## Abstract

In this work, a thermodynamic model based on an endoreversible engine approach is developed to analyze the performance of heat engines operating under different thermodynamic cycles. The model considers finite heat transfer rate, variable heat source and sink temperatures, and irreversibilities associated with the expansion and compression. Expressions for the maximum power and efficiency at maximum power output are obtained as a function of hot and cold reservoir temperatures, the equivalent isentropic efficiency of compression and expansion components, and the effective conductance ratio between heat exchangers. In all cases, the Curzon–Ahlborn efficiency is retrieved at constant reservoir temperatures and neglected compression–expansion irreversibilities. The proposed model allows assessing the effect of isentropic efficiencies and heat exchanger design and operation characteristics for different thermodynamic cycles.

## 1 Introduction

The development of efficient heat engines is a fundamental alternative to counteract the accelerated depletion of non-renewable energy sources during the last decades. The design and optimization of efficient heat engines require thermodynamic models that describe the engine performance, and how this performance is being impacted by the engine characteristics and operating factors. The maximum possible efficiency attained by heat engines operating between two energy reservoirs is set by the Carnot efficiency (ηC). It corresponds to the efficiency of heat engines operating in the reversible limit under quasi-static processes that results in zero power output [1]. In an effort to account for some of the main irreversibilities, models have been developed considering internally reversible engines where all irreversibilities occur during the energy exchange processes. From this approach, which considers the irreversibility of finite rate heat transfer, the efficiency at maximum power output (ηCA), also known as Curzon—Ahlborn efficiency, has been obtained [13]. This result was first presented by Chambadal [2] and Novikov [3]. Similar to ηC, ηCA is independent of the system characteristics, working fluid properties, or operation regimes, as it only depends on hot and cold reservoir temperatures. It was found that in the limit of low irreversibility, ηCA is bounded by ηc/2 and ηc/(2 − ηc) [4], i.e., ηc/2 ≤ ηCAηc/(2 − ηc), and the bounds correspond to the extremes of the ratio of dissipation in the heat exchange processes. These studies have found that ηCA exhibits some degree of generality as it describes relatively well the efficiency of thermal cycles including Otto, Brayton, Stirling, and Ericsson cycles [5,6] and several real thermal plants [79].

In addition to the irreversibility of finite heat transfer rate, real thermal engines also present irreversibilities associated with the working fluid friction against turbine blades or friction between the piston and cylinder, pressure drop at heat exchangers, and heat leaks that decrease the efficiency and power output. Several models of efficiency at maximum power output that consider different sources of irreversibility such as finite compression ratio, irreversibilities inside the working fluid, heat leak loss, and fluid flow irreversibilities have been proposed [1020]. Bejan [16] presented an analogy between the power conversion in a heat engine (thermal power conversion) and the extraction of power in a fluid flow driven by a pressure difference. It was shown in that work that, when the relationships between pressure difference and flow rates are linear, the energy conversion efficiency at maximum power is analogous to the Carnot efficiency. For nonlinear relationships between flow rate and pressure drops, the paper presents the conditions for operation at maximum power. In particular, for a turbine, the study looks at the effects of the pressure drops that occur in the ducting of the stream to the first expansion stage, the pressure drop across the turbine itself (characterized by an isentropic efficiency), and the discharge pressure drop. Maximum power can be obtained by selecting optimally the flow rate or the pressure drops associated with ducting the flow into and out of the turbine. Similar analyses are presented for compressors and pumps. Grazzini [17] presented the maximum available work and the maximum efficiency for an engine with heat transfer and internal irreversibilities and no heat leak. The engine cycle consisted of constant thermal-capacity heating and cooling processes together with polytropic and irreversible compression and expansion branches. The heat source and heat sink were modeled as temperature changing fluid streams, whereas the internal irreversibility was given by an entropy-production amount dependent on the extent of the polytropic processes. In a different work, Ibrahim et al. [18] conducted a power output optimization of Carnot and closed Brayton cycles using Lagrange multipliers. In that work, the efficiency at maximum power output for both cycles was analyzed and compared under the same specific boundary conditions. Based on this analysis, the closed Brayton cycles can produce more power than the Carnot cycle. Also, in that work, a generic expression for the efficiency at maximum power output for both cycles, as a function of an entropy change parameter during heat addition and rejection, was developed. A finite-time optimization for an ideal Rankine cycle was proposed by Lee and Kim [19]. In that work, a general expression for the efficiency at maximum power output as a function of the reservoir, sink, and pinch-temperature difference of the heating and cooling fluids was obtained. In another study, Lee and Kim [20] considered a model of a heat engine consisting of two isothermal processes and two adiabatic processes. The adiabatic processes (expansion and compression) were treated as irreversible and the isothermal processes of heat addition and heat rejection were driven by fluids of finite capacity rates. By modeling the heat exchangers, their model captured the temperature variation of the hot and cold streams that drive the cycle. They maximized the power output, and corresponding efficiency, with respect to temperatures of the isothermal processes, and showed the dependence of power output on heat conductance, temperature levels, and internal irreversibility in the expansion/compression. They also showed that it is possible to maximize two times the power with respect to temperature levels and conductance allocation. In most of these studies, expressions for the efficiency at maximum power output smaller than ηCA have been obtained including parameters accounting for the additional source of irreversibility. As expected, ηCA is recovered from the efficiency expressions when the additional irreversibilities (other than the finite heat transfer rate) are neglected.

Besides the efficiency at maximum power output, several indicators such as maximum efficient power [8,21], maximum power density [22], ecological criterium [23], as well as methods including thermo-economic analysis [24], exergoeconomic analysis and optimization [25], entropy generation minimization [26], and ecological and exergetic performance optimization [27] have been developed to assess the performance of thermal engines and analyze the effect of system configuration, new designs, and operating parameters. For instance, the ecological criterium proposed by Angulo-Brown [23] allows the quantification of heat engines performance based on the trade-off between the power produced and the power loss in the heat engine as a result of the entropy generation. Different studies have performed analysis and optimization based on this criterion for diverse thermal cycles including Brayton, Stirling, and Ericsson [2831]. The development of the generalized model capable of describing the behavior of any thermal cycle has been constantly pursued. Chen et al. [32] developed a model based on finite-time thermodynamics and ecological optimization to produce general expressions for the power, efficiency, and entropy generation rate of Diesel, Otto, Brayton, Atkinson, among other characteristic thermodynamic cycles. An exergoeconomic analysis and optimization were performed for a heat engine cycle consisting of two constant thermal-capacity heating, two constant thermal-capacity cooling, and two adiabatic branches [25]. The heat engine model in that work included heat transfer irreversibilities, as well as heat leakage, and internal irreversibilities due to compression and expansion. Importantly, in that study, the heat source and heat sink temperatures were considered constant, while the internal irreversibilities were treated through a coefficient greater than one relating the heat released by the engine under irreversible conditions to the heat released in the reversible case.

Besides the traditional derivation of the Curzon–Ahlborn efficiency, some alternative approaches have been used in the last years to get the same result based on different models [33], which contribute to highlight the features and limitations of the result. For example, Van den Broeck [34] obtained the result based on linear irreversible thermodynamics theory that does not use the explicit assumption of heat transfer processes or a reversible compartment. Also, another approach was used by Esposito et al [35], which is commonly referred to as low dissipation model.

In this work, a comprehensive model based on the endoreversible engine approach was developed to analyze the performance of several types of heat engines, including internal combustion engines, steam cycles, gas turbines, and organic Rankine cycles. The model considers finite heat transfer rate, variable heat source and sink temperature, and irreversibilities associated with the fluid friction losses, which are lumped into isentropic efficiencies of compression and expansion components. The model presented in this paper shares commonalities with models available in the literature addressing maximization of efficiency at maximum power output. Also, it presents similarities regarding the exploration of thermal conductance allocation, temperatures distribution, and the investigation of several sources of irreversibilities, as well as the limit for reversible operation. Despite these similarities, the new approach uses a general dimensionless model in which an effort is made to present the optimization results in terms of variables that are commonly available for actual engines. This includes the equivalent efficiency that accounts for irreversibilities for the compression and expansion components through the commonly available isentropic efficiencies, while most of the models use an irreversibility parameter that, although containing the same information, is rarely used in practical situations. In addition, we present the results in terms of an effective conductance ratio between heat exchangers (rM). This parameter compares the conductance of the heat exchangers and the characteristics of the fluids involved including specific heat capacity and mass flow rate.

## 2 Model

A schematic diagram of a heat engine operating between hot and cold thermal reservoirs is presented in Fig. 1. In addition to finite heat transfer rate between the engine and the two reservoirs, variable reservoir temperatures and isentropic efficiencies in the compression and expansion components are considered. The inclusion of variable reservoir temperatures adds a level of reality to the model because most heat engines are driven by streams at hot and cold heat exchangers whose temperature varies. In addition, this consideration allows the incorporation of the effects of hot and cold mass flow rates and thermophysical properties of the external streams. From the diagram, heat input at a rate QH is transferred to the high-pressure working fluid in a heat exchanger (heat input heat exchanger) increasing its temperature. The fluid is then expanded in the expansion component (turbine or piston–cylinder) to produce power (Wtr). A fraction of the energy gets dissipated as heat in this component due to irreversibilities (friction between piston and cylinder or turbine blades and the working fluid), which are accounted through the isentropic efficiency (ηt). This loss is transferred to the cold thermal reservoir at a rate of QLt, see Fig. 1(a). After expansion, the fluid dissipates heat to the cold reservoir QLc through a heat exchanger (heat dissipation heat exchanger) and then passes through the compression component (compressor, pump, or piston–cylinder) where its pressure increases. The isentropic compression efficiency considers the irreversibilities in this component. After compression, additional increase in the fluid temperature and entropy occurs with respect to the ideal process. This additional heat (QLP) remains in the fluid and is represented in Fig. 1(a) by the arrow pointing the engine’s heat input QH. Finally, the high-pressure fluid enters the heat input heat exchanger, and the cycle is repeated. To simplify the model and analysis, irreversibilities in the compression device are merged with those for the expansion component, see Fig. 1(b). This diagram presents the reversible net power output (Wi) of the heat engine and the real net power output Wr that considers energy losses in the compression and expansion processes. In this case, the isentropic efficiencies are lumped together into the isentropic equivalent efficiency (ηeq) defined in Sec. 2.1.

Fig. 1
Fig. 1
Close modal

### 2.1 Isentropic Efficiencies.

Based on the diagram presented in Fig. 1(a), the real and reversible net power output (Wr and Wi) for the heat engine can be, respectively, expressed as follows:
$Wr=Wrt−Wrp=QH−QLc−QLt$
(1)
$Wi=Wit−Wip=QH−QLc$
(2)
The isentropic compression and expansion efficiencies are defined by
$ηp=WipWrp$
(3)
$ηt=WrtWit$
(4)
Similarly, the real and ideal net power outputs for the engine can be related with an isentropic equivalent efficiency ηeq that accounts for the losses in the compression and expansion components:
$ηeq=WrWi$
(5)
Replacing Eqs. (2)(5) into Eq. (1), the real power output yields:
$Wr=ηeq(Wrtηt−ηpWrp)$
(6)
Defining the power ratio Wrp/Wrt as a parameter that accounts for the fraction of the produced power that is used to run the compression component and using Eqs. (1) and (6), the following expression for the ηeq can be obtained:
$ηeq=ηt(1−Wrp/Wrt)1−ηtηpWrp/Wrt$
(7)

The behavior of ηeq as a function of Wrp/Wrt is presented in Fig. 2, for different values of ηp and ηt. From Eq. (7) and Fig. 2, it is clear that if Wrp = 0 or WrpWrt, then Wrp/Wrt ≅ 0 and ηeqηt. This situation represents a heat engine where the power in the compression device is negligible with respect to the power produced in the turbine. This could describe the operation of some cycles using a pump to increase the pressure in the fluid; consequently, in this type of cycles, the isentropic efficiency for the turbine has a higher impact on ηeq; if Wrp = Wrt, then Wrp/Wrt = 1 and ηeq = 0. For this case, the compression device consumes all the power produced in the turbine and the cycle is “zero” efficient. In real engines 0 < Wrp/Wrt < 1, and in particular for a Rankine cycle, Wrp could represent around 5% of Wrt. In Brayton cycles, for instance, Wrp could represent more than 40% of Wrt. From Fig. 2, for lower values of Wrp/Wrt, ηt exerts a strong impact in ηeq and defines the initial value for the curves. As expected, the effect of ηp is small for low Wrp/Wrt and its impact on ηeq becomes stronger as Wrp/Wrt increases. For a cycle characterized by a value of Wrp/Wrt ≈ 0.05, ηeq is practically independent on ηp and proportional to ηt. In turn, for a cycle having a Wrp/Wrt ≈ 0.4, ηeq could be as low as 0.65 for ηp = ηt = 0.8. It is worth noting that ηeq incorporates a combined effect of ηt and ηp. For instance, ηeq resulting from equal values for ηt and ηp (different than 1.0) is lower than such common value. In addition, as the role of the compression is magnified by increasing Wrp/Wrt, this combined effect increases, leading to a reduced ηeq. From Fig. 2, the range for ηeq goes from around 0.6 to 1.0, which covers the most common types of heat engines like Brayton (ηeq = 0.65–0.8), Rankine and Organic Rankine cycles (ηeq = 0.8–0.9), and internal combustion engines ηeq = 0.6–0.8. Then, the results are presented in terms of three values of ηeq (1.0, 0.8, and 0.6) that are representative of this range.

Fig. 2
Fig. 2
Close modal

### 2.2 Engine Model.

The heat transfer rate between the hot/cold reservoirs and the reversible compartment can be, respectively, expressed in terms of the logarithmic mean temperature difference as follows:
$QH=(UA)HTHi(1−τH)−(DH−τH)ln((1−τH)/(DH−τH))=(m˙cp)HTHi(1−DH)$
(8)
$QL=(UA)LTLi(1/τL−1)−(1/τL−1/DL)ln((1/τL−1)/(1/τL−1/DL))=(m˙cp)LTLi(1/DL−1$
(9)
where τH = THc/THi, τL = TLi/TLc, DH = THo/THi, and DL = TLi/TLo.
From the right side of Eqs. (8) and (9),
$ln(1−τHDH−τH)=(UA)H(m˙cp)H⇒1−τHDH−τH=eNTUH$
(10)
$ln(1/τL−11/τL−1/DL)=(UA)L(m˙cp)L⇒1/τL−11/τL−1/DL=eNTUL$
(11)
with $NTUH=(UA)H/(m˙cp)H$ and $NTUL=(UA)L/(m˙cp)L$ being, respectively, the number of transfer units for the heat input and heat dissipation heat exchangers.
Replacing Eqs. (10) and (11) into the left side of Eqs. (8) and (9), respectively, the heat input and total heat rejection rates (including the engine heat rejection and the heat loss) yield:
$QH=(UA)HTHi(1−e−NTUH)NTUH(1−τH)$
(12)
$QL=QLc+QLt=(UA)LTLi(1−e−NTUL)NTUL(1/τL−1)$
(13)
The following expression for the heat loss (QLt) is obtained from Eqs. (1), (2), and (5):
$QLt=(1−ηeq)(QH−QLc)$
(14)
In turn, the engine heat rejection rate (QLc) can be found from Eqs. (12)(14):
$QLc=(UA)LTLi1ηeq(1−e−NTUL)NTUL(1/τL−1)−(UA)HTHi(1−e−NTUH)NTUH(1−τH)(1−ηeq)ηeq$
(15)
The operation of the heat engine is constrained to the reversible operation of the Carnot engine, for which the entropy balance yields:
$QHTHc=QLcTLc$
(16)
Replacing Eqs. (8) and (15) into Eq. (16) and using the expressions of Eqs. (10) and (11) result in the following equation that relates τH, τL, and τ:
$(1τH−1)=rMηeq(1−1τL)−1τ1τL(1−τH)(1−ηeq)ηeq$
(17)

In Eq. (17), τ corresponds to the cold and hot input reservoir temperature ratio (τ = TLi/THi), $M=(1−e−NTUL)NTUH/(1−e−NTUH)$$NTUL$, and r = (UA)L/(UA)H are parameters that, respectively, represent design and operating characteristics (M) and the ratio between conductances (r) of heat input and heat dissipation heat exchangers. Both parameters can be lumped together into a single parameter $rM=[(m˙cp)L(1−e−NTUL)]/[(m˙cp)H(1−e−NTUH)]$, which can be seen as an effective conductance ratio between heat exchangers that compares the conductance and fluid characteristics including specific heat capacity and mass flow rate.

Solving for τL, Eq. (17) yields:
$τL=rM−(ηeq/τH)(1−τH)rM+((1−ηeq)/τ)(1−τH)$
(18)
From Eqs. (2) and (5),
$Wr=ηeq(QH−QLc)$
(19)
Replacing the values of QH, QLC (Eqs. (12) and (15)), and τL (Eq. (18)) into Eq. (19), the power output for the real heat engine becomes
$Wr=(UA)HTHi(1−e−NTUH)NTUHηeq(1−τH)[rMτH+τH−rMτ−1]τH(rM+ηeq)−ηeq$
(20)
Defining the non-dimensionalization factor Qmax as the maximum possible heat transfer rate between the hot and cold reservoirs (when no heat engine is present):
$Qmax=14(UA)sTHi(1−τ)εsNTUs$
(21)
where ɛs(UA)s/NTUs = ɛH(UA)H/NTUH + (UA)LɛL/NTUL = (1 + rM)ɛH(UA)H/NTUH accounts for the total effective conductance in both heat exchangers, being $εL=1−e−NTUL$ and $εH=1−e−NTUH$.
Normalizing the power output (Eq. (20)) and the heat input (Eq. (12)) using the definition of Qmax, expressions for dimensionless power output $(W*)$ and dimensionless heat input $(QH*)$ are obtained:
$W*=4ηeq(1−τH)[rMτH+τH−rMτ−1](1−τ)(1+rM)[τH(rM+ηeq)−ηeq]$
(22)
$QH*=4(1−τH)(1−τ)(1+rM)$
(23)
Finally, from Eqs. (22) and (23), the heat engine efficiency can be found as
$η=W*QH*=ηeq[rMτH+τH−rMτ−1]τH(rM+ηeq)−ηeq$
(24)

### 2.3 Efficiency at Maximum Power Output.

The maximum power output $(Wmax*)$ can be found by maximizing $W*$ with respect to τH, i.e., $dW*/dτH=0$:
$Wmax*=4rMηeq(rM+ηeq)2(ηeq(τ−1)+(rMτ+1)−rM+1)2(rM+1)(1−τ)$
(25)
with the corresponding τH,opt determined as
$τH,opt=ηeq+(rM/(rM+1))(rM+1)[ηeq(τ−1)+(rMτ+1)]rM+ηeq$
(26)
Defining τc as the cold to hot thermal engine temperature ratio (τc = TLc/THc) and using the definition of τ, τL, and τH, τc = τ/(τLτH). After replacing τH,opt in this last equation, the corresponding τc,opt is determined as
$τc,opt=(rM+1)[ηeq(τ−1)+(rMτ+1)]−(1−ηeq)(rM+ηeq)$
(27)
Finally, from the expression for $Wmax*$ (Eq. (25)), the efficiency at maximum power output $(η@Wmax)$ can be found as follows:
$η@Wmax*=Wmax*QH*=ηeq(rM+1)(rM+ηeq)[1−ηeq(τ−1)+(rMτ+1)(rM+1)]$
(28)

It can be easily verified that the Curzon–Ahlborn efficiency $(η@Wmax=1−τ)$ can be retrieved for a reversible engine with constant reservoir temperatures and ideal operation of compression and expansion components, i.e., ηeq = 1 and rM = 1.

Equations (25)(28) are general expressions for $Wmax*$, τH,opt, τc,opt, and $η@Wmax*$, respectively. These expressions incorporate the effect of variable heating and cooling flow temperatures and isentropic efficiencies at compression and expansion components. A summary of these expressions is presented in Table 1 for the heat engine considering constant and variable heating and cooling flow temperatures, and also with and without compression–expansion irreversibilities.

Table 1

Expressions for τc,opt, $Wmax*$, and $η@Wmax*$ considering different model characteristics

## 3 Results and Analysis

In this section, results from the model presented in Sec. 2 are illustrated for different values of heat engine parameters τ, rM, and ηeq. As explained before, there is an optimal value of τc (Eq. (27)) at which the power output achieves a maximum value (Eq. (25)). Figure 3 presents the behavior of τc,opt as a function of τ for three selected values of ηeq = 0.6, 0.8, and 1.0 and five values with different order of magnitude for rM. From this figure, higher τ values (lower difference between cold and hot reservoir temperatures) lead to higher τc,opt. For the ideal case where ηeq = 1.0, when τ approaches 1.0, τc,opt also approaches 1.0, see Fig. 3(a), which implies a reduction in the temperature difference (potential) between reservoirs, and consequently a decrement in the power output. In the limit, when τ = τc,opt = 1, W and η are equal to zero. It is important to note that τc,opt is independent of the parameter rM for ηeq = 1.0. For values of ηeq = 0.8 (Fig. 3(b)) and ηeq = 0.6 (Fig. 3(c)), τc,opt is slightly higher for lower values of τ, when compared with the ideal scenario of Fig. 3(a); this means that for the same τ and ηeq < 1.0, irreversibilities in compression and expansion processes reduce the difference between cold and hot engine temperatures decreasing its power and efficiency. As expected, the larger the irreversibilities (smaller values of ηeq), the higher τc,opt for any value of τ < 1.0, as can be seen in Figs. 3(b) and 3(c). From these figures, it can be appreciated that the parameter rM becomes important as the irreversibilities in the compression and expansion processes increase. This effect is more significant for lower τ values, i.e., for heat engines, such as Brayton engines, that operate with a larger difference between cold and hot reservoir temperatures, when compared with engines whose operation is framed by higher τ like organic Rankine cycles. Higher values of rM tend to reduce τc,opt for the same ηeq, which means that a larger effective conductance in the cold dissipation heat exchanger should help compensate the detriment in engine’s performance due to irreversibilities.

Fig. 3
Fig. 3
Close modal

In order to better appreciate the effect of rM on τc,opt, Fig. 4 displays curves of τc,opt for five orders of magnitude values of rM, considering three ηeq and τ = 0.5. For ηeq = 1.0, τc,opt = 0.707, and, as it was mentioned, it is independent of rM. As ηeq reduces, τc,opt increases along the entire rM range; however, τc,opt reduces as rM increases, approaching to the corresponding value of τc,opt when ηeq = 1.0, in this case 0.707. It is worth noting that for τ = 0.5, the maximum change in τc,opt in the analyzed rM range is about 0.02 (from τc,opt = 0.707 for ηeq = 1.0 to τc,opt = 0.728 for ηeq = 0.6), but this change is greater for lower τ values and, in the limit when τ → 0.0, τc,opt approaches this maximum change of about 0.386 (from τc,opt = 0.0 for ηeq = 1.0 to τc,opt = 0.386 for ηeq = 0.6), as can be seen in Fig. 3.

Fig. 4
Fig. 4
Close modal

Figure 5 shows how $Wmax*$ changes with τ for different orders of magnitude in the value of rM (from 10−2 to 102) and three different values of ηeq 0.6, 0.8, and 1. In general terms, $Wmax*$ decreases with τ as expected, which is a result of smaller temperature differences between the heat source and the heat sink. In the extreme case τ = 1, when both heat source and sink have the same temperature, $Wmax*$ is naturally equal to zero for all situations, as the power generation’s driving effect for the engine vanishes.

Fig. 5
Fig. 5
Close modal

The largest possible value of $Wmax*=1$ occurs when ηeq = 1, rM = 1, and τ = 0 (equivalent to TL = 0 or TH = ∞). In this case $Wmax*$ becomes equal to the heat transfer between the reservoirs when no heat engine is present. In any other situation, $Wmax*$ is a fraction of such bound on the amount of energy that could be transferred between heat source and heat sink, which is due to a reduction in the heat-to-power conversion efficiency when τ increases (closer TL and TH), a decrease in ηeq, or because the balance in heat exchanger inventory between hot and cold sides is substantially altered.

As described in the model, rM represents an effective conductance ratio between heat input and heat dissipation heat exchangers. Figure 5 shows how taking rM away from 1 (order of magnitude 100) brings down $Wmax*$. As can be verified, for rM equal to 10−1 or 101, $Wmax*$ is reduced in more than 50% for any value of τ and ηeq. For rM equal to 10−2 or 102, $Wmax*$ reduction is even greater. In order to better capture this behavior, Figs. 6(a) and 6(b) present $Wmax*$ as a function of rM for constant values of τ and ηeq. As can be seen, there is always a maximum value of $Wmax*$ for rM around 1. If the locus of maxima in Figs. 6(a) and 6(b) are isolated, the corresponding values of rMopt as a function of τ and ηeq can be found, as shown in Fig. 6(c). Values of rMopt are almost entirely between 1 and 1.2 for the considered values in this work. If τ or ηeq go very low (lower than approximately 0.2 and 0.6, respectively), rMopt can start rising beyond 1.2 but in any case not much higher than 1.4. It is worth remembering that a value of rM higher than 1 means a larger heat exchanger (larger effective conductance) in the cold side when compared with the hot side. rMopt becomes significant in practical implementations when a decision has to be made on the allocation of conductance of heat exchangers mainly driven by economic considerations.

Fig. 6
Fig. 6
Close modal

Table 2 helps emphasizing the importance of heat exchangers’ conductance allocation. Different values of $Wmax*$ are presented as a function of rM and τ together with the percentual deviation with respect to the maximum possible $Wmax*$ ($Wmax,max*$) for each case. As can be seen, choosing rM equal to 1.0 produces a $Wmax*$ close to the optimum for all cases considered. If one of the heat exchangers is twice large than the other, the percentual deviation respect to $Wmax,max*$ is around 10%. These relative deviations are more significant for lower values of τ.

Table 2

Effect of optimizing heat exchanger conductance allocation, ηeq = 0.8

τrM$Wmax*$
$Wmax,max*−Wmax*Wmax,max*$ (%)
0.250.50.2105413.65
10.243390.18
rMopt = 1.088$Wmax,max*=0.24382$
20.222658.68
0.500.50.1157312.27
10.131870.04
rMopt = 1.040$Wmax,max*=0.13192$
20.118769.98
0.750.50.0500311.56
10.056576.02 × 10−3
rMopt = 1.015$Wmax,max*=0.05657$
20.0505410.67
τrM$Wmax*$
$Wmax,max*−Wmax*Wmax,max*$ (%)
0.250.50.2105413.65
10.243390.18
rMopt = 1.088$Wmax,max*=0.24382$
20.222658.68
0.500.50.1157312.27
10.131870.04
rMopt = 1.040$Wmax,max*=0.13192$
20.118769.98
0.750.50.0500311.56
10.056576.02 × 10−3
rMopt = 1.015$Wmax,max*=0.05657$
20.0505410.67

Figure 7 illustrates the effects of ηeq and rM on the dependence of the efficiency at maximum power as a function of the temperature ratio τ. When τ = 1, the temperature gradient that drives the engine disappears, and at that point, independently of ηeq and rM, $Wmax*$ goes to zero. This is also true for the Carnot efficiency. $Wmax*$ increases monotonically as τ decreases and peaks in the limit τ → 0 when the low temperature reservoir tends to a temperature of absolute zero or the high temperature grows to infinity. Notice that differently from what occurred with the maximum power (see Fig. 5), in the case of ηeq = 1, the value of the rM has no effect on $η@Wmax*$ (thus all curves collapse into one in Fig. 7(a)). As the compression or the expansion component efficiencies decrease, or the compressing power becomes more important with respect to the expansion power, ηeq decreases and so does $η@Wmax*$. When ηeq < 1, the efficiency at maximum power drops as rM decreases.

Fig. 7
Fig. 7
Close modal

Figure 8 reemphasizes the impact of the heat exchangers effective conductance ratio. In the range considered, which spans five orders of magnitude, it can be observed that for large τ, the efficiency at maximum power is essentially insensitive to the allocation of heat exchanger conductance and effectiveness represented by rM. When it does impact (τ < 0.5) the behavior is essentially a smooth transition between a low limit value for rM ≪ 1 and a higher value for rM ≫ 1, with the inflection occurring near the point when the heat exchangers have equal allocation of effective conductance $(rM∼1)$. Observe in Fig. 8 a trend that is common in heat engines; the impact of the cold heat exchanger is greater than the hot one in terms of efficiency. For example, in a Rankine cycle, a small variation in the condenser temperature has a larger impact on efficiency than the same variation in the boiler [7]. It can also be seen that the effect of rM on efficiency is not as high as it is for $Wmax*$ (see Figs. 5 and 6). The meaning of a very large rM is allocating most of the available area to the cold heat exchanger, which implies TLCTLi (see Fig. 1). After some point TLC is so close to TLi that any additional area allocated to the cold heat exchanger no longer increase the heat engine efficiency (the curve levels off for large values of rM). Similar situation occurs in the lower end value where most of the area is allocated to the hot heat exchanger and THCTHi.

Fig. 8
Fig. 8
Close modal

## 4 Conclusions

A thermodynamic model considering finite heat transfer rate, variable heat source and sink temperature, and irreversibilities associated with the compression–expansion processes was developed to analyze the performance of heat engines operating under different thermodynamic cycles. Expressions for the optimum ratio between cold and hot thermal engine temperatures (τc,opt), maximum power $(Wmax*)$, and efficiency at maximum power output $(η@Wmax*)$ are obtained as a function of the parameters τ (cold and hot reservoirs temperature ratio), rM (effective conductance ratio between heat input and heat dissipation heat exchangers), and ηeq (equivalent isentropic efficiency). In all cases, the Curzon–Ahlborn efficiency is retrieved for rM = 1.0 and ηeq = 1.0. It was found that rM has no effect on τc,opt, $Wmax*$, and $η@Wmax*$ when ηeq = 1.0. As the isentropic efficiencies in the compression or the expansion components decrease, as a consequence of irreversibilities, ηeq decreases leading to reductions in $Wmax*$, $η@Wmax$, and in the difference between cold and hot engine temperatures (increment in τc,opt). When ηeq < 1, τc,opt decreases and $η@Wmax*$ increases with rM, respectively. In turn, $Wmax*$ drops for rM away from 1.0 in more than one order of magnitude. Optimum rM values (rMopt) leading to maximum power output are found to be between 1 and 1.4 for the analyzed ranges of τ and ηeq. rMopt shifts to the upper limit of that range as irreversibilities increase (ηeq reduces) and/or for engines operating within higher temperature difference between cold and hot reservoirs (lower τ). In this case, the compression power is usually a significant portion of the net power. A value of rMopt slightly greater than 1.0 means that in order to achieve maximum power, the heat dissipation heat exchanger must be larger when compared with the heat exchanger at the hot side.

## Acknowledgment

This research was primarily supported by the Mechanical and Thermal Engineering Sciences Center at National Renewable Energy Laboratory as a part of the Laboratory Directed Research and Development (LDRD) Program funded by the U.S. Department of Energy (DOE).

The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government.

Obie I. Abakporo acknowledges the support of the Florida A&M University Title III Fellowship and the facility usage of Florida State University Center of Advanced Power Systems.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article can be obtained from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment.

## Nomenclature

A =

effective heat transfer area, m2

U =

overall heat transfer coefficient, kW m−2 K−1

$m˙$ =

mass flow rate, kg/s

cp =

specific heat at constant pressure, kJ kg−1 K−1

DH =

THo to THi temperature ratio

DL =

TLi to TLo temperature ratio

QH =

heat input, kW

QLc =

heat rejection, kW

QLt =

heat loss, kW

QL =

total heat rejection, kW

THi =

inlet hot reservoir temperature, K

TLi =

inlet cold reservoir temperature, K

THc =

hot heat engine temperature, K

TLc =

cold heat engine temperature, K

THo =

exit hot reservoir temperature, K

TLo =

exit cold reservoir temperature, K

Wr =

net power output, kW

Wrt =

expansion power, kW

Wrp =

compression power, kW

Wi =

reversible net power output, kW

Wit =

reversible expansion power, W

Wip =

reversible compression power, kW

$W*$ =

dimensionless power output

$QH*$ =

dimensionless heat input rate

rM =

effective conductance ratio between cold and hot side heat exchangers

NTU =

number of transfer units

ɛ =

effectiveness

η =

thermal efficiency

$η@Wmax*$ =

efficiency at maximum power

ηt =

isentropic efficiency expansion component

ηp =

isentropic efficiency compression component

ηeq =

equivalent isentropic efficiency

τ =

cold to hot reservoirs temperature ratio

τc =

cold to hot thermal engine temperature ratio

τH =

THc to THi temperature ratio

τL =

TLi to TLc temperature ratio

H =

high temperature

L =

low temperature

max =

maximum

opt =

optimum

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