Optimization, Stability, and Entropy in Endoreversible Heat Engines

Global Stability of the Curzon-Ahlborn Engine with a Working Substance That Satisfies the van der Waals Equation of State

In this paper, we show an analysis of the global stability of a Curzon-Ahlborn engine considering that the working substance of the engine satisfies the Van der Waals equation of state, which is more general than the ideal gas case. We use the Lyapunov stability theory for the case where the engine operates at a maximum power output. We analyze the steady state of the intermediate temperatures as well as the asymptotic behavior of the performance of the engine. Additionally, we study the relationship between the inherent time delay by analyzing the dynamic properties of the system and the stability of the steady state. We present illustrative graphs of the obtained results. Finally, we include a brief discussion of the obtained results and appropriate conclusions.

Heat-Only-Driven Vuilleumier Refrigeration

Vuilleumier refrigerators are devices which provide cooling power by using heat from a source at a temperature above the ambient. This feature makes Vuilleumier refrigerators particularly useful in situations where waste heat is abundant but electrical energy is scarce even as auxiliary operating power. For an operation by heat only with no need of auxiliary power, the Vuilleumier refrigerators must be designed in such a way that the mechanical losses occurring during operation are compensated by power gained from pressure differences. Here, we study the optimal design of such a device with heat-only operation maximizing the cooling power.

Optimized Cooling Power of a Vuilleumier Refrigerator with Limited Regeneration

Vuilleumier refrigerators provide cooling power by utilizing a heat source at temperatures above the ambient. This is particularly helpful in situations where waste heat is available and other power sources are limited. Vuilleumier refrigerators come in different technical configurations; here we analyze the thermodynamic performance of a configuration utilizing two displacer pistons with integrated regenerators. More specifically, we optimize the cooling power by optimizing the piston movement for a range of operation speeds. The optimization is based on the AS motion class for cyclic dynamics and uses an endoreversible model for the refrigerator. Our focus is on the influence of the regeneration extent present, and we find performance gains of about 17% for high regeneration extent and of about 28% for lower regeneration extent.

Cooling Cycle Optimization for a Vuilleumier Refrigerator

Vuilleumier refrigerators are a special type of heat-driven cooling machines. Essentially, they operate by using heat from a hot bath to pump heat from a cold bath to an environment at intermediate temperatures. In addition, some external energy in the form of electricity can be used as an auxiliary driving mechanism. Such refrigerators are, for example, advantageous in situations where waste heat is available and cooling power is needed. Here, the question of how the performance of Vuilleumier refrigerators can be improved is addressed with a particular focus on the piston motion and thus the thermodynamic cycle of the refrigerator. In order to obtain a quantitative estimate of the possible cooling power gain, a special class of piston movements (the AS motion class explained below) is used, which was already used successfully in the context of Stirling engines. We find improvements of the cooling power of more than 15%.

A Class of Reduced-Order Regenerator Models

We present a novel class of reduced-order regenerator models that is based on Endoreversible Thermodynamics. The models rest upon the idea of an internally reversible (perfect) regenerator, even though they are not limited to the reversible description. In these models, the temperatures of the working gas that alternately streams out on the regenerator’s hot and cold sides are defined as functions of the state of the regenerator matrix. The matrix is assumed to feature a linear spatial temperature distribution. Thus, the matrix has only two degrees of freedom that can, for example, be identified with its energy and entropy content. The dynamics of the regenerator is correspondingly expressed in terms of balance equations for energy and entropy. Internal irreversibilities of the regenerator can be accounted for by introducing source terms to the entropy balance equation. Compared to continuum or nodal regenerator models, the number of degrees of freedom and numerical effort are reduced considerably. As will be shown, instead of the obvious choice of variables energy and entropy, if convenient, a different pair of variables can be used to specify the state of the regenerator matrix and formulate the regenerator’s dynamics. In total, we will discuss three variants of this endoreversible regenerator model, which we will refer to as ES, EE, and EEn-regenerator models.

Optimal Heat Exchanger Area Distribution and Low-Temperature Heat Sink Temperature for Power Optimization of an Endoreversible Space Carnot Cycle

Using finite-time thermodynamics, a model of an endoreversible Carnot cycle for a space power plant is established in this paper. The expressions of the cycle power output and thermal efficiency are derived. Using numerical calculations and taking the cycle power output as the optimization objective, the surface area distributions of three heat exchangers are optimized, and the maximum power output is obtained when the total heat transfer area of the three heat exchangers of the whole plant is fixed. Furthermore, the double-maximum power output is obtained by optimizing the temperature of a low-temperature heat sink. Finally, the influences of fixed plant parameters on the maximum power output performance are analyzed. The results show that there is an optimal temperature of the low-temperature heat sink and a couple of optimal area distributions that allow one to obtain the double-maximum power output. The results obtained have some guidelines for the design and optimization of actual space power plants.

Performance Analysis and Four-Objective Optimization of an Irreversible Rectangular Cycle

Based on the established model of the irreversible rectangular cycle in the previous literature, in this paper, finite time thermodynamics theory is applied to analyze the performance characteristics of an irreversible rectangular cycle by firstly taking power density and effective power as the objective functions. Then, four performance indicators of the cycle, that is, the thermal efficiency, dimensionless power output, dimensionless effective power, and dimensionless power density, are optimized with the cycle expansion ratio as the optimization variable by applying the nondominated sorting genetic algorithm II (NSGA-II) and considering four-objective, three-objective, and two-objective optimization combinations. Finally, optimal results are selected through three decision-making methods. The results show that although the efficiency of the irreversible rectangular cycle under the maximum power density point is less than that at the maximum power output point, the cycle under the maximum power density point can acquire a smaller size parameter. The efficiency at the maximum effective power point is always larger than that at the maximum power output point. When multi-objective optimization is performed on dimensionless power output, dimensionless effective power, and dimensionless power density, the deviation index obtained from the technique for order preference by similarity to an ideal solution (TOPSIS) decision-making method is the smallest value, which means the result is the best.

Optimizing Power and Thermal Efficiency of an Irreversible Variable-Temperature Heat Reservoir Lenoir Cycle

Applying finite-time thermodynamics theory, an irreversible steady flow Lenoir cycle model with variable-temperature heat reservoirs is established, the expressions of power (P) and efficiency (η) are derived. By numerical calculations, the characteristic relationships among P and η and the heat conductance distribution (uL) of the heat exchangers, as well as the thermal capacity rate matching (Cwf1/CH) between working fluid and heat source are studied. The results show that when the heat conductances of the hot- and cold-side heat exchangers (UH, UL) are constants, P-η is a certain “point”, with the increase of heat reservoir inlet temperature ratio (τ), UH, UL, and the irreversible expansion efficiency (ηe), P and η increase. When uL can be optimized, P and η versus uL characteristics are parabolic-like ones, there are optimal values of heat conductance distributions (uLP(opt), uLη(opt)) to make the cycle reach the maximum power and efficiency points (Pmax, ηmax). As Cwf1/CH increases, Pmax-Cwf1/CH shows a parabolic-like curve, that is, there is an optimal value of Cwf1/CH ((Cwf1/CH)opt) to make the cycle reach double-maximum power point ((Pmax)max); as CL/CH, UT, and ηe increase, (Pmax)max and (Cwf1/CH)opt increase; with the increase in τ, (Pmax)max increases, and (Cwf1/CH)opt is unchanged.

Performance Optimizations with Single-, Bi-, Tri-, and Quadru-Objective for Irreversible Diesel Cycle

Applying finite time thermodynamics theory and the non-dominated sorting genetic algorithm-II (NSGA-II), thermodynamic analysis and multi-objective optimization of an irreversible Diesel cycle are performed. Through numerical calculations, the impact of the cycle temperature ratio on the power density of the cycle is analyzed. The characteristic relationships among the cycle power density versus the compression ratio and thermal efficiency are obtained with three different loss issues. The thermal efficiency, the maximum specific volume (the size of the total volume of the cylinder), and the maximum pressure ratio are compared under the maximum power output and the maximum power density criteria. Using NSGA-II, single-, bi-, tri-, and quadru-objective optimizations are performed for an irreversible Diesel cycle by introducing dimensionless power output, thermal efficiency, dimensionless ecological function, and dimensionless power density as objectives, respectively. The optimal design plan is obtained by using three solution methods, that is, the linear programming technique for multidimensional analysis of preference (LINMAP), the technique for order preferences by similarity to ideal solution (TOPSIS), and Shannon entropy, to compare the results under different objective function combinations. The comparison results indicate that the deviation index of multi-objective optimization is small. When taking the dimensionless power output, dimensionless ecological function, and dimensionless power density as the objective function to perform tri-objective optimization, the LINMAP solution is used to obtain the minimum deviation index. The deviation index at this time is 0.1333, and the design scheme is closer to the ideal scheme.

Optimal Control for a Hydraulic Recuperation System Using Endoreversible Thermodynamics

Energy savings in the traffic sector are of considerable importance for economic and environmental considerations. Recuperation of mechanical energy in commercial vehicles can contribute to this goal. One promising technology rests on hydraulic systems, in particular for trucks which use such system also for other purposes such as lifting cargo or operating a crane. In this work the potential for energy savings is analyzed for commercial vehicles with tipper bodies, as these already have a hydraulic onboard system. The recuperation system is modeled based on endoreversible thermodynamics, thus providing a framework in which realistic driving data can be incorporated. We further used dissipative engine setups for modeling both the hydraulic and combustion engine of the hybrid drive train in order to include realistic efficiency maps. As a result, reduction in fuel consumption of up to 26% as compared to a simple baseline recuperation strategy can be achieved with an optimized recuperation control.

Comparative Assessment of Various Low-Dissipation Combined Models for Three-Terminal Heat Pump Systems

Thermally driven heat pump systems play important roles in the utilization of low-grade thermal energy. In order to evaluate and compare the performances of three different constructions of thermally driven heat pump and heat transformer, the low-dissipation assumption has been adopted to establish the irreversible thermodynamic models of them in the present paper. By means of the proposed models, the heating loads, the coefficients of performance (COPs) and the optimal relations between them for various constructions are derived and discussed. The performances of different constructions are numerically assessed. More importantly, according to the results obtained, the upper and lower bounds of the COP at maximum heating load for different constructions are generated and compared by the introduction of a parameter measuring the deviation from the reversible limit of the system. Accordingly, the optimal constructions for the low-dissipation three-terminal heat pump and heat transformer are determined within the frame of low-dissipation assumption, respectively. The optimal constructions in accord with previous research and engineering practices for various three-terminal devices are obtained, which confirms the compatibility between the low-dissipation model and endoreversible model and highlights the validity of the application of low-dissipation model for multi-terminal thermodynamic devices. The proposed models and the significant results obtained enrich the theoretical thermodynamic model of thermally driven heat pump systems and may provide some useful guidelines for the design and operation of realistic thermally driven heat pump systems.

Optimization Modeling of Irreversible Carnot Engine from the Perspective of Combining Finite Speed and Finite Time Analysis

An irreversible Carnot cycle engine operating as a closed system is modeled using the Direct Method and the First Law of Thermodynamics for processes with Finite Speed. Several models considering the effect on the engine performance of external and internal irreversibilities expressed as a function of the piston speed are presented. External irreversibilities are due to heat transfer at temperature gradient between the cycle and heat reservoirs, while internal ones are represented by pressure losses due to the finite speed of the piston and friction. Moreover, a method for optimizing the temperature of the cycle fluid with respect to the temperature of source and sink and the piston speed is provided. The optimization results predict distinct maximums for the thermal efficiency and power output, as well as different behavior of the entropy generation per cycle and per time. The results obtained in this optimization, which is based on piston speed, and the Curzon–Ahlborn optimization, which is based on time duration, are compared and are found to differ significantly. Correction have been proposed in order to include internal irreversibility in the externally irreversible Carnot cycle from Curzon–Ahlborn optimization, which would be equivalent to a unification attempt of the two optimization analyses.

Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle

An improved irreversible closed modified simple Brayton cycle model with one isothermal heating process is established in this paper by using finite time thermodynamics. The heat reservoirs are variable-temperature ones. The irreversible losses in the compressor, turbine, and heat exchangers are considered. Firstly, the cycle performance is optimized by taking four performance indicators, including the dimensionless power output, thermal efficiency, dimensionless power density, and dimensionless ecological function, as the optimization objectives. The impacts of the irreversible losses on the optimization results are analyzed. The results indicate that four objective functions increase as the compressor and turbine efficiencies increase. The influences of the latter efficiency on the cycle performances are more significant than those of the former efficiency. Then, the NSGA-II algorithm is applied for multi-objective optimization, and three different decision methods are used to select the optimal solution from the Pareto frontier. The results show that the dimensionless power density and dimensionless ecological function compromise dimensionless power output and thermal efficiency. The corresponding deviation index of the Shannon Entropy method is equal to the corresponding deviation index of the maximum ecological function.