Article Efficiency at Maximum Power of the Low-Dissipation Hybrid Electrochemical–Otto Cycle
David Diskin 1 and Leonid Tartakovsky 1,2,* 1 Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa 3200003, Israel; [email protected] 2 Grand Technion Energy Program, Technion—Israel Institute of Technology, Haifa 3200003, Israel * Correspondence: [email protected]
Received: 24 June 2020; Accepted: 24 July 2020; Published: 1 August 2020
Abstract: A novel analytical method was developed for analysis of efficiency at maximum power of a hybrid cycle combining electrochemical and Otto engines. The analysis is based on the low-dissipation model, which relates energy dissipation with energy transfer rate. Efficiency at maximum power of a hybrid engine operating between two reservoirs of chemical potentials is evaluated. The engine is composed of an electrochemical device that transforms chemical potential to electrical work of an Otto engine that uses the heat generated in the electrochemical device and its exhaust effluent for mechanical work production. The results show that efficiency at maximum power of the hybrid cycle is identical to the efficiency at maximum power of an electrochemical engine alone; however, the power is the product of the electrochemical engine power and the compression ratio of the Otto engine. Partial mass transition by the electrochemical device from the high to the low chemical potential is also examined. In the latter case, heat is generated both in the electrochemical device and the Otto engine, and the efficiency at maximum power is a function of the compression ratio. An analysis performed using the developed method shows, for the first time, that, in terms of a maximal power, at some conditions, Otto cycle can provide better performance that the hybrid cycle. On the other hand, an efficiency comparison at maximum power with the separate Otto-cycle and chemical engine results in some advantages of the hybrid cycle.
Keywords: finite time thermodynamics; efficiency at maximal power; electro-chemical reaction; low-dissipation; Otto cycle
1. Introduction According to equilibrium thermodynamics, the maximal efficiency of a cycle operating between two reservoirs at thermodynamic potential PH and PL (PH > PL) is the Carnot efficiency, η = 1 PL/PH. C − However, finite size engines that work at equilibrium conditions (i.e., undergo only reversible processes) cannot produce finite power. Therefore, non-equilibrium thermodynamics that predicts efficiency as a function of power is meaningful. The first to analyze heat engines in a finite power regime were Novikov in 1957 [1] and Curzon-Ahlborn in 1975 [2]. Their approach was to model the interactions between the engine and its reservoirs with energy transfer resistance. The engine itself was modeled as a reversible system, called lately and expended to be an endoreversible cycle [3]. An equivalent approach of engine analyses at maximum power is based on the low-dissipation assumption and was suggested by Esposito et al. [4]. They found, with this approach, the bounds of the efficiency at maximal power for heat engines. Guo et al. [5] expended those limits to chemical engines and non-Carnot heat cycles (Otto, Brayton, etc.). To the best of our knowledge, the combination of chemical engine and Otto cycle was not analyzed by the low-dissipation approach for finite power. This combination is called a “hybrid cycle” hereafter. A schematic outline of the hybrid cycle is shown
Energies 2020, 13, 3961; doi:10.3390/en13153961 www.mdpi.com/journal/energies Energies 2020, 13, 3961 2 of 10
in Figure1. Electrochemical engines use the high chemical potential (µH) of a substance (for example fuel-oxygen mixture) as the high-potential reservoir to generate electrical work. Petrescu et al. [6] studied the main differences between electrochemical engines and heat engines from the viewpoint of efficiency at maximum power. Pavelka et al. [7] and Vagner et al. [8] commented on the maximum work of electrochemical engines that undergo heat interactions. They concluded that, in general, the maximum power of electrochemical engines cannot be evaluated by exergy analysis or entropy minimization. The low-dissipation assumption is consistent with this conclusion. The Otto cycle implemented in internal combustion engines (ICEs) uses the same high chemical potential µH carrier to generate mechanical work by converting the chemical potential to thermal energy with subsequent gas expansion. The conversion of chemical potential to thermal energy in ICE is a spontaneous process and is thus completely irreversible. Hybrid cycles that combine electrochemical engine (fuel cell) and a bottoming heat engine have been studied and developed since 1990s [9,10]. The combination of a high efficiency fuel cell and the recovery of the waste heat and fuel by the bottoming heat cycle is a promising path towards clean and efficient energy use. Most of the combinations include a solid oxide fuel cell (SOFC) and a Rankine [11] or a Brayton cycle [12,13]. Lately, research on an ICE in combination with a SOFC was initiated as well [14,15]. In these studies, integration of engine experiment results with a basic fuel cell model were performed. Exergy analysis showed efficiency dependence on basic parameters such as compression ratio and anode off-gas temperature. Thermo-economic analysis for this configuration was also performed [16]. An efficient onboard storage and utilization of fuel as the high chemical potential carrier is another important aspect to be addressed when fuel cell usage is considered. Fuel cells are known to be fuel sensitive, and fuel reforming processes are implemented frequently in fuel cell power pack designs [17]. When a hybrid cycle involving a fuel cell and an ICE is considered, usage of fuel reforming in combination with waste heat recovery known as Thermochemical Recuperation (TCR) could be beneficial. Recently, utilization of fuel reforming through TCR in ICEs was proven to be energetically efficient [18–20]. Hence, a potential of its implementation in a hybrid Fuel-Cell-ICE cycle should be evaluated. Chuahy and Kokjohn [21] analyzed a hybrid powertrain including a diesel engine and a fuel cell with diesel fuel reforming to produce hydrogen and reported on the high thermal efficiency (above 70%) of the system. The main goal of the previously published studies investigating the hybrid fuel cell–ICE cycle was an achievable efficiency gain. However, as stated earlier, approaching Carnot efficiency will result in the loss of power. Hence, a question arises of whether the combination of electrochemical engine and ICE can provide power gain in addition to efficiency improvement. Considering this, the reported study aims at the analysis of the power–efficiency relation of the hybrid cycle combining chemical and Otto engines and a comparison with the pure Otto cycle and chemical engine operation alone. For this purpose, a novel analytical method of analysis of the power–efficiency tradeoff for this hybrid cycle was suggested for the first time. This method enables better understanding of the advantages and drawbacks of the hybrid cycle compared to the other cycles in terms of the power–efficiency relationship. Energies 2020 13 Energies 2020,, 13,, 3961 x FOR PEER REVIEW 33 ofof 1011
Figure 1. Schematic outline of a fuel cell–internal combustion engine (ICE) hybrid cycle. Air is Figure 1. Schematic outline of a fuel cell–internal combustion engine (ICE) hybrid cycle. Air is compressed in the ICE and then reacts in the electrochemical device (fuel cell) to produce electrical power. compressed in the ICE and then reacts in the electrochemical device (fuel cell) to produce electrical Hydrogen-rich fuel cell exhaust gases combust and expand in the ICE to produce mechanical work. power. Hydrogen-rich fuel cell exhaust gases combust and expand in the ICE to produce mechanical 2. Low-Dissipationwork. Model of the Hybrid Cycle
2. Low-DissipationIn this analysis, Model we assume of the that Hybrid the only Cycle irreversible processes take place in the electrochemical engine, and the bottoming Otto cycle is reversible. This assumption means that the mechanical processesIn this are analysis, reversible, wei.e., assume the compression that the only and irreve expansionrsible processes are isentropic. take place Moreover, in the theelectrochemical processes in theengine, Otto and cycle the are bottoming assumed toOtto be relativelycycle is reversib fast comparedle. This toassumption the chemical means energy that transfer, the mechanical thus the highprocesses and low are chemicalreversible, potential i.e., the compression reservoirs could and beexpansion considered are asisentropic. the isothermal Moreover, boundaries the processes of the chemicalin the Otto engine. cycle are The assumed low-dissipation to be relatively model assumesfast compared that the toirreversibility the chemical energy in the process transfer, of thus energy the conversionhigh and low between chemical the potential reservoirs reservoirs and the hybridcould be engine considered is proportional as the isothermal to 1/t, where boundariest is the of time the ofchemical the energy engine. conversion The low-dissipation process. The model meaning assumes of this that assumption the irreversibility is that the in relaxationthe process time of energy in the hybridconversion engine between is relatively the reservoirs short compared and the hybrid to the energy engine transmissionis proportional time. to 1/ Accordingt , where tot this is the model, time theof the cycle energy will becomeconversion reversible process. for Thet meaning, as expected. of this assumption The low-dissipation is that the model relaxation expresses time energy in the → ∞ absorptionhybrid engine from is therelatively high chemical short compared potential to reservoir the energy in antransmission electrochemical time. engineAccording as [5 to] this model, the cycle will become reversible for t →∞, as expected. The low-dissipation model expresses energy absorption from the high chemical potential reservoirσH in an electrochemical engine as [5] EH = µH∆N 1 (1) − tH σ =Δμ −H and the energy rejected to the low chemicalENHH potential reservoir1 as (1) tH σL and the energy rejected to the low chemicalEL = potentialµL∆N 1 reservoir+ as (2) tL σ Here, µH, µL are the chemical potentials,=Δμ∆N is the + totalL mass transmitted between the chemical ENLL1 (2) reservoirs, σ , σ are the irreversibility factors that include the information on the irreversibility of H L tL the energy transmission between the high and the low chemical potential reservoirs, respectively. μμ Δ TheHere, expressions HL, are include the chemical the term potentials, of reversible N energy is theconversion total mass (transmittedµH∆N, µL∆ Nbetween) and the the dissipation chemical σσ termreservoirs, caused byH , theL are irreversible the irreversibility energy transferfactors that processes. include the Obviously, information according on the irreversibility to the first law, of thethe workenergy that transmission extracted from between the chemical the high engine and the is thelow di chemicalfference betweenpotential thereservoirs, above absorbed respectively. and rejected energy, μ ΔN μ ΔN The expressions include the term of reversible energy conversion ( H , L ) and the σH σL dissipation term caused byWC the= EirreversibleH EL = µ Henergy∆N 1 transferµ processes.L∆N 1 + Obviously, according to the(3) − − tH − tL first law, the work that extracted from the chemical engine is the difference between the above absorbed and rejected energy,
Energies 2020, 13, 3961 4 of 10 while the reversible work is W = µH∆N µL∆N (4) C,rev − We conclude that the difference between the reversible work and the irreversible work is the waste heat. σH σL WC,rev WC = QC = µH∆N + µL∆N (5) − tH tL Waste heat can be recovered to produce useful work. In the case of the bottoming Otto cycle, heat is recovered to increase thermal energy of the working substance after the compression stroke. By neglecting chemical potentials dependence on temperature, thermal energy increase can be expressed by Z TH σ σ CdT = µ ∆N H + µ ∆N L (6) H t L t T1 H L where C is the heat capacity, T1 is the temperature after compression, TH is the temperature before the expansion stroke, and by assuming constant heat capacity, TH can be expressed as 1 σH σL TH = T1 + µH∆N + µL∆N (7) C tH tL
Following the assumption of isentropic compression, T1 dependence on the compression ratio r can be expressed as γ 1 T1/TL = r − (8)
Here, TL is the temperature of the low potential reservoir, γ is the heat capacities ratio. The work produced by the reversible Otto engine is [5] ! TH WOtto = C(TH T1) CTL 1 (9) − − T1 −
By combining Equations (6) and (9), we can obtain the work of the Otto engine. The total work of the hybrid cycle will be a sum of the Otto engine mechanical work and the electrochemical engine electrical work. Obviously, the total power will be the total work of the cycle divided by the time of the cycle. ! ! σH σH TL σL σL TL WOtto = µH∆N + µL∆N (10) tH − tH T1 tL − tL T1 ! ! σH TL σL TL Wtot = WOtto + WC = µH∆N 1 µL∆N 1 + (11) − tH T1 − tL T1 ! ! µH∆N σH TL µL∆N σL TL Ptot = Wtot/(tH + tL) = 1 1 + (12) tH + tL − tH T1 − tH + tL tL T1 We also notice that the expression for power is similar to the expression for power of a chemical engine (without heat recovery by a bottoming cycle) and the difference is the TL/T1 ratio (which depends on the compression ratio of the Otto cycle) that multiply the irreversibility terms σH,L/tH,L. This difference is reflected also in the efficiency term, ! ! W σ T µL σ T η = tot = 1 H L 1 L L (13) µH∆N − tH T1 − µH − tL T1
The similarity in the efficiency expression turns to identity in the efficiency at maximum power and the same expression, as derived previously by Guo et al. [5], is obtained for the hybrid cycle.
ηCµ ηm = q (14) 2 η / 1 + a(1 η ) − Cµ − Cµ Energies 2020, 13, 3961 5 of 10
Here, ηCµ is the Carnot efficiency of the chemical engine, a = σL/σH. However, the maximum power of the hybrid engine is different from that of the chemical engine, and it is a function of the TL/T1 ratio 2 µH∆N ηCµ T1 T1 P = = P (15) max 4σ q T max,C T H 1 + (1 η )a L L − Cµ
Here, Pmax,C is the maximum power of a chemical engine obtained by Guo et al. [5]. The simple multiplication of the chemical engine power and the TL/T1 ratio is a consequence of the negligible time consumed by the processes in the Otto cycle.
3. Low-Dissipation Model for Partial Conversion Notably, in the analysis presented in the previous section, all the high-chemical-potential carrier (for example, hydrogen and air mixture) was assumed to be transformed to the low-potential carrier in the chemical engine. In this section, we analyze the more realistic case of partial conversion in the chemical engine, where a part of the high-chemical-potential carrier is expelled from the electrochemical engine and reused in the Otto cycle for combustion. Again, the processes in the Otto cycle are assumed to be relatively fast compared to the energy transfer in the electrochemical engine. However, in this case, the process of chemical potential conversion to thermal energy also exists. Nevertheless, the fact that in an ICE the energy conversion process from high to low chemical potential is uncontrolled (the process is spontaneous, in contrast to the energy conversion process in the electrochemical engine) justifies the relatively fast process assumption. In such a case, the temperature TH is calculated by the same steps as in Equation (7) (Section2), with the addition of contribution of the remaining high-potential carrier combustion.
1 σH 1 σL 1 TH = T1 + µH∆N + µL∆N + NL(µH µL) (16) C tH C tL C −
Here, NL is the remaining high-potential carrier mass that was not transformed to the low-potential one in the chemical engine. For simplification and without losing generality, we analyze hereinafter the case of striving for zero low chemical potential
1 σH 1 TH = T1 + µH∆N + NLµH (17) C tH C
Work and power then are extracted in similar steps expressed by Equations (9)–(12), and the efficiency for maximum power as a function of ∆N is obtained.
1 TL TL ηm = 1 ∆N(∆N + NL(1 )) NL (18) − 2 − T1 − T1
Recognizing that for normalized mass NL = 1 ∆N and dNL = d∆N, we can obtain the optimal − − conversion in the electrochemical engine for the maximal efficiency. 1 T1 NL,max = 1 T 2 TL − 1 3 1 T1 1 3 (19) ∆Nmax = ≤ TL ≤ 2 − 2 TL
In the case of a high compression ratio that corresponds to T1/TL > 3, the maximum power is obtained for a complete conversion in the ICE, i.e., no electrochemical reaction—∆N = 0. For the typical chemical potential carriers, as hydrogen and air, and adiabatic index γ = 1.4, the compression ratio needed for the redundancy of the electrochemical engine is about 15. In other words, when a maximum power is the main objective of powertrain development, Otto-cycle-based ICE with a compression ratio above 15 would be beneficial. Notably, higher compression ratios would be required at smaller γ Energies 2020, 13, 3961 6 of 10
values typical for ICE exhaust gases. The situation is changed when the powertrain efficiency is of importance together with the maximum power level. The maximal efficiency as a function of the T1/TL ratio in the range of the hybrid operation is
1 1 T1 1 TL T1 ηm,maxNL = + + , 1 3 (20) 4 8 TL 8 T1 ≤ TL ≤
4. Results and Discussion Efficiency at a maximum power sets a mark for energy conversion configurations. Of course, this mark is not a limit [22,23], therefore efficiency vs power curve is a valuable figure worthy of exergy-economic discussion [24]. We begin the discussion with the case of full conversion of the high-potential carrier in the electrochemical engine (waste heat recovery only by the Otto cycle). The first question is whether the hybrid cycle could provide better performance than Otto cycle and what level of hybridization (i.e., fraction from the work supplied by the electrochemical engine) is needed for achieving maximum power. The answer is that the level of hybridization for maximum power depends on the T1/TL ratio (corresponds to compression ratio of the Otto cycle). Figure2 shows the power and efficiency of the hybrid cycle as a function of the hybridization level. Here, 0 means Otto cycle alone, and 1 means electrochemical engine alone. Four examples of different T1/TLratio values are presented. Maximum power and efficiency at maximum power are marked. It is clearly seen that in the range of 1 < T1/TL < 2, the maximum power is achieved by a hybrid cycle, and the necessary level of hybridization decreases with increase of the T1/TL ratio. Notably, for T1/TL > 2 the maximum power is achieved with Otto cycle alone. The result for efficiency at maximum power Energies 2020, 13, x FOR PEER REVIEW 7 of 11 (calculated according to Equation (14)) is also reflected in Figure2. As seen, for the degenerated case of a , ηm 0.5 regardless of the T1/TL ratio. However, the maximum power is proportional to the marks→ ∞for maximum→ power. As expected, the temperature TH decreases as TT1 / L ratio decreases, T /T ratio (see Equation (15)). It is important to underline, that the limiting T /T value (when the however,1 L the values of T remain high (above 2000 K) even for the lowest considered1 L TT/ ratio maximum power is achievedH with Otto cycle alone) is substantially lower for the case1 of fullL mass values. conversion in the electrochemical engine (T /TL 2) as compared to the hybrid mass conversion in 1 ≥ both the electrochemical and the Otto-cycle engines (T /TL 3). 1 ≥
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
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0.3 0.3
0.2 0.2
0.1 0.1
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction of Chemical Work from Total Work
Figure 2. Efficiency and normalized power of the cycle as a function of the electro-chemical work fraction. Figure 2. Efficiency and normalized power of the cycle as a function of the electro-chemical work fraction.
Figure 3. Temperature before expansion in the Otto cycle ( TH ) as a function of the chemical work
fraction for different TT1 / L ratios. is a mark for TH at maximum power for each TT1 / L ratio.
Energies 2020, 13, x FOR PEER REVIEW 7 of 11
marks for maximum power. As expected, the temperature TH decreases as TT1 / L ratio decreases,
however, the values of TH remain high (above 2000 K) even for the lowest considered TT1 / L ratio values.
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
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0.2 0.2 Energies 2020, 13, 3961 7 of 10 0.1 0.1
0 0 As noted earlier0 in 0.1 this paper, 0.2 the 0.3 main 0.4 reason 0.5 for the 0.6 low 0.7efficiency 0.8 of contemporary 0.9 1 ICEs compared to the Otto cycle efficiencyFraction is the of practical Chemical limitation Work from of Total the Work temperature before expansionT . H Hybrid operation with electrochemical work extraction can decrease this temperature. Figure3 shows this temperatureFigure 2. Efficiency for the hybridand normalized cycle as apower function of the of cycle the hybridization as a function of level the togetherelectro-chemical with the work marks for maximumfraction. power. As expected, the temperature TH decreases as T1/TL ratio decreases, however, the values of TH remain high (above 2000 K) even for the lowest considered T1/TL ratio values.
FigureFigure 3. 3.Temperature Temperature before before expansion expansion in in the the Otto Otto cycle cycle ( T(HTH) as) as a a function function of of the the chemical chemical work work fraction for different T /T ratios. is a mark for T at maximum power for each T /T ratio. fraction for different 1 TT1 /L L ratios. is a markH for TH at maximum power for 1eachL TT1 / L ratio.
An additional important question is whether the T1/TL ratio range discussed earlier is relevant for the compression ratios used in contemporary ICE designs, and whether the hybrid cycle has an advantage in terms of maximum power for these compression ratios. Figure4 shows the level of hybridization needed for maximum power as a function of the compression ratio for different adiabatic indexes γ. For γ = 1.4, the range of compression ratios is narrow, and modern ICEs use substantially larger compression ratio values. For smaller adiabatic index γ values, the range of compression ratios for optimal hybridization level is significantly larger. Notably, γ values of the expanding burned gases in ICE usually lie in the 1.2–1.3 range [25]. Next, the discussion deals with a partial conversion of the high-potential carrier in the electrochemical engine, i.e., the bottoming Otto-cycle engine recovers waste heat and converts an unused part of the high-potential carrier to mechanical work. Figure5 shows the e fficiency and the optimal conversion of a high-potential carrier in the chemical engine as a function of T1/TL calculated using Equations (19) and (20). As seen, the optimal carrier conversion from the high to low potential in a chemical engine for maximum power is a linear function of the T1/TL ratio. As mentioned earlier, for T /TL 3 the maximum power performance is achieved for Otto cycle 1 ≥ alone, and this result is different from that obtained in the case of the full high-potential carrier conversion in the electrochemical engine—T /TL 2. The reason for this difference lies in the nature 1 ≥ of energy dissipation and sources of irreversibility. The irreversibility that occurs in the electrochemical engine is proportional to the high-potential carrier conversion in it (Equation (5)). On the other hand, the irreversibility of the Otto cycle is a function of the T1/TL only. Additionally, the efficiency at maximum power for partial high-potential carrier conversion in the chemical engine is greater compared to the case of full high-potential carrier conversion, and equals the Otto-cycle efficiency for Energies 2020, 13, x FOR PEER REVIEW 8 of 11
An additional important question is whether the TT1 / L ratio range discussed earlier is relevant for the compression ratios used in contemporary ICE designs, and whether the hybrid cycle has an Energiesadvantage2020, 13in, 3961terms of maximum power for these compression ratios. Figure 4 shows the level8 of 10of hybridization needed for maximum power as a function of the compression ratio for different adiabatic indexes γ . For γ =1.4 , the range of compression ratios is narrow, and modern ICEs use T1/TL = 3. However, it is important to note that the latter conclusion is correct only for a specific case substantially larger compression ratio values. For smaller adiabatic index γ values, the range of of a (a = σL/σH) considered in this study. Figure6 presents the temperature before expansion compression→ ∞ ratios for optimal hybridization level is significantly larger. Notably, γ values of the TH and the efficiency at maximum power as a function of the optimal electrochemical high-potential carrierexpanding conversion burned fraction.gases in ICE In thatusually case lieT Hinis the higher 1.2–1.3 than range in Figure[25]. 3, making it more di fficult to implement in a practical engine.
Energies 2020, 13, x FOR PEER REVIEW 9 of 11 FigureFigure 4.4. Electro-chemical energy conversion fraction for maximal power as a functionfunction ofof thethe compressioncompression ratio.ratio.
1 1 Next, the discussion deals with a partial conversion of the high-potential carrier in the m, Nmax electrochemical engine, i.e., the bottoming Otto-cycle engine recovers waste Nheat and converts an max unused part of the high-potential carrier to mechanical work. Figure 5 shows the efficiency and the 0.8 0.8 optimal conversion of a high-potential carrier in the chemical engine as a function of TT1 / L calculated using Equations (19) and (20). As seen, the optimal carrier conversion from the high to low
potential 0.6in a chemical engine for maximum power is a linear function of the TT1 0.6/ L ratio. As ≥ mentioned earlier, for TT1 /3L the maximum power performance is achieved for Otto cycle alone, and this result is different from that obtained in the case of the full high-potential carrier conversion 0.4 ≥ 0.4 in the electrochemical engine—TT1 /2L . The reason for this difference lies in the nature of energy dissipation and sources of irreversibility. The irreversibility that occurs in the electrochemical engine is proportional0.2 to the high-potential carrier conversion in it (Equation (5)). On the other0.2 hand, the
irreversibility of the Otto cycle is a function of the TT1 / L only. Additionally, the efficiency at maximum power for partial high-potential carrier conversion in the chemical engine is greater compared to0 the case of full high-potential carrier conversion, and equals the Otto-cycle0 efficiency for = 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 TT1 /3L . However, it is important to note thatT theT latter conclusion is correct only for a specific case / L →∞ = σσ 1 of a (/)a LH considered in this study. Figure 6 presents the temperature before FigureFigure 5. Efficiency 5. Efficiency and and optimal optimal mass mass conversion conversion from from high high to tolow low chemical chemical potential potential in inthe the chemical chemical expansion TH and the efficiency at maximum power as a function of the optimal electrochemical engine as a function of T /T . engine as a function of TT1 /1 L L. high-potential carrier conversion fraction. In that case TH is higher than in Figure 3, making it more difficult to implement in a practical engine.
3500 1 T H
max 0.8 3000 ] K [ 0.6
2500
0.4 Temperature
2000 0.2
1500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N max
Figure 6. Temperature TH and efficiency at maximum power as a function of the optimal mass conversion fraction.
5. Conclusions A novel analytical method was developed for the first time for analysis of efficiency at maximum power of a hybrid cycle combining an electrochemical engine and the Otto cycle. The analysis is based on the low-dissipation model, which relates energy dissipation with energy transfer rate. The assumptions made in this study restrict the conclusions of this analysis to an extreme case that technological difficulties make hard to implement nowadays. However, from the perspective of future development, it is interesting to conclude that, in terms of a maximum power, at some conditions, an ICE can provide better performance than a hybrid cycle. On the other hand, efficiency gain can be achieved with the hybrid cycle—at the expense of power. Nonetheless, a more detailed
Energies 2020, 13, x FOR PEER REVIEW 9 of 11
1 1
m, Nmax N max 0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 T /T 1 L
Figure 5. Efficiency and optimal mass conversion from high to low chemical potential in the chemical
engine as a function of TT1 / L . Energies 2020, 13, 3961 9 of 10
3500 1 T H
max 0.8 3000 ] K [ 0.6
2500
0.4 Temperature
2000 0.2
1500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 N max
Figure 6. Temperature TH and efficiency at maximum power as a function of the optimal mass Figure 6. Temperature TH and efficiency at maximum power as a function of the optimal mass conversion fraction. conversion fraction. 5. Conclusions 5. Conclusions A novel analytical method was developed for the first time for analysis of efficiency at maximum powerA novel of aanalytical hybrid cyclemethod combining was developed an electrochemical for the first time engine for analysis and the of Ottoefficiency cycle. at Themaximum analysis poweris based of a hybrid on the cycle low-dissipation combining an model, electrochemical which relates engine energy and dissipationthe Otto cycle. with The energy analysis transfer is based rate. onThe the assumptionslow-dissipation made model, in this which study restrictrelates theenergy conclusions dissipation of this with analysis energy to transfer an extreme rate. case The that assumptionstechnological made diffi cultiesin this makestudy hard restrict to implement the conclu nowadays.sions of this However, analysis from to thean extreme perspective case of that future technologicaldevelopment, difficulties it is interesting make hard to conclude to implem that,ent in nowadays. terms of a maximumHowever, power,from the at perspective some conditions, of futurean ICE development, can provide it better is interesting performance to conclude than a hybrid that, cycle.in terms On of the a othermaximum hand, epower,fficiency at gainsome can conditions,be achieved an ICE with can the provide hybrid better cycle—at performance the expense than of a power. hybridNonetheless, cycle. On the aother more hand, detailed efficiency analysis gainthat can considers be achieved finite-time with the Otto hybrid cycle cycle—at duration th ande expense heat losses of power. can provide Nonetheless, much a more more information. detailed A numerical investigation would be necessary in the latter case rather than an analytical solution.
Author Contributions: Conceptualization, D.D. and L.T.; methodology, D.D.; software, D.D.; formal analysis, D.D.; investigation, D.D.; writing—original draft preparation, D.D.; writing—review and editing, L.T.; visualization, D.D.; supervision, L.T.; project administration, L.T. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Grand Technion Energy Program. Conflicts of Interest: The authors declare no conflict of interest.
References
1. Novikov, I.I. Efficiency of an atomic power generating installation. Sov. J. At. Energy 1957, 3, 1269–1272. [CrossRef] 2. Curzon, F.L.; Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 1975, 43, 22–24. [CrossRef] 3. Hoffmann, K.H.; Burzler, J.M.; Schubert, S. Endoreversible thermodynamics. CiteSeer 1997, 22, 311. 4. Esposito, M.; Kawai, R.; Lindenberg, K.; Van den Broeck, C. Efficiency at maximum power of low-dissipation Carnot engines. Phys. Rev. Lett. 2010, 105, 150603. [CrossRef] 5. Guo, J.; Wang, J.; Wang, Y.; Chen, J. Universal efficiency bounds of weak-dissipative thermodynamic cycles at the maximum power output. Phys. Rev. E 2013, 87, 012133. [CrossRef][PubMed] Energies 2020, 13, 3961 10 of 10
6. Petrescu, S.; Maris, V.; Costea, M.; Boriaru, N.; Stanciu, C.; Dura, I. Comparison between Fuel Cells and Heat Engines. I. A Similar Approach in the Framework of Thermodynamics with Finite Speed. Rev. Chim. 2013, 64, 739–746. 7. Pavelka, M.; Klika, V.; Vágner, P.; Maršík, F. Generalization of exergy analysis. Appl. Energy 2015, 137, 158–172. [CrossRef] 8. Vágner, P.; Pavelka, M.; Maršík, F. Pitfalls of exergy analysis. J. Non-Equilib. Thermodyn. 2017, 42, 201–216. [CrossRef] 9. Zabihian, F.; Fung, A. A review on modeling of hysbrid solid oxide fuel cell systems. Int. J. Eng. 2009, 3, 85–119. 10. Winkler, W.; Nehter, P.; Williams, M.C.; Tucker, D.; Gemmen, R. General fuel cell hybrid synergies and hybrid system testing status. J. Power Sources 2006, 159, 656–666. [CrossRef] 11. Dunbar, W.R.; Lior, N.; Gaggioli, R.A. Exergetic advantages of topping rankine power cycles with fuel cell units. Am. Soc. Mech. Eng. Adv. Energy Syst. Div. (AES) 1990, 21, 63–68. 12. Harvey, S.P.; Richter, H.J. Improved gas turbine power plant efficiency by use of recycled exhaust gases and fuel cell technology. Proc. ASME Winter Annu. Meet. 1993, 30, 199–207. 13. Gorla, R.S. Probabilistic analysis of a solid-oxide fuel-cell based hybrid gas-turbine system. Appl. Energy 2004, 78, 63–74. [CrossRef] 14. Kim, J.; Kim, Y.; Choi, W.; Ahn, K.Y.; Song, H.H. Analysis on the operating performance of 5-kW class solid oxide fuel cell-internal combustion engine hybrid system using spark-assisted ignition. Appl. Energy 2020, 260, 114231. [CrossRef] 15. Park, S.H.; Lee, Y.D.; Ahn, K.Y. Performance analysis of an SOFC/HCCI engine hybrid system: System simulation and thermo-economic comparison. Int. J. Hydrog. Energy 2014, 39, 1799–1810. [CrossRef] 16. Lee, Y.D.; Ahn, K.Y.; Morosuk, T.; Tsatsaronis, G. Exergetic and exergoeconomic evaluation of an SOFC-Engine hybrid power generation system. Energy 2018, 145, 810–822. [CrossRef] 17. Pettersson, L.J.; Westerholm, R. State of the art of multi-fuel reformers for fuel cell vehicles: Problem identification and research needs. Int. J. Hydrog. Energy 2001, 26, 243–264. [CrossRef] 18. Tartakovsky, L.; Sheintuch, M. Fuel reforming in internal combustion engines. Prog. Energy Combust. Sci. 2018, 67, 88–114. [CrossRef] 19. Eyal, A.; Tartakovsky, L. Second-law analysis of the reforming-controlled compression ignition. Appl. Energy 2020, 263, 114622. [CrossRef] 20. Poran, A.; Thawko, A.; Eyal, A.; Tartakovsky,L. Direct injection internal combustion engine with high-pressure thermochemical recuperation–Experimental study of the first prototype. Int. J. Hydrog. Energy 2018, 43, 11969–11980. [CrossRef] 21. Chuahy, F.D.; Kokjohn, S.L. Solid oxide fuel cell and advanced combustion engine combined cycle: A pathway to 70% electrical efficiency. Appl. Energy 2019, 235, 391–408. [CrossRef] 22. Andresen, B. Comment on A fallacious argument in the finite time thermodynamic concept of endoreversibility [J. Appl. Phys. 83, 4561 (1998)]. J. Appl. Phys. 2001, 90, 6557–6559. [CrossRef] 23. Chen, J.; Yan, Z.; Lin, G.; Andresen, B. On the Curzon–Ahlborn efficiency and its connection with the efficiencies of real heat engines. Energy Convers. Manag. 2001, 42, 173–181. [CrossRef] 24. Ocampo-García, A.; Barranco-Jiménez, M.A.; Angulo-Brown, F. Thermodynamic and thermoeconomic optimization of coupled thermal and chemical engines by means of an equivalent array of uncoupled endoreversible engines. Eur. Phys. J. Plus 2018, 133, 342. [CrossRef] 25. Heywood, J.B. Combustion Engine Fundamentals; McGraw-Hill: New York, NY, USA, 1988.
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