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The maximum power cycle operating between a heat source and heat sink with finite heat capacity rates

Osama M. Ibrahim*, Raed I. Bourisli

Department of Mechanical Engineering, College of Engineering and Petroleum, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait

Abstract

The objective of this study is to identify the thermodynamic cycle that produces the maximum possible power output from a heat source and sink with finite heat capacity rates. Earlier efforts used sequential Carnot cycles governed by heat transfer rate equations to determine the Maximum Power (MP) cycle, where its performance and shape were used as criteria to evaluate alternative power cycles and working fluids. The maximum power output of the sequential Heat Transfer Limited (HTL) Carnot cycles is realized by maximizing the sum of the net power output of all cycles subject to the entropy balance constraints. In this study, a hypothesis is proposed in which the heat capacity rates of the heat addition and rejection processes of the proposed MP cycle are assumed to match the ones for the heat source and sink, respectively. The result is a simple thermodynamic model that approximately defines the performance and shape of the proposed MP cycle, which are compared and verified with the shape and performance of optimized sequential HTL Carnot cycles with closely matching results.

Introduction

The Carnot cycle is an ideal reversible thermodynamic cycle that sets the upper limit of the thermal efficiency and produces the maximum possible work output from a heat source and heat sink with infinite heat capacities. The reversible heat transfer processes of the Carnot cycle require energy transfer with infinitesimal temperature differences. Considering a realistic power plant with finite-size heat exchanges, however, the rate of heat transfer of the reversible processes approaches zero, resulting in zero power output. Several early researchers studied the effect of heat transfer rate equations on power cycles. They recognized the existence of a maximum power point for the Heat Transfer Limited (HTL) Carnot cycle. Feidt [1] and Vaudrey et al. [2] brought to attention much earlier work by Moutier, Serrier, and Reitlinger. Chambadal [3], Novikov [4], and El-Wakil [5] were also among the first to consider the effect of heat transfer rate constraints on the performance of the Carnot cycle. These earlier studies show the existence of a maximum power point where the thermal cycle efficiency at maximum power is given by a simple expression that depends only on the square root of the temperature ratio of the heat sink and heat source.

Curzon and Ahlborn [6] independently derived the same expression for the thermal cycle efficiency at the maximum power of a Carnot engine limited by heat transfer rate equations. Curzon and Ahlborn's paper motivated many subsequent studies, e.g., [7-19]. Besides the external irreversibilities due to the rate of heat transfer to and from the working fluid, several of these studies proposed simple models to account for internal irreversibilities within the cycle [10,12]. Bejan [13] confirmed the fact that, for power plants, the maximum power is equivalent to the minimum entropy generation rate when the internal and external entropy generation rates are taken into account. Blaise et al. [14] studied the working fluid properties' influence on the maximum power of an irreversible finite dimension Carnot engine with changing phase working fluid. At the maximum power, they determined the optimum vaporization and condensation temperatures and the optimum allocation of the total heat transfer area between the boiler and condenser.

The HTL Carnot cycle considered in most of these early studies was assumed to operate between an isothermal heat source and sink with infinite thermal capacity. Actual power plants, however, operate between a hot stream heat source and a cold stream heat sink with finite heat capacity rates. Follow-up studies [12-16] considered a simplified HTL Carnot cycle coupled to heat reservoirs with finite heat capacity rates, as shown in Fig. 1 in a T-Ṡ plane. The cycle operates between 𝑇 and 𝑇 . A simplified heat exchanger model was used to determine the rate of heat transfer supplied to the cycle, �̇� , and rejected from the cycle, �̇� . All irreversible losses are associated with heat transfer in the heat exchangers and the discarding of the outlets of the hot and cold streams into the surroundings, with no internal irreversibilities within the cycle.

Fig. 1. The HTL Carnot cycle operating between a heat source and sink with finite heat capacity rates.

The following equations give the steady-state energy and entropy balances on the HTL Carnot cycle:

�̇� = �̇� − �̇� = 𝜀 �̇� 𝑇 , − 𝑇 − 𝜀 �̇� 𝑇 − 𝑇 , (1)

̇ −

̇ =

̇ ,

− ̇

, = 0 (2)

where �̇� is the net power output, 𝑇 , and 𝑇 , are the inlet temperatures, and �̇� and �̇� are the heat capacity rates of the hot and cold streams, respectively; 𝜀 and 𝜀 are the heat exchanger effectiveness of the hot and cold sides, which are evaluated using the equations listed in Table A-1, Appendix A.

The maximum power, �̇� , , and the thermal cycle efficiency at maximum power, 𝜂 ∗ , are obtained

analytically as [12],

�̇� , = ̇ ̇

̇ ̇ 𝑇 , − 𝑇 , (3)

𝜂∗ = 1 − ,

, (4)

Several researchers expanded the maximum power concept to other heat power cycles, such as the Otto, Diesel, and Brayton cycle [12, 20-22]. Of interest to the current study is the HTL Brayton cycle coupled to a heat source and sink with finite heat capacity rates. A simplified closed Brayton cycle was considered, as shown in Fig. 2 in a T-Ṡ plane. Like the HTL Carnot cycle, all irreversible losses of the HTL Brayton cycle are associated with the heat transfer in the heat exchangers and the discarding of the outlets of the hot and cold streams into the surroundings; there are no internal losses within the cycle itself. The working fluid is assumed to have a constant heat capacity rate, �̇� .

The following equations give the steady-state energy and entropy balances on the HTL Brayton cycle:

�̇� = �̇� − �̇� = 𝜀 �̇� 𝑇 , − 𝑇 − 𝜀 �̇� 𝑇 − 𝑇 , = �̇� (𝑇 − 𝑇 ) − �̇� (𝑇 − 𝑇 ) (5)

�̇� 𝑙𝑛 − �̇� 𝑙𝑛 = 0 (6)

Fig. 2. The HTL Brayton cycle operating between a heat source and sink with finite heat capacity rates.

The optimization of the power output of the HTL Brayton cycle was obtained analytically as [12],

�̇� , = ̇

, ̇ , ̇

, ̇

, ̇ , / ̇ 𝑇 , − 𝑇 , (7)

𝜂∗ = 1 − ,

, (8)

where �̇� , is the maximum power, 𝜂 ∗ is the thermal cycle efficiency at maximum power, �̇� ,

is the smaller value of �̇� and �̇� , and �̇� , is the smaller value of �̇� and �̇� . The equations of the heat exchanger effectiveness of the hot and cold streams, 𝜀 and 𝜀 , are listed in Table A-1, Appendix A.

The HTL Carnot and Brayton cycles result in the same simple equation for the thermal efficiency at the maximum power, as expressed by Equations 4 and 8. It is also worth mentioning that as the heat capacity of the working fluid of the Brayton cycle, �̇� approaches infinity, the HTL Brayton cycle approaches the HTL Carnot cycle, i.e., Equations 3 and 7 become identical.

The Carnot reversible cycle sets the limits for the maximum possible efficiency and work output for heat engines. The question then arises, does the HTL Carnot cycle set the limits for the maximum possible power output? To answer this question, the maximum power of the HTL Carnot and Brayton cycles was compared for the same parameters and thermal boundary conditions, i.e., the same heat exchanger sizes, and the same inlet temperatures and heat capacity rates of the hot and cold streams [12]. The maximum power ratio of the Brayton cycle, 𝜔 , , is defined as,

𝜔 , = ̇ , ̇ ,

(9)

The maximum power ratio of the Brayton cycle, 𝜔 , , is plotted against �̇� /�̇� , for different values of heat capacity rate ratio of �̇� /�̇� , as shown in Fig. 3a. The plots are created for arbitrary values of the number of transfer units of the hot and cold side heat exchangers, with 𝑁𝑇𝑈 =𝑁𝑇𝑈 =3. The four plots are created for fixed �̇� /�̇� values of 0.1, 0.5, 1, 2, 5, and 100. The results show power ratios higher than one, indicating that the HTL Brayton cycle can produce more power than the maximum power of the HTL Carnot cycle. It is also observed that the maximum power ratio occurs when �̇� /�̇� ≈ �̇� /�̇� or �̇� ≈ �̇� . For relatively high values of �̇� /�̇� , the plotline becomes a horizontal line aligned with a work ratio of one, indicating that the maximum power output of both cycles are equal, i.e., �̇� , = �̇� , .

Fig. 3. (a) The maximum power ratio of the Brayton cycle, 𝜔 , , vs. �̇� /�̇� , for different values of heat capacity rate ratio of �̇� /�̇� ; (b) the values of 𝜔 , at the maximum power points of the curves in Fig. 3a vs. �̇� /�̇� .

In Fig. 3b, the values of 𝜔 , at the maximum power points of the curves in Fig. 3a are plotted against �̇� /�̇� . It is clear from Figs. 3(a and b) that the HTL Carnot cycle does not always set the upper limit for the maximum power. This fact raises an important question, what is the heat power cycle that sets the upper limit for the maximum power for specified parameters and thermal boundary conditions. This study aims at answering this question. A previous effort used sequential Carnot cycles to determine the maximum work and the shape of the maximum work cycle, in a T-S plane, from a heat source with finite heat capacity and a heat sink with infinite heat capacity [23]. Sequential HTL Carnot cycles were also used to determine the maximum power from a heat source and sink with finite heat capacity rates

[12]. Follow-up studies used the maximum power cycle's shape and performance as criteria to study and evaluate alternative cycles and working fluids, e.g., [24-27].

This paper starts with reviewing, in detail, two thermodynamic models by (1) Ondrechen et al. [23] to identify the reversible cycle which extracts the maximum work from a heat source with finite heat capacity and a heat sink with infinite heat capacity; and (2) by Ibrahim et al. [12] to identify the heat power cycle that sets the upper limits for the maximum power from a heat source and sink with finite heat capacity rates. Both studies used sequential Carnot cycles to achieve their objectives. Summaries of their thermodynamic models of the sequential Carnot cycles are then presented, where the differences between work and power optimization are explained. Finally, a hypothesis is proposed to determine the heat power cycle that sets the upper limit for the maximum power. The performance and shape of the proposed Maximum Power (MP) cycle are compared to an equivalent sequence of HTL Carnot cycles

1. Maximum work by sequential Carnot cycles

Ondrechen et al. [23] investigated the maximum work by a single Carnot cycle and then generalized their analysis to the maximum work by sequential Carnot cycles. They considered a heat source with finite heat capacity and initial temperature, 𝑇 , , and a heat sink with infinite heat capacity and constant temperature, 𝑇 . They analyzed a single Carnot cycle operating between 𝑇 , and 𝑇 , , as shown in Fig. 4, where 𝑇 , < 𝑇 , , to provide a temperature difference, a driving force for irreversible heat transfer, 𝑄 . , from the heat source. The heat transfers from the heat source to the hot side of the Carnot cycle until the heat source temperature reaches 𝑇 , . For the cold side of the heat cycle, the heat rejection, 𝑄 . , is reversible where the low temperature of the cycle 𝑇 , = 𝑇 .

Fig. 4. A single Carnot cycle coupled with a heat source with finite heat capacity and initial temperature, 𝑇 , , and a heat sink with infinite heat capacity and constant temperature, 𝑇 .

The analysis presented by Ondrechen et al. [23] assumes an endoreversible cycle where the thermal efficiency and the heat transfer to the single Carnot cycle are given by,

𝜂 = ,

= 1 − ,

(10)

𝑄 , = 𝐶(𝑇 , − 𝑇 , ) (11)

where C is the heat capacity of the heat source.

The work, 𝑊 , of the single Carnot cycle is then given as follows:

𝑊 = 𝑄 , 𝜂 = 𝐶 𝑇 , − 𝑇 , 1 − ,

(12)

To maximize the work, 𝑊 , of Equation 12, 𝐶, 𝑇 , , and 𝑇 are assumed constants. The optimum value of the high cycle temperature, which results in the maximum work, is then given as,

𝑇 , = 𝑇 , 𝑇 (13)

where the maximum work and the equivalent thermal efficiency, for the single Carnot cycle, are then given as,

𝑊 , = 𝐶 𝑇 , − 𝑇 (14)

𝜂∗ = 1 − ,

(15)

The total work produced by a sequence of N-Carnot cycles is then given as follows:

𝑊 = ∑ 𝑊 = ∑ 𝐶(𝑇 , − 𝑇 , ) 1 − ,

(16)

As shown in Fig. 5, the high cycle temperature of the preceding cycle, 𝑇 , , is equal to the heat source temperature for the subsequent cycle, 𝑇 , , i.e.,

𝑇 , = 𝑇 , (17)

Using Equation 17, Equation 16 is rearranged as follows:

𝑊 = 𝐶𝑁𝑇 + 𝐶𝑇 , − 𝐶𝑇 , − 𝐶 ∑ ,

, (18)

Fig. 5. Example of a sequence of 15 Carnot cycles coupled with a heat source with finite heat capacity and initial temperature, 𝑇 , , and heat sink with infinite heat capacity and constant temperature, 𝑇 .

Ondrechen et al. [23] derived an elegant analytical solution for the total maximum work of a sequence of N-Carnot cycles,

𝑊 , = 𝐶 𝑁𝑇 + 𝑇 , − (𝑁 + 1)𝑇 ,

/( ) (19)

The thermal efficiency at the maximum total work of the collective N-cycles is then given by,

𝜂∗ = ,

, , = 1 −

, /

, / , /

(20)

For 𝑁 = 1, Equations 19 and 20 reduce to Equations 14 and 15.

Ondrechen's paper did not provide a clear difference between the definition of the thermal cycle efficiency and the overall cycle efficiency. The thermal cycle efficiency, as expressed by Equations 15 and 20, is defined as the ratio between the work output and the heat input to the cycle, which is

consistent with the definition of the Carnot efficiency. The overall cycle efficiency, on the other hand, is defined as the ratio of the work output and the maximum possible heat input. The overall efficiency at the maximum work of N-Carnot cycles is then expressed as,

𝜂 , ∗ =

( , ) = 1 −

( ) , /

, / (21)

Equations 15, 20, and 21 were not explicitly provided in these forms by Ondrechen et al. [23], but it is easy to derive them from the information provided in their paper.

As the number of cycles approaches infinity, the total maximum work and the equivalent thermal efficiency are reduced to,

𝑊ꝏ, = 𝐶 𝑇 , − 𝑇 − 𝑇 ln , (22)

𝜂ꝏ ∗ = 1 −

, ln

, (23)

The maximum work of a sequence of N-Carnot cycles is normalized by the maximum work of a single Carnot cycle and plotted in Fig. 6. Also shown in Fig. 6 is the normalized work as the number of cycles, N, approaches infinity, which represents the upper limit of the maximum work. The results show that the work output increases with the number of cycles asymptotically approaching the maximum work as 𝑁 → ∞.

Fig. 6. The normalized maximum work, 𝑊 , /𝑊 , , of a sequence of N-Carnot cycles vs. the number of cycles in the sequence, N. Also shown is the normalized maximum work as the number of Carnot cycles approaches infinity, 𝑊ꝏ, /𝑊 , .

The thermal efficiency of a single cycle, N-Carnot cycles, and an infinite number of Carnot cycles, as expressed by Equations 15, 20, and 23, are plotted in Fig. 7. Also plotted, for comparison, are the overall efficiency, as expressed by Equation 21, and the Carnot efficiency, 𝜂 = 1 − 𝑇 /𝑇 , , representing the upper limit of the thermal efficiency. The thermal efficiency of a single cycle or multiple cycles in sequence is given or well approximated by the well-known expression, 𝜂∗ = 1 − 𝑇 /𝑇 , . The overall efficiency, 𝜂 ,

∗ , however, increases with the number of cycles and asymptotically approaches the thermal efficiency 𝜂ꝏ

∗ , as 𝑁 → ∞, where the total heat transfer from the heat source reaches its maximum. Fig. 6 and Fig. 7 were re-created with some modifications using the same initial ratio of the heat source and sink temperatures used by Ondrechen et al. [23], where 𝑇 , /𝑇 = 1.1.

In summary, Ondrechen et al. [23] provided an analytical solution for the maximum work from a single Carnot cycle, which was then generalized for a sequence of Carnot cycles operating between a heat source with finite heat capacity and a sink with infinite heat capacity. Their model of the single Carnot cycle starts with a temperature gap between the heat source and the cycle resulting in irreversible heat

addition, while the cycle and the heat sink are assumed to have the same temperature with reversible heat rejection. These two different assumptions result in a simplified model for the single Carnot cycle. As the number of cycles approaches infinity, however, the temperature gap diminishes, and a perfect temperature match between the heat source and the Carnot cycles is achieved. The results define the shape of a reversible maximum work cycle with an ideal match with the finite capacity heat source on the hot side and with the infinite heat capacity heat sink on the cold side, as shown in Fig. 8.

Fig. 7. The thermal efficiency of a single cycle, 𝜂∗ , N-Carnot cycles, 𝜂∗ , and an infinite number of cycles, 𝜂ꝏ

∗ , vs. the number of cycles, 𝑁. Also shown, for comparison, are the overall efficiency, 𝜂 ,

∗ , and the Carnot efficiency, 𝜂 .

Fig. 8. The reversible cycle produces the maximum possible work from a heat source with finite heat capacity and a heat sink with infinite heat capacity.

2. Maximum Power by sequential Carnot cycles

Ibrahim et al. [12] considered a sequence of HTL Carnot cycles to determine the maximum possible power from a heat source and sink with finite heat capacity rates. They used the single HTL Carnot cycle model described in the introduction of this paper as a building block to construct the sequence of N-HTL Carnot cycles. In contrast with the work optimization by Ondrechen el al. [23], summarized in the previous section, the power optimization, in this section, includes heat rate equations and a simple model for heat exchanger based on the effectiveness-NTU method.

The steady-state energy and entropy balances of a sequence of HTL Carnot cycles, shown in Fig. 9, can be generalized as follows:

�̇� = ∑ �̇� , = ∑ �̇� , − �̇� , = ∑ 𝜀 �̇� 𝑇 , , − 𝑇 , − 𝜀 �̇� 𝑇 , − 𝑇 , , (24)

�̇� = ̇

, −

̇ ,

, =

̇ , , ,

, −

̇ , , ,

, = 0 (i=1 to N) (25)

where �̇� is the total work of N-HTL Carnot cycles in sequence, and “𝑖” is the number of a cycle in the sequence, and �̇� is a constraint function satisfying the entropy balance of an internally reversible HTL Carnot cycle.

The thermal efficiency of a sequence of HTL Carnot cycles, 𝜂 , is defined as,

𝜂 = ∑ ̇ ,

∑ ̇ , =

̇ ,

(26)

The inlet temperature of the hot stream of the subsequent, 𝑇 , , , is related to the outlet temperature to the preceding cycle, 𝑇 , , , by,

𝑇 , , = (1 − 𝜀 )𝑇 , , + 𝜀 𝑇 , (i=2 to N) (27)

where 𝑇 , is the high temperature of the preceding cycle.

Considering the counterflow heat exchanger configuration, as shown in Fig. 9, the inlet temperature of the cold stream of the preceding (𝑇 , , ) is related to the outlet temperature to the subsequent cycle (𝑇 , , ) by,

𝑇 , , = (1 − 𝜀 )𝑇 , , + 𝜀 𝑇 , (i=1 to N-1) (28)

where 𝑇 , is the low temperature of the subsequent cycle.

Fig. 9. Example of a sequence of 15 Carnot cycles coupled with a heat source and sink with finite heat capacity rates and inlet temperatures of 𝑇 , , and 𝑇 , , .

The power optimization of the objective function, �̇� , as expressed by Equation 24 is subject to the entropy balance constraint functions as expressed by Equation 25, where 𝑇 , , , 𝑇 , , , �̇� , �̇� , 𝜀 , and 𝜀 are assumed constants. The heat exchanger effectiveness is calculated with the equations listed in Table A-1, Appendix A, assuming the 𝑁𝑇𝑈 and 𝑁𝑇𝑈 are equally divided among the cycles in the sequence, i.e.,

𝑁𝑇𝑈 , = (29a)

𝑁𝑇𝑈 , = (29b)

The values of 𝑇 , and 𝑇 , that result in the total maximum power of the HTL Carnot cycles in series are found using the Lagrange multipliers optimization method, where Lagrange multipliers, 𝜆 for i=1 to N, are defined as:

, = ∑ 𝜆

, (i=1 to N) (30a)

, = ∑ 𝜆

, (i=1 to N) (30b)

Evaluating the partial derivatives and rearranging the equations lead to,

−1 + ∑ [𝜀 (1 − 𝜀 ) ] = − , ,

, + ∑

( )

, (i=1 to N-1) (31a)

−1 = − , ,

, (i=N) (31b)

−1 = − , ,

, (i=1) (32a)

−1 + ∑ 𝜀 (1 − 𝜀 ) = − , ,

, + ∑

( )

, (i=2 to N) (32b)

The maximum power, the thermal efficiency at maximum power, and the shape of the N-HTL Carnot cycles in series are determined numerically by solving Equations 24 to 29, in addition to Equations 31 and 32.

As an example, consider the following case study where the objective is to obtain the maximum power of N-HTL sequential cycles operating between hot and cold flowing streams with an inlet temperature ratio 𝑇 , /𝑇 , = 2 and a heat capacity rate ratio �̇� /�̇� = 5. The number of transfer units of the hot side and cold side heat exchangers are assumed to be equal, i.e., 𝑁𝑇𝑈 = 𝑁𝑇𝑈 = 3.

Fig. 10 shows the results of the power ratio, 𝜔 , , as the number of the HTL Carnot cycles varies from 1 to 100. The power ratio, in this case, is defined as the ratio of the maximum power of the sequence of N-HTL Carnot cycles, �̇� , , to the maximum power of a single HTL cycle, �̇� , , i.e.,

𝜔 , = ̇ , ̇ ,

(33)

The maximum power initially increases significantly as the number of HTL cycles increases from 1 to 5, then asymptotically reaches a limiting value as the number of cycles approaches infinity. Also shown in Fig. 10 are the thermal efficiency at the maximum power and the Carnot efficiency. The thermal efficiency is slightly changing with the number of cycles, and its values are well approximated by the simple expression, 𝜂∗ = 1 − 𝑇 , /𝑇 , .

Fig. 10. Maximum power ratio, 𝜔 , , the Carnot efficiency, 𝜂 , and the thermal efficiencies at maximum power, 𝜂∗ and 𝜂∗ , vs. the number of cycles in sequence, for 𝑇 , /𝑇 , = 2, �̇� /�̇� = 5, and 𝑁𝑇𝑈 = 𝑁𝑇𝑈 = 3.

Examples of multiple HTL Carnot cycles in sequence are shown in Fig. 11 in T-S diagrams, where 𝑇 = 𝑇/𝑇 , and 𝑆̅ = �̇�/�̇� are the normalized temperature and normalized entropy transfer rate. The

multiple HTL cycles are generated at maximum power for 𝑇 , , /𝑇 , , = 2, �̇� /�̇� = 5, 𝑁𝑇𝑈 = 𝑁𝑇𝑈 = 3. The shape and performance of the sequential HTL Carnot cycles evolved into the shape and performance of the MP cycle as the number of cycles approaches infinity. Although few HTL Carnot cycles in sequence reach the maximum possible power, as shown in Fig. 10, their shape, on the other hand, does not converge to the expected smooth shape of the MP cycle. 100-HTL Carnot cycles in sequence, however, are an excellent approximation to the shape and performance of the MP cycle, which is used as a baseline for comparison. We also observed the parallel matching temperature profiles between the heat source and the hot side of the cycle, and the heat sink and the cold side of the cycle. The parallel matching temperature profiles are indicating possible matching heat capacity rates between the cycle and heat source and sink. The shape of the maximum work cycle, reported by Ondrechen et al. [23], has also revealed a perfect match with the heat source and sink, as shown in Fig. 8. These observations form the foundation of our proposed hypothesis to approximately identify the MP cycle, as introduced in the next section.

Fig. 11. The shape of Multiple HTL Carnot cycles in sequence in T-S diagrams for 𝑇 , /𝑇 , = 2, �̇� /�̇� = 5, 𝑁𝑇𝑈 = 𝑁𝑇𝑈 = 3.

3. The Maximum Power (MP) cycle

In this section, a hypothesis is introduced where the heat capacity rates of the heat addition and rejection processes of the proposed MP cycle are assumed to match the ones for the heat source and heat sink, respectively. As shown in Fig. 12, The proposed MP cycle consists of four thermodynamic processes: (1) heat addition, where the working fluid and the heat source have the same heat capacity rate, �̇� ; (2) adiabatic expansion; (3) heat rejection where the working fluid and the heat sink have the same heat capacity rate, �̇� ; and (4) adiabatic compression. A simplified heat exchanger model was used to determine the rate of heat supplied to the cycle and the rate of heat rejected from the cycle. All irreversible losses in the proposed MP cycle are associated with heat transfer in the heat exchangers and the discarding of the outlets of the hot and cold streams into the surroundings; there are no irreversible internal losses within the cycle itself. The proposed MP cycle is referred to hereafter, in this paper, as the MP cycle.

Fig. 12. The MP cycle with equal heat capacity rates with the heat source and sink.

The steady-state energy and entropy balances of the MP cycle are then given by,

�̇� = �̇� − �̇� = 𝜀 �̇� 𝑇 , − 𝑇 − 𝜀 �̇� 𝑇 − 𝑇 , (34)

�̇� 𝑙𝑛(𝑇 /𝑇 ) − �̇� 𝑙𝑛(𝑇 /𝑇 ) = 0 (35)

Satisfying the energy balances across the heat exchangers leads to the following equations: �̇� = 𝜀 �̇� 𝑇 , − 𝑇 = �̇� 𝑇 , − 𝑇 , = �̇� (𝑇 − 𝑇 ) (37)

�̇� = 𝜀 �̇� 𝑇 − 𝑇 , = �̇� 𝑇 , − 𝑇 , = �̇� (𝑇 − 𝑇 ) (38)

The MP cycle thermal efficiency is defined as,

𝜂 = ̇

̇ (39)

Equations 34 to 39 define the thermodynamic model of the MP cycle. Equations 34 to 39 represent a set of equations with one degree of freedom. Arbitrary values of the parameters and thermal boundary conditions are specified, i.e., 𝑇 , /𝑇 , = 2; 𝑁𝑇𝑈 = 5, and �̇� /�̇� = 1, 5, and 10. The equations are then solved for values of the thermal efficiency between 0 and 0.5, where 𝜂 = 0.5, in this case, is the upper limit of the thermal efficiency.

A tradeoff between the MP cycle power ratio, 𝜔 = �̇� / �̇� , , and thermal cycle efficiency, 𝜂 , is shown in Fig. 13. It is clear from the three power plots that when the thermal efficiency is zero, the power is zero. The power is also zero when the cycle operates at the Carnot efficiency, “the maximum efficiency.” The results also show that the thermal cycle efficiency at maximum power is not always exactly equal, but very close to the value obtained by the simple expression, 𝜂∗ = 1 − 𝑇 , /𝑇 , . The

maximum power ratio of the MP Cycle is higher than one, indicating that the MP cycle produces more power than the HTL Carnot cycle.

Fig. 13. Power ratio, 𝜔 , vs. thermal efficiency, 𝜂 , for the MP cycle for 𝑇 , /𝑇 , = 2 and 𝑁𝑇𝑈 = 5.

The power optimization of the MP cycle can also be obtained analytically using the Lagrange multipliers. The analytical solution starts with solving for 𝑇 and 𝑇 from Equations 37 and 38, then substituting into Equations 34 and 35.

�̇� = 𝜀 �̇� 𝑇 , − 𝑇 − 𝜀 �̇� (𝑇 − 𝜀 𝑇 , )/(1 − 𝜀 ) − 𝑇 , (40)

�̇� = ( ) ,

̇ / ̇

− ( , )/( )

= 0 (41)

where �̇� is the objective function and �̇� is the constraint function. The values of 𝑇 and 𝑇 that result in the maximum power are found using Lagrange multipliers, where the Lagrange multiplier, λ, is defined as,

̇ =

̇ (42a)

̇ = λ

̇ (42b)

Evaluating the partial derivatives and rearranging the equations lead to,

-�̇� = −λ , (43a)

-�̇� = −λ , (43b)

Dividing Equation 43a by Equation 43b results in the following relation:

𝑇 𝑇 𝑇 , = 𝑇 𝑇 𝑇 , (44)

For the case when �̇� /�̇� = 1, the HTL Brayton cycle is the MP cycle. The maximum power is expressed as a special case of Equation 7, and the thermal efficiency at maximum power is given by Equation 8, i.e.,

�̇� , =

�̇� 𝑇 , − 𝑇 , (�̇� /�̇� = 1) (45)

𝜂∗ = 1 − ,

, (�̇� /�̇� = 1) (46)

For �̇� /�̇� > 1, simple expressions for the maximum power and the thermal cycle efficiency at maximum power are not clear to discern. The maximum power and the thermal cycle efficiency at maximum power, in this case, are determined numerically by solving Equations 34-39 in addition to Equation 44.

4. Comparisons between the MP cycle and 100-HTL sequential Carnot cycles

The performance and shape of the proposed MP cycle are compared to those of a sequence of 100-HTL Carnot cycles optimized for maximum power. The goal is to verify the values of the maximum power, the thermal efficiency at maximum power, and the shape of the MP cycle with the ones obtained by optimizing a sequence of 100-HTL cycles for the same thermal parameters and boundary conditions.

Comparisons between the maximum power ratio of the MP cycle (𝜔 , ) and 100-HTL Carnot Cycles, 𝜔 , , are presented in Fig. 14a. The results are generated for �̇� /�̇� =1, 5, and 10, and NTU between 1 and 10. Three ratios of the heat source and sink inlet temperatures are considered: 𝑇 , /𝑇 , = 1.5, 2, and 3. The deviation, 𝛿 , of the maximum power of the MP cycle from the maximum power of 100-HTL Carnot Cycles is shown in Fig. 14b. The results show that the deviation is within −1.2%. It is noticed that the deviation increases as the ratio of the inlet temperatures and heat capacity rates of the heat source and sink increase, with 𝑇 , /𝑇 , = 3, �̇� /�̇� =10, and 𝑁𝑇𝑈 ≈ 3.5 having the highest deviation. It is also noticed, for the curves in Fig. 14b, that the deviation reaches a maximum value and then decreases as the NTU increases.

Fig. 14. (a) Comparisons between the maximum power ratio of the MP cycle, 𝜔 , , and 100- HTL Carnot Cycles in sequence, 𝜔 , ; (b) power deviation, 𝛿 = (𝜔 , − 𝜔 , )/𝜔 , .

Comparisons between the thermal efficiency at the maximum power of the MP cycle and a sequence of 100-HTL Carnot cycles are presented in Fig. 15a. Like Fig. 14a, the results are generated for selected values of �̇� /�̇� , NTU, and 𝑇 , /𝑇 , . It is noteworthy that the values of the efficiency at the maximum power of both the MP cycle and the 100-HTL cycles are clustered close together, but not exactly the same as the values obtained from the simple expression, 𝜂∗ = 1 − 𝑇 , /𝑇 , . The percent deviation, 𝛿 , of the thermal efficiency at maximum power of the MP cycle from the equivalent ones of 100-HTL Carnot Cycles are in Fig. 15b. The deviations for all the cases considered in Fig 15a are within −0.8 %. Similar to the maximum power deviation, shown in Fig. 14b, the efficiency deviation, shown in Fig. 15b, increases as the ratio of the inlet temperatures and heat capacitance rates of the heat source and sink increase, with 𝑇 , /𝑇 , = 3, �̇� /�̇� =10, and 𝑁𝑇𝑈 ≈ 3.5 having the highest deviation,

but still within −0.8 %. Like Fig 14b, the efficiency deviation curves in Fig. 15b have maximum values and then show reducing trends as the NTU increases.

Fig. 15. (a) Comparisons between the thermal efficiency at the maximum power of the MP cycle (𝜂∗ ) and 100- HTL Cycles in sequence (𝜂∗ ); (b) efficiency deviation, 𝛿 = (𝜂∗ − 𝜂∗ )/𝜂∗ .

Comparisons between the shapes of the MP cycle and 100-HTL sequential Carnot Cycles in T-S diagrams are presented in Figs. 16-18. Each figure shows 4 cycles, which are created for a specified value of 𝑇 , /𝑇 , , while studying the effects of increasing �̇� /�̇� from 1 to 10, and NTU from 3 to 10. Figs. 16a to 16d show a close match between the shapes of the MP cycle and 100-HTL sequential Carnot Cycles for 𝑇 , /𝑇 , =1.5. Similarly, Figs.17 and 18 show similar matches between the two cycles for 𝑇 , /𝑇 , = 2 and 3. The exceptions are Fig. 17c and Fig. 18c, for �̇� /�̇� = 10 and 𝑁𝑇𝑈 = 3, where a small but noticeable mismatch at the hot side of the cycles is observed. The mismatch on the hot side decreases significantly as the NTU increases from 3 to 10.

It is worth noting that the power optimizations of the proposed MP cycle and the 100-HTL sequential Carnot cycles result in approximate performance and shape of the ultimate MP cycle. That is to say; the two approaches do not precisely identify the ultimate MP cycle and locate the global optimum point for the maximum power. Instead, they provide reasonable and approximate solutions near the global optimum. In this paper, we assume the ultimate MP cycle exists, and its best approximation is given by a sequence of 100-HTL Carnot cycles, which helps to provide a baseline and a reference for comparisons.

The differences between the proposed MP cycle's shapes and performance and the 100-HTL sequential Carnot cycles are attributed to the assumptions behind these two approaches used to search and define the ultimate MP cycle. Crucial factors that lead to slightly different shapes and performance are the heat transfer processes and the corresponding heat exchanger models. In a single HTL cycle in the sequence, the high and low-temperature heat addition and rejection occur at constant temperatures. 100- HTL Carnot cycles result in step-like temperature heat addition and rejection processes, modeled as 100 heat exchangers. In contrast, the proposed MP cycle's heat addition and rejection processes occur at variable temperatures represented by continuous and smooth functions, modeled as a single heat exchanger. The different heat transfer processes in these two approaches are modeled by different heat exchange effectiveness equations, leading to slightly different shape and performance.

Fig. 16. Comparisons between the shapes of the MP cycle and 100-HTL Carnot Cycles in T-S diagrams for 𝑇 , /𝑇 , = 1.5.

Fig. 17. Comparisons between the shapes of the MP cycle and 100-HTL Carnot Cycles in T-S diagrams for 𝑇 , /𝑇 , = 2.

Fig. 18. Comparisons between the shapes of the MP cycle and 100-HTL Carnot Cycles in T-S diagrams for 𝑇 , /𝑇 , = 3.

5. Conclusions

A hypothesis was introduced where the heat capacity rates of the heat addition and rejection processes of the MP cycle are assumed to match the ones for the heat source and heat sink. A simplified heat exchanger model was used to determine the rate of heat supplied to and rejected from the cycle. The maximum power and the thermal efficiency at the maximum power of the proposed MP cycle were verified and compared with those of the HTL sequential Carnot cycles within ±1% for the parameters and boundary conditions considered in this study.

The shape of the MP cycle is changing to provide the maximum possible power from the specified parameters and the thermal boundary conditions. When comparing the shape of the MP cycle to the HTL sequential Carnot cycles, a close match at a low ratio of inlet temperatures of the heat source and sink was observed. As the ratio of the inlet temperatures increases, a slight mismatch in the cycles' hot- side temperature profiles was observed at a low NTU of 3. At a higher NTU value of 10, the mismatch in the cycles' hot-side temperature profiles reduces significantly. The cold side for all cycles considered in this study, however, has a perfect match, independent of the ratios of the inlet temperatures and heat capacity rates of the heat source and sink.

In conclusion, the proposed hypothesis leads to a simple thermodynamic model that approximately determines the maximum power, efficiency at the maximum power, and shape of the MP cycle operating between a heat source and heat sink with finite heat capacity rates.

Appendix A

Table A-1. Counterflow heat exchanger effectiveness equations for the HTL Carnot, Brayton, and MP cycles [28, 29].

Heat power cycle Heat exchanger Equation

HTL Carnot cycle Hot-side heat exchanger 𝜀 = 1 − exp (−𝑁𝑇𝑈 )

Cold-side heat exchanger 𝜀 = 1 − exp (−𝑁𝑇𝑈 )

HTL Brayton cycle*

MP cycle

Hot-side heat exchanger 𝜀 =

[ ( )]

[ ( )] (𝐶𝑟 < 1)

𝜀 = (𝐶𝑟 = 1)

Cold-side heat exchanger 𝜀 =

[ ( )]

[ ( )] (𝐶𝑟 < 1)

𝜀 = (𝐶𝑟 = 1)

𝑁𝑇𝑈 and 𝑁𝑇𝑈 are the number of heat transfer units of the hot and cold side heat exchangers.

𝐶𝑟 = ̇

̇ , ; 𝐶𝑟 =

̇ ,

�̇� , is the larger value of �̇� and �̇� , while �̇� , is the larger value of �̇� and �̇� .

*The number of transfer units, 𝑁𝑇𝑈 and 𝑁𝑇𝑈 , in the HTL Bryton cycle case, are based on the minimum heat capacity rates.

Nomenclatures C Heat capacity (mC ), kJ/K

Ċ Heat capacity rate (ṁC ), kW/K C Specific heat capacity, kW/Kg-K N Number of sequential cycles NTU Number of Transfer Units (UA/Ċ) ṁ Mass flow rate, kg/s Q̇ Heat transfer rate, kW Ṡ Entropy transfer rate, kW/K S Normalized entropy transfer rate (S = Ṡ/Ċ ) T temperature, K T Normalized temperature (T = T/T , ) UA Heat exchanger conductance, kW/K Ẇ Power (work per unit time), kW Greek symbols δ Deviation ε Heat exchanger effectiveness (ε = �̇�/�̇� ) η Thermal efficiency (η = �̇�/�̇� ) ω Power ratio (𝜔 = �̇�/�̇� , ) Subscripts B Baryton cycle C Carnot cycle e efficiency H Heat source h hot side in inlet/in L Heat sink l cold side max maximum min minimum out outlet/out p Power MP Maximum power cycle wf working fluid N Number of heat cycles in sequential arrangement ꝏ Infinity refers to the number of heat cycles in sequential arrangement Superscripts * at maximum power Overbar 𝒈 normalized

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