent related to natural convection [23], with i = 1 at the hot isotherm and i = 2 at the cold isotherm of the cycle.

Maximum Power Output
In order to investigate the efficiency when a heat engine is working at Maximum Power Output, let us consider that, for the Dulong-Petit heat transfer law, the MP, after some algebraic steps, is given by: where the superscript DP denotes that we are using the Dulong-Petit heat transfer law. Then, the efficiency of the engine at Maximum Power Output with a Dulong-Petit heat yields:

Maximum Power Density
Now, for the MPD, similar to the previous section, we proceed to maximize the power density for the cited heat transfer law. Therefore, it is easy to obtain: The condition d W DP PD has a maximum value, being: Then, the efficiency of engine at maximum power density regime yields: Entropy 2018, 20, 637 6 of 9

Maximum Efficient Power
Let us analyze the Maximum Efficient Power regime. As before, we proceed to maximize the MEP. Then, using the definition given byYilmaz [17], we obtain: The condition d . W DP EP /dθ = 0 provides the value of θ DP MEP , where . W DP EP has a maximum value: Then, the efficiency of the engine at Maximum Efficient Power regimes is,

Dulong-Petit Heat Transfer Law Case
Now, using the same procedure as in the previous section, a comparison of efficiencies at MP, MPD, and MEP is shown in Figure 3, in which a similar relation of the efficiencies, as shown in Equation (27), is fulfilled:

Dulong-Petit Heat Transfer Law Case
Now, using the same procedure as in the previous section, a comparison of efficiencies at MP, MPD, and MEP is shown in Figure 3, in which a similar relation of the efficiencies, as shown in Equation (27), is fulfilled: Entropy 2018, 20, 637 7 of 9

Dulong-Petit Heat Transfer Law Case
Now, using the same procedure as in the previous section, a comparison of efficiencies at MP, MPD, and MEP is shown in Figure 3, in which a similar relation of the efficiencies, as shown in Equation (27), is fulfilled:

Conclusions
This paper presents a finite-time thermodynamic optimization based on three different optimization criteria-Maximum Power Output (MP), Maximum Efficient Power (MEP) and Maximum Power Density (MPD)-for a simplified Curzon-Alhborn engine proposed by Agrawal [20]. Despite the model being very simple, it captures the behaviordeterminedby Yilmaz et al. [24], in the sense that, for a Newton heat transfer law, we observed that > > when ≤ 0.19 ,and > > when 0.19 < < 1 . Moreover, when = 0.19 , = .This is interesting because the same result was obtained by Yilmaz et al. [24] but using a more robust model

Conclusions
This paper presents a finite-time thermodynamic optimization based on three different optimization criteria-Maximum Power Output (MP), Maximum Efficient Power (MEP) and Maximum Power Density (MPD)-for a simplified Curzon-Alhborn engine proposed by Agrawal [20]. Despite the model being very simple, it captures the behavior determined by Yilmaz et al. [24], in the sense that, for a Newton heat transfer law, we observed that η MPD > η MEP > η MP when τ ≤ 0.19, and η MEP > η MPD > η MP when 0.19 < τ < 1. Moreover, when τ = 0.19, η MPD = η MEP . This is interesting because the same result was obtained by Yilmaz et al. [24] but using a more robust model that even included irreversibilities. The latter shows that the oversimplified model proposed by Agrawal could be used to model some real heat engines. When τ = 0.68, η MPD = η MP but is less than η MEP . This result is not the same as that obtained by Yilmaz et al. [24] but is close to the one reported. In our case, additionally, we changed the heat transfer law, so instead of using a Newton law, we proposed a Dulong-Petit heat transfer law. For this case, we observed that η MEP > η MPD > η MP when 0 < τ ≤ 0.68 and η MEP > η MPD ≥ η MP when 0.68 < τ < 1.
The behavior of the Maximum Power Density can be explained by involving different operation parameters of a thermal engine related to its design restrictions. This depends on the high power or high efficiency of the heat engine. Notably, when the Dulong-Petit heat transfer law was used, the interval η MEP > η MPD > η MP was greater than in the Newton heat transfer law case.
The above evaluation can be seen more clearly in Figure 4, where, for a specific τ value, the thermal efficiency at the MEP (η MEP ) condition was greater than the other conditions, MPD and MP. However, if the parameter τ changed, the behavior of the efficiency also changed. Moreover, some authors [25,26] showed that a good trade-off between the engine performance and its dynamic behavior occurs when 0.32 ≤ τ ≤ 0.64, providing important guidance when real power plants are designed. This indicates that τ plays an important role in engine performance as other authors have stated [27,28].
Author Contributions: All authors made substantial contributions to the analysis and conclusions presented in this work. All authors have read and approved the final manuscript.
Funding: This research received no external funding.