Efficiency at Maximum Power of the Low-Dissipation Hybrid Electrochemical–Otto Cycle

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.

1. Introduction

According to equilibrium thermodynamics, the maximal efficiency of a cycle operating between two reservoirs at thermodynamic potential $P_H$ and $P_L$ $(P_H>P_L)$ is the Carnot efficiency, ${\eta }_C=1-P_L/P_H$. 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 in Figure 1. Electrochemical engines use the high chemical potential $({\mu }_H)$ 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 ${\mu }_H$ 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 [18,19,20]. 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.

2. Low-Dissipation Model of the Hybrid Cycle

In this analysis, we assume that the only irreversible processes take place in the electrochemical engine, and the bottoming Otto cycle is reversible. This assumption means that the mechanical processes are reversible, i.e., the compression and expansion are isentropic. Moreover, the processes in the Otto cycle are assumed to be relatively fast compared to the chemical energy transfer, thus the high and low chemical potential reservoirs could be considered as the isothermal boundaries of the chemical engine. The low-dissipation model assumes that the irreversibility in the process of energy conversion between the reservoirs and the hybrid engine is proportional to $1/t$, where $t$ is the time of the energy conversion process. The meaning of this assumption is that the relaxation time in the hybrid engine is relatively short compared to the energy transmission time. According to this model, the cycle will become reversible for $t\to \infty$, as expected. The low-dissipation model expresses energy absorption from the high chemical potential reservoir in an electrochemical engine as [5]

$$E_H={\mu }_H\Delta N\left(1-\frac{{\sigma }_H}{t_H}\right)$$

(1)

and the energy rejected to the low chemical potential reservoir as

$$E_L={\mu }_L\Delta N\left(1+\frac{{\sigma }_L}{t_L}\right)$$

(2)

Here, ${\mu }_H,{\mu }_L$ are the chemical potentials, $\Delta N$ is the total mass transmitted between the chemical reservoirs, ${\sigma }_H,{\sigma }_L$ are the irreversibility factors that include the information on the irreversibility of the energy transmission between the high and the low chemical potential reservoirs, respectively. The expressions include the term of reversible energy conversion (${\mu }_H\Delta N$, ${\mu }_L\Delta N$) and the dissipation term caused by the irreversible energy transfer processes. Obviously, according to the first law, the work that extracted from the chemical engine is the difference between the above absorbed and rejected energy,

$$W_C=E_H-E_L={\mu }_H\Delta N\left(1-\frac{{\sigma }_H}{t_H}\right)-{\mu }_L\Delta N\left(1+\frac{{\sigma }_L}{t_L}\right)$$

(3)

while the reversible work is

$$W_{C,rev}={\mu }_H\Delta N-{\mu }_L\Delta N$$

(4)

We conclude that the difference between the reversible work and the irreversible work is the waste heat.

$$W_{C,rev}-W_C=Q_C={\mu }_H\Delta N\frac{{\sigma }_H}{t_H}+{\mu }_L\Delta N\frac{{\sigma }_L}{t_L}$$

(5)

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

$${\displaystyle {\int }_{T_1}^{T_H}CdT}={\mu }_H\Delta N\frac{{\sigma }_H}{t_H}+{\mu }_L\Delta N\frac{{\sigma }_L}{t_L}$$

(6)

where $C$ is the heat capacity, $T_1$ is the temperature after compression, $T_H$ is the temperature before the expansion stroke, and by assuming constant heat capacity, $T_H$ can be expressed as

$$T_H=T_1+\frac{1}{C}\left({\mu }_H\Delta N\frac{{\sigma }_H}{t_H}+{\mu }_L\Delta N\frac{{\sigma }_L}{t_L}\right)$$

(7)

Following the assumption of isentropic compression, $T_1$ dependence on the compression ratio $r$ can be expressed as

$$T_1/T_L=r^{\gamma -1}$$

(8)

Here, $T_L$ is the temperature of the low potential reservoir, $\gamma$ is the heat capacities ratio. The work produced by the reversible Otto engine is [5]

$$W_{Otto}=C(T_H-T_1)-C{T}_L\left(\frac{T_H}{T_1}-1\right)$$

(9)

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.

$$W_{Otto}={\mu }_H\Delta N\left(\frac{{\sigma }_H}{t_H}-\frac{{\sigma }_H}{t_H}\frac{T_L}{T_1}\right)+{\mu }_L\Delta N\left(\frac{{\sigma }_L}{t_L}-\frac{{\sigma }_L}{t_L}\frac{T_L}{T_1}\right)$$

(10)

$$W_{tot}=W_{Otto}+W_C={\mu }_H\Delta N\left(1-\frac{{\sigma }_H}{t_H}\frac{T_L}{T_1}\right)-{\mu }_L\Delta N\left(1+\frac{{\sigma }_L}{t_L}\frac{T_L}{T_1}\right)$$

(11)

$$P_{tot}=W_{tot}/(t_H+t_L)=\frac{{\mu }_H\Delta N}{t_H+t_L}\left(1-\frac{{\sigma }_H}{t_H}\frac{T_L}{T_1}\right)-\frac{{\mu }_L\Delta N}{t_H+t_L}\left(1+\frac{{\sigma }_L}{t_L}\frac{T_L}{T_1}\right)$$

(12)

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 $T_L/T_1$ ratio (which depends on the compression ratio of the Otto cycle) that multiply the irreversibility terms ${\sigma }_{H,L}/t_{H,L}$. This difference is reflected also in the efficiency term,

$$\eta =\frac{W_{tot}}{{\mu }_H\Delta N}=\left(1-\frac{{\sigma }_H}{t_H}\frac{T_L}{T_1}\right)-\frac{{\mu }_L}{{\mu }_H}\left(1-\frac{{\sigma }_L}{t_L}\frac{T_L}{T_1}\right)$$

(13)

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.

$${\eta }_m=\frac{{\eta }_{C\mu }}{2-{\eta }_{C\mu }/\left[1+\sqrt{a(1-{\eta }_{C\mu })}\right]}$$

(14)

Here, ${\eta }_{C\mu }$ is the Carnot efficiency of the chemical engine, $a={\sigma }_L/{\sigma }_H$. However, the maximum power of the hybrid engine is different from that of the chemical engine, and it is a function of the $T_L/T_1$ ratio

$$P_{\max }=\frac{{\mu }_H\Delta N}{4{\sigma }_H}{\left[\frac{{\eta }_{C\mu }}{1+\sqrt{(1-{\eta }_{C\mu })a}}\right]}^2\frac{T_1}{T_L}=P_{\max ,C}\frac{T_1}{T_L}$$

(15)

Here, $P_{\max ,C}$ is the maximum power of a chemical engine obtained by Guo et al. [5]. The simple multiplication of the chemical engine power and the $T_L/T_1$ 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 $T_H$ is calculated by the same steps as in Equation (7) (Section 2), with the addition of contribution of the remaining high-potential carrier combustion.

$$T_H=T_1+\frac{1}{C}{\mu }_H\Delta N\frac{{\sigma }_H}{t_H}+\frac{1}{C}{\mu }_L\Delta N\frac{{\sigma }_L}{t_L}+\frac{1}{C}{N}_L({\mu }_H-{\mu }_L)$$

(16)

Here, $N_L$ 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

$$T_H=T_1+\frac{1}{C}{\mu }_H\Delta N\frac{{\sigma }_H}{t_H}+\frac{1}{C}{N}_L{\mu }_H$$

(17)

Work and power then are extracted in similar steps expressed by Equations (9)–(12), and the efficiency for maximum power as a function of $\Delta N$ is obtained.

$${\eta }_m=1-\frac{1}{2}\Delta N(\Delta N+N_L(1-\frac{T_L}{T_1}))-N_L\frac{T_L}{T_1}$$

(18)

Recognizing that for normalized mass $N_L=1-\Delta N$ and $d{N}_L=-d\Delta N$, we can obtain the optimal conversion in the electrochemical engine for the maximal efficiency.

$$\begin{array}{l}{N}_{L,\max }=\frac{1}{2}\left(\frac{T_1}{T_L}-1\right)\\ \Delta N_{\max }=\frac{3}{2}-\frac{1}{2}\frac{T_1}{T_L}\end{array}\}1\le \frac{T_1}{T_L}\le 3$$

(19)

In the case of a high compression ratio that corresponds to $T_1/T_L>3$, the maximum power is obtained for a complete conversion in the ICE, i.e., no electrochemical reaction—$\Delta N=0$. For the typical chemical potential carriers, as hydrogen and air, and adiabatic index $\gamma =1.4$, 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 $\gamma$ 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 $T_1/T_L$ ratio in the range of the hybrid operation is

$${\eta }_{m,\max N_L}=\frac{1}{4}+\frac{1}{8}\frac{T_1}{T_L}+\frac{1}{8}\frac{T_L}{T_1},1\le \frac{T_1}{T_L}\le 3$$

(20)

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 $T_1/T_L$ ratio (corresponds to compression ratio of the Otto cycle). Figure 2 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 $T_1/T_L$ratio values are presented. Maximum power and efficiency at maximum power are marked. It is clearly seen that in the range of $1<T_1/T_L<2$, the maximum power is achieved by a hybrid cycle, and the necessary level of hybridization decreases with increase of the $T_1/T_L$ ratio. Notably, for $T_1/T_L>2$ the maximum power is achieved with Otto cycle alone. The result for efficiency at maximum power (calculated according to Equation (14)) is also reflected in Figure 2. As seen, for the degenerated case of $a\to \infty$, ${\eta }_m\to 0.5$ regardless of the $T_1/T_L$ ratio. However, the maximum power is proportional to the $T_1/T_L$ ratio (see Equation (15)). It is important to underline, that the limiting $T_1/T_L$ value (when the maximum power is achieved with Otto cycle alone) is substantially lower for the case of full mass conversion in the electrochemical engine $(T_1/T_L\ge 2)$ as compared to the hybrid mass conversion in both the electrochemical and the Otto-cycle engines $(T_1/T_L\ge 3)$.

As noted earlier in this paper, the main reason for the low efficiency of contemporary ICEs compared to the Otto cycle efficiency is the practical limitation of the temperature before expansion$T_H$. Hybrid operation with electrochemical work extraction can decrease this temperature. Figure 3 shows this temperature for the hybrid cycle as a function of the hybridization level together with the marks for maximum power. As expected, the temperature $T_H$ decreases as $T_1/T_L$ ratio decreases, however, the values of $T_H$ remain high (above 2000 K) even for the lowest considered $T_1/T_L$ ratio values.

An additional important question is whether the $T_1/T_L$ 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. Figure 4 shows the level of hybridization needed for maximum power as a function of the compression ratio for different adiabatic indexes $\gamma$. For $\gamma =1.4$, the range of compression ratios is narrow, and modern ICEs use substantially larger compression ratio values. For smaller adiabatic index $\gamma$ values, the range of compression ratios for optimal hybridization level is significantly larger. Notably, $\gamma$ 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. Figure 5 shows the efficiency and the optimal conversion of a high-potential carrier in the chemical engine as a function of $T_1/T_L$ 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 $T_1/T_L$ ratio. As mentioned earlier, for $T_1/T_L\ge 3$ 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 in the electrochemical engine—$T_1/T_L\ge 2$. 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 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 $T_1/T_L$ 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 $T_1/T_L=3$. However, it is important to note that the latter conclusion is correct only for a specific case of $a\to \infty$ $(a={\sigma }_L/{\sigma }_H)$ considered in this study. Figure 6 presents the temperature before expansion $T_H$ and the efficiency at maximum power as a function of the optimal electrochemical high-potential carrier conversion fraction. In that case $T_H$ is higher than in Figure 3, making it more difficult to implement in a practical engine.

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 analysis that considers finite-time Otto cycle duration and heat losses can provide much more information. A numerical investigation would be necessary in the latter case rather than an analytical solution.