applied sciences Article Optimizing Power and Thermal Efficiency of an Irreversible Variable-Temperature Heat Reservoir Lenoir Cycle Ruibo Wang 1,2,3, Lingen Chen 1,2,3,* , Yanlin Ge 1,2,3 and Huijun Feng 1,2,3 1 Institute of Thermal Science and Power Engineering, Wuhan Institute of Technology, Wuhan 430205, China; [email protected] (R.W.); [email protected] (Y.G.); [email protected] (H.F.) 2 Hubei Provincial Engineering Technology Research Center of Green Chemical Equipment, Wuhan 430205, China 3 School of Mechanical & Electrical Engineering, Wuhan Institute of Technology, Wuhan 430205, China * Correspondence: [email protected] or [email protected] Abstract: 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 (h) are derived. By numerical calculations, the characteristic relationships among P and h and the heat conductance distribution (uL) of the heat exchangers, as well as the thermal capacity rate matching (Cw f 1/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-h is a certain “point”, with the increase of heat reservoir inlet temperature ratio (t), UH, UL, and the irreversible expansion efficiency (he), P and h increase. When uL can be optimized, P and h versus uL characteristics are parabolic-like ones, there are optimal values of heat conductance distributions (u , u ) to make the cycle reach the maximum power and efficiency points (Pmax, hmax). LP(opt) Lh (opt) As Cw f 1/CH increases, Pmax-Cw f 1/CH shows a parabolic-like curve, that is, there is an optimal value of C /CH ((C /CH) ) to make the cycle reach double-maximum power point ((Pmax) ); as Citation: Wang, R.; Chen, L.; Ge, Y.; w f 1 w f 1 opt max C /C , U , and h increase, (P ) and (C /C ) increase; with the increase in t, (P ) Feng, H. Optimizing Power and L H T e max max w f 1 H opt max max increases, and (C /C ) is unchanged. Thermal Efficiency of an Irreversible w f 1 H opt Variable-Temperature Heat Reservoir Lenoir Cycle. Appl. Sci. 2021, 11, 7171. Keywords: finite-time thermodynamics; irreversible steady-flow Lenoir cycle; cycle power; thermal https://doi.org/10.3390/ efficiency; heat conductance distribution; thermal capacity rate matching app11157171 Academic Editor: Xiaolin Wang 1. Introduction Received: 14 July 2021 As a further extension of traditional irreversible process thermodynamics, finite-time Accepted: 30 July 2021 Published: 3 August 2021 thermodynamics (FTT) [1–11] has been applied to analyze and optimize performances of actual thermodynamic cycles, and great progress has been made. FTT has been applied in Publisher’s Note: MDPI stays neutral micro- and nano-cycles [12–15], thermoelectric devices [16,17], thermionic devices [18,19], with regard to jurisdictional claims in gas turbine cycles [20–22], internal combustion cycles [23,24], cogeneration plants [25,26], published maps and institutional affil- thermoradiative cell [27], chemical devices [28,29], and economics [30,31]. iations. According to the nature of the cycle, the researched heat engine (HEG) cycles include steady flow cycles [32–37] and reciprocating cycles [38–48]. For the steady flow HEG cycle, considering the temperature change of the heat reservoir (HR) can make the cycle closer to the actual working state of the HEG, therefore, some scholars have studied the steady flow cycles with variable temperature HR [49–53]. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. The Lenoir cycle (LC) model [54] was proposed by Lenoir in 1860. From the perspec- This article is an open access article tive of the cycle process, the LC lacks a compression process. It looks like a triangle in distributed under the terms and the cycle T-s diagram. It is a typical atmosphere pressure compression HEG cycle, the conditions of the Creative Commons compression process required by the HEG during operation is realized by atmosphere Attribution (CC BY) license (https:// pressure and it can be used in aerospace, ships, vehicles, and power plants in engineering creativecommons.org/licenses/by/ practice. Georgiou [55] first used classical thermodynamics to study the steady flow Lenoir 4.0/). cycle (SFLC) and compared its performance with that of a steady flow Carnot cycle. Appl. Sci. 2021, 11, 7171. https://doi.org/10.3390/app11157171 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 15 cycle Ts- diagram. It is a typical atmosphere pressure compression HEG cycle, the com- pression process required by the HEG during operation is realized by atmosphere pres- sure and it can be used in aerospace, ships, vehicles, and power plants in engineering Appl. Sci. 2021, 11, 7171 practice. Georgiou [55] first used classical thermodynamics to study the steady flow2 ofLe- 15 noir cycle (SFLC) and compared its performance with that of a steady flow Carnot cycle. Compared to the classical thermodynamics, the finite time process of the finite rate heat exchangeCompared (HEX) to the between classical the thermodynamics, system and the theenvironment finite time and process the finite of the size finite device rate areheat considered exchange (HEX)in the FTT between [1–11,5 the6–59], system therefore, and the the environment result obtained and is the closer finite to size the deviceactual HEGare considered performance in the FTT [1–11,56–59], therefore, the result obtained is closer to the actual HEGConsidering performance the heat transfer loss, Shen et al. [60] established an endoreversible SFLC modelConsidering with constant-temperature the heat transfer HRs loss, by Shen applying et al. FTT [60] theory, established analyzed an endoreversible the influences ofSFLC HR modeltemperature with constant-temperature ratio and total heat conductance HRs by applying (HTC) FTT on the theory, power analyzed output the( P influ-) and efficiencyences of HR (η ) temperature characteristics, ratio and and obtained total heat the conductancemaximum P (HTC) and maximum on the power η and output the corresponding(P) and efficiency optimal (h) characteristics, HTC distributions. and obtained Based on the the maximum NSGA-IIP algorithm,and maximum Ahmadih and et al.the [61] corresponding optimized the optimal ecological HTC performa distributions.nce coefficient Based on and the NSGA-IIthermoeconomic algorithm, perfor- Ah- mancemadi et of al. the [61 endoreversible] optimized the SFLC ecological with constant-temperature performance coefficient HRs. Based and thermoeconomic on the Ref. [60], Wangperformance et al. [62] of thefurther endoreversible considered SFLCthe internal with constant-temperature irreversibility loss, established HRs. Based the on irre- the versibleRef. [60], SFLC Wang model et al. [and62] furtheroptimized considered its P and the η internal performance. irreversibility loss, established the irreversible SFLC model and optimized its P and h performance. The above-mentioned were all studies on the SFLC with constant temperature HR. The above-mentioned were all studies on the SFLC with constant temperature HR. Based on Refs. [60–62], an irreversible SFLC with a variable temperature HR will be es- Based on Refs. [60–62], an irreversible SFLC with a variable temperature HR will be tablished in this paper, and the influence of internal irreversibility, HR inlet temperature established in this paper, and the influence of internal irreversibility, HR inlet temperature ratio, thermal thermal capacity capacity rate rate (TCR) (TCR) matching, matching, and and total total HTC HTC on on cycle cycle performance performance will will be studied.be studied. 2. Cycle Cycle Model Model and Thermodynamic Performance Figure 11 showsshows the TTs--s diagramdiagram ofof anan irreversibleirreversible variablevariable temperature HR SFLC. SFLC. Process 112!→ 2 ((331!→ 1)) is is a a constant constant volume volume (pressure) (pressure) endothermic endothermic (exothermic) (exothermic) one, and process 223!→ 3 is is an an irreversible irreversible expansion expansion one one (2 !( 23→3s iss the is correspondingthe corresponding isentropic isentropic one). one).Assuming Assuming the cycle the cycle working working fluid fluid is an is ideal an ideal gas, asgas, well as well as the as inlet the inlet (outlet) (outlet) temperature temper- atureof the of hot- the and hot- the and cold-side the cold-side fluid arefluidTHin are(T HoutTHin )(T andHout )T andLin(T LoutTLin ().TLout ). Figure 1. Cycle TTs-s- diagram. diagram. The irreversible expansion efficiency (he) is defined as [41,44,46,51]: T2 − T3 he = (1) T2 − T3s Appl. Sci. 2021, 11, 7171 3 of 15 Assuming the heat transfer between the working fluid and HR obeys the law of Newton heat transfer, according to the ideal gas properties and the theory of HEX, the cycle heat absorption and heat release rates are, respectively: QH = Cw f 1(T2 − T1) = CHminEH1(THin − T1) (2) QL = Cw f 2(T3 − T1) = CLminEL1(T3 − TLin) (3) . where CH(CL) and Cw f 1(Cw f 2) are heat source and working fluid TCRs (Cw f 1 = mCv, . Cw f 2 = mCP = kCw f 1), respectively, m is the working fluid mass flow rate, Cv(CP) is the constant volume (pressure) specific heat, k is the specific heat ratio. EH1 and EL1 are the effectiveness of hot- and cold-side HEXs, respectively: 1 − e−NH1(1−CHmin/CHmax) E = (4) H1 −N (1−C /C ) 1 − (CHmin/CHmax)e H1 Hmin Hmax 1 − e−NL1(1−CLmin/CLmax) E = (5) L1 −N (1−C /C ) 1 − (CLmin/CLmax)e L1 Lmin Lmax where NH1