Optimizing Power and Thermal Efficiency of an Irreversible Variable-Temperature Heat Reservoir Lenoir Cycle

: Applying finite-time thermodynamics theory, an irreversible steady flow Lenoir cycle model with variable-temperature heat reservoirs is established, the expressions of power ( P ) and efficiency ( η ) are derived. By numerical calculations, the characteristic relationships among P and η and the heat conductance distribution ( L u ) of the heat exchangers, as well as the thermal capacity rate matching ( 1 / wf H C C ) between working fluid and heat source are studied. The results show that when the heat conductances of the hot- and cold-side heat exchangers ( H U , L U ) are constants, - P η is a certain “point”, with the increase of heat reservoir inlet temperature ratio ( τ ), H U , L U , and the irreversible expansion efficiency ( e η ), P and η increase. When L u can be optimized, P and η versus L u characteristics are parabolic-like ones, there are optimal values of heat conductance distributions ( ( ) P L opt u , ( ) L opt u η ) to make the cycle reach the maximum power and efficiency points ( max P , max η ). As 1 / wf H C C increases, max 1 -/ wf H P C C shows a parabolic-like curve, that is, there is an optimal value of 1 / wf H C C ( 1 ( / ) wf H opt C C ) to make the cycle reach dou-ble-maximum power point ( max max ( ) P ); as / L H C C , T U , and e η increase, max max ( ) P and 1 ( / ) wf H opt C C increase; with the increase in τ , max max ( ) P increases, and 1 ( / ) wf H opt C C is un-changed.

wf H opt C C
The Lenoir cycle (LC) model [54] was proposed by Lenoir in 1860. From the perspective of the cycle process, the LC lacks a compression process. It looks like a triangle in the cycle -T s diagram. It is a typical atmosphere pressure compression HEG cycle, the compression process required by the HEG during operation is realized by atmosphere pressure and it can be used in aerospace, ships, vehicles, and power plants in engineering practice. Georgiou [55] first used classical thermodynamics to study the steady flow Lenoir cycle (SFLC) and compared its performance with that of a steady flow Carnot cycle. Compared to the classical thermodynamics, the finite time process of the finite rate heat exchange (HEX) between the system and the environment and the finite size device are considered in the FTT [1][2][3][4][5][6][7][8][9][10][11][56][57][58][59], therefore, the result obtained is closer to the actual HEG performance Considering the heat transfer loss, Shen et al. [60] established an endoreversible SFLC model with constant-temperature HRs by applying FTT theory, analyzed the influences of HR temperature ratio and total heat conductance (HTC) on the power output ( P ) and efficiency (η ) characteristics, and obtained the maximum P and maximum η and the corresponding optimal HTC distributions. Based on the NSGA-II algorithm, Ahmadi et al. [61] optimized the ecological performance coefficient and thermoeconomic performance of the endoreversible SFLC with constant-temperature HRs. Based on the Ref. [60], Wang et al. [62] further considered the internal irreversibility loss, established the irreversible SFLC model and optimized its P and η performance.
The above-mentioned were all studies on the SFLC with constant temperature HR. Based on Refs. [60][61][62], an irreversible SFLC with a variable temperature HR will be established in this paper, and the influence of internal irreversibility, HR inlet temperature ratio, thermal capacity rate (TCR) matching, and total HTC on cycle performance will be studied.  The irreversible expansion efficiency ( e η ) is defined as [41,44,46,51]:

Cycle Model and Thermodynamic Performance
Assuming the heat transfer between the working fluid and HR obeys the law of Newton heat transfer, according to the ideal gas properties and the theory of HEX, the cycle heat absorption and heat release rates are, respectively: where H C ( L C ) and ), respectively, m  is the working fluid mass flow rate, v C ( P C ) is the constant volume (pressure) specific heat, k is the specific heat ratio.
According to the second law of thermodynamics, one obtained: From Equations (2) and (3), the expressions of 2 T and 3 T are, respectively: From Equations (1) and (10)-(12), the expression of 1 T can be obtained as: From to Equations (2), (3), and (11)-(13), the expressions of P and η can be obtained as: When =1 e η , substituting into Equation (13), the expression of 1 T for an endoreversible SFLC with variable temperature HR can be obtained as: Combining Equations (4)- (9) and (14)- (16), by the numerical solution, the relationship between the P and η characteristics of the variable temperature HR endoreversible SFLC can be obtained. (4), (5) and (13)-(15) yields the expressions of the effectiveness of the two HEXs, P , η , and 1 T for an irreversible SFLC with constant temperature HR [62]: When

Cycle Performance Analysis When the HTC of Hot-and Cold-Side HEXs Is Constant
Determining the relevant parameters according to the Refs. [53,[60][61][62]:

Cycle Performance Optimization When the HTC Distributions of the Two HEXs Can Be Optimized
Assuming that the sum of the HTCs of the two HEXs are a constant value: (1 ) where / L L T u U U = and 0 1 L u < < . Combining Equations (4)-(7), (22) and (23) Combining Equations (13)- (15) and (24)- (25), the relationships between P and η versus L u of the irreversible SFLC with variable temperature HR can be obtained.

Conclusions
In this paper, an irreversible SFLC model with variable temperature HR is estab- (2) With the increase of

Acknowledgements:
The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.