Energetic and exergetic efficiency comparison of recuperative transcritical organic Rankine cycles under different pure and mixture fluids

Abstract

A thermodynamic model of basic and recuperative transcritical organic Rankine cycles (TRCs) associated with using five pure and six mixed fluids as working fluids has been developed. This model can be employed to investigate the effects of recuperators at inlet expander temperatures (T_exp,in) of 150–200°C and inlet expander pressures (P_exp,in) of 3.7–6.9 MPa. The ratio of change (ROC) of the first‐ and second‐law efficiencies (η_I and η_II) was positively correlated with the heat transfer rate of the recuperator and exhibited an opposite trend for a specific volume ratio. ROC was substantially affected by operating parameters and working fluid. However, the recuperator heat transfer rate was negligibly affected by the mixture temperature glide (T_glide). A universal empirical equation of η_I,II was proposed for both TRC configurations. The equation, a function of the specific volume ratio, can predict the system efficiencies for pure and mixed fluids, even if a mixture has an arbitrary mole fraction. As T_glide of R600a‐base mixture and R245fa/R134a was lower than 16 K, they had low error rates and low standard deviations. Finally, the equation was highly accurate in η_II prediction, particularly at a T_exp,in of 160°C and 170°C and a P_exp,in of 4.9–5.8 MPa.

INTRODUCTION

Since the growing industrialization of the twentieth century, energy consumption has sharply increased, and in the 21st century, environmental problems such as acid rain, ozone layer depletion, and global warming can no longer be ignored. Methods of improving energy efficiency are crucial for energy efficiency policies and environmental protection. The organic Rankine cycle (ORC) is an effective method of recovering low-grade waste heat for electricity generation. Tchanche et al.¹ reported on operational ORC power plants with capacities ranging from 125 kW to 6 MW. Numerous authors have studied ORC systems; their focuses were on the selection of organic fluids,^2,3 parameter optimization^4,5 and efficiency optimization.⁶ Among currently available renewable energy sources, geothermal energy seems to be the most attractive energy source because it is stable, unaffected by weather conditions, and easily utilized by conventional technologies. However, most geothermal resources are obtained at temperatures below 150°C and are therefore referred to as “low temperature” or “low enthalpy” geothermal resources.⁷ A low-temperature geothermal source can deliver useful energy through a binary device. Approximately 44% of existing geothermal units are binary plants due to their low average capacity, but their total energy production is less than 10% of the world's geothermal energy production.⁸

The performance comparison between sub- and trans-critical ORCs has been investigated at various heat sources. Xu et al.⁹ examined the effects of heat source temperature, critical temperature, and evaporating pressure on sub- and trans-critical ORCs. They found that the net power output of a transcritical ORC decreases rapidly with the increase of evaporating pressure when the gas temperature is close to the working fluid's critical temperature. Braimakis et al.¹⁰ found that the mixtures can have notably positive effects on cycle efficiency and output power under specific heat source temperatures. Relative to subcritical ORC, the transcritical ORC can improve exergetic efficiency.¹¹ Li et al.¹² found most research focused on the improvement of the thermal efficiency and net power output of the modified ORC configurations. Lei et al.¹³ built an ORC system to investigate the characteristics of the evaporator and condenser under different operating conditions.

From the perspective of environmental protection, carbon dioxide is a promising natural working fluid in binary cycles. Therefore, CO₂ and its mixtures have been studied by researchers for transcritical cycles. Wu et al.¹⁴ found that except for R161/CO₂ and R152a/CO₂, CO₂-based mixtures were unsuitable for transcritical power cycles with low cooling sources. Liu et al.¹⁵ investigated transcritical cycles by using CO₂/R134a with various composition ratios for waste heat recovery from diesel engines. Wang and Dai¹⁶ studied a transcritical ORC (TRC) with CO₂ and ORCs with different organic fluids as the bottoming cycles. They found that the performance level of the CO₂-TRC was superior to those of the ORCs at lower compressor pressure ratios. Dai et al.¹⁷ found that relative to pure CO₂, CO₂-based zeotropic mixtures can improve thermal performance and lower operating pressure. Pan et al.¹⁸ indicated that thermal oil outlet temperatures were affected by heating pressure and mass fraction associated with R290/CO₂ mixture in TRC. Vélez et al.^19,20 studied the use of a low-temperature heat source for power generation through a CO₂ transcritical power cycle. Their theoretical results revealed the irreversibilities of the evaporator and condenser were reduced with operating pressure because of the increase of the fluid's pump outlet temperature causing an increase in the exergetic efficiency. Gayer et al.²¹ found that a pressure level of approximately 13.5 MPa maximized the thermal and exergetic efficiencies in a CO₂ transcritical cycle without a regenerator at a pressure range from 9 to 15 MPa. Recently, operating conditions affecting turbine performance on Brayton cycles and CO₂ transcritical cycles have been investigated.^22,23 Hsieh²⁴ indicated the pseudocritical temperature of CO₂ in TRC should be increased to approach half the expander inlet temperature. Although CO₂-based mixtures have been investigated in the literature, the associated system costs were markedly high because the supercritical pressure of CO₂ is higher than that of other organic mixtures.

As expected from previous studies,^10,25,26 an internal heat exchanger (IHE), which is commonly called a recuperator, can increase thermal and exergy efficiency levels. A study reported a heat source exhaust temperature of an evaporator that was higher than that of the corresponding nonrecuperative cycle.²⁷ Consequently, an ORC adapting recuperator configuration can be used to increase power output at constant thermal input. However, brine controlled at a fixed mass flow rate cannot provide increased power output even with a recuperator.²⁷ Dong et al.²⁸ suggested that a supercritical ORC with a recuperator would be suitable for solar heating power systems with biomass. Sánchez and Silva²⁹ indicated the size of a heat exchanger (namely evaporator and condenser) can be reduced by a recuperator. Tian et al.³⁰ indicated a high impact of regenerative configurations on improving net power output, especially for split dual regenerative TRC. However, Ramy et al.³¹ indicated the recuperator had negative effects on compactness and performance. Some published studies^27,32,33 have proposed dry and isentropic fluids with ORCs that had recuperators to recover more sensible heat. Wang and Zhao³² commented on mixed fluids and suggested a recuperator with the ORC to increase system efficiency because of the effect of the mixture's condensing temperature glide. Lu et al.³⁴ indicated that the application of IHE may lead to larger, smaller, or the same net power with different temperature glides and restrictive conditions. Few studies have provided evidence on the relationships between performance and the temperature glide of a mixed fluid in a recuperator; limited evidence is available regarding the relationship between the heat transfer rate of the recuperator and temperature glide. Recently, neural network methods were used to optimize and predict the system performance of the ORC. Yang et al.³⁵ presented a back propagation neural network methodology to predict and optimize ORC performance for diesel engine waste heat recovery. Zhi et al.³⁶ presented a TRC model that optimized and predicted thermal and exergy efficiencies by using an artificial neural network (ANN) for low-grade heat recovery. Kılıç and Arabacı³⁷ presented an approach to predict the performance analysis of an ORC by using refrigerants R123, R125, R227, R365mfc, and SES36 by using an ANN and adaptive neuro-fuzzy inference system. Luo et al.³⁸ developed an approach by using ANNs to predict the working fluid properties of ORC. Therefore, predicting system performance is increasingly attracting attention. However, system performance is predicted by training and learning algorithms through well-known methods. Furthermore, the predicted method cannot provide a realistic relationship between working fluid properties and operating parameters.

Although the efficiency of a TRC/ORC can be improved by a recuperator, various effects of system parameters and types of working fluid on recuperators are still unclear, especially for TRCs with mixed working fluids. When a TRC system has a recuperator, it has complicated system piping and the construction cost of the system is higher than that of a system without a recuperator. Therefore, it is crucial to clarify how the evaporating conditions and a recuperator might affect the efficiency of TRC systems with pure and mixed working mediums. This paper presents an analysis of the influences of inlet expander pressure, inlet expander temperature, and condensing temperature for five pure and six mixed fluids. If one were to introduce a recuperator, one would typically consider the rates of change in the first- and second-law efficiencies, and the specific power output to evaluate the effects of the recuperator on TRC performance. The data of this study proved that the first- and second-law efficiencies for the organic fluids, and performance of the recuperator were correlated with specific volume ratios across the expander and dimensionless evaporating parameters of inlet expander pressure and temperature. The error and standard deviation between the predicted correlation equation and analysis data were calculated.

THEORETICAL ANALYSIS

The TRC system is schematized in Figure 1. Figure 2 illustrates noteworthy designed components and state points, in which the state points (6) and (7) are represented the outlets of the hot and cold side of the recuperator, respectively. In this study, five pure fluids (namely R245fa, R600a, R134a, R1234yf, and R290) and six mixed fluids (namely R245fa/R134a, R245fa/R1234yf, R245fa/R290, R600a/R134a, R600a/R1234yf, and R600a/R290) were used as working fluids to investigate the effects of system parameters and a recuperator on TRC performance. Figure 2 shows the T-s diagram for a TRC system, which indicates the temperature glide of the zeotropic mixtures in the condenser. The temperature glide provides a better temperature match between cooling water and working fluid during the condensing process, which means the mixed fluids can improve irreversibility relative to pure fluids. In the present study, the properties of the working fluid were processed in REFPROP 9.0³⁹ and the thermodynamic model was coded in MATLAB. The given parameters of TRCs are listed in Table 1.

Figure 1. Schematic of (A) basic and (B) recuperative TRCs. TRC, transcritical organic Rankine cycle.

Figure 2. T-s diagram of the mixture of basic and recuperative TRCs. TRC, transcritical organic Rankine cycle.

Table 1 Given parameters of TRCs

Turbine isentropic efficiency	εexp,is	80%
Pump isentropic efficiency	εpump,is	65%
Recuperator effectiveness	εrec	80%
Generator/motor efficiency	εg,m	90%
Temperature difference of pinch point in the evaporator	ΔTpp,eva	5 K
Temperature difference of pinch point in the condenser	ΔTpp,cond	5 K
Inlet temperature of the expander	Texp,in	150–200°C
Inlet pressure of the expander	Pexp,in	3.7–6.9 MPa
Temperature of geothermal fluid	Th,in	160–210°C
Mass flowrate of geothermal fluid	$${\dot{m}}_{h}$$	30 kg/s
Inlet temperature of cooling source	Tc,in	32°C
Ambient temperature	T0	32°C

Abbreviation: TRC, transcritical organic Rankine cycle.

In a typical system using a TRC, the working fluid achieves a supercritical state because it is pressurized by a pump. A large pressure difference exists between the evaporator and the condenser. The system's power source (typically a generator) supplies the requisite quantities of power to the working fluid pump. The net thermal efficiency (first-law efficiency) can be approximated by $${\eta }_{I}$$ by following the formula: [Image Omitted. See PDF] where $${\dot{W}}_{\mathrm{net}}$$ is the shaft power of the expander minus the working fluid pump power consumption. The heat transfer equations of the evaporator ($${Q}_{\mathrm{eva}}$$) for basic and recuperative TRC systems are given in Table 2.

Table 2 Mathematical models of the components for basic and recuperative TRCs

Items	Basic TRC	Recuperative TRC
Heat transfer rate in evaporator	$${\dot{Q}}_{\mathrm{eva}}={\dot{m}}_{{\rm{f}}}({h}_{3}-{h}_{2})$$	$${\dot{Q}}_{{\rm{eva}}}={\dot{m}}_{f}({h}_{3}-{h}_{7})$$
Heat transfer rate in condenser	$${\dot{Q}}_{\mathrm{cond}}={\dot{m}}_{f}({h}_{4}-{h}_{1})$$	$${\dot{Q}}_{\mathrm{cond}}={\dot{m}}_{{\rm{f}}}({h}_{6}-{h}_{1})$$
Heat transfer rate in recuperator	NA	$${\dot{Q}}_{\mathrm{rec}}={\dot{m}}_{{\rm{f}}}({h}_{4}-{h}_{6})={\dot{m}}_{f}({h}_{7}-{h}_{2})$$
Mass flowrate of working fluid	$${\dot{m}}_{{\rm{f}}}=\frac{{\dot{m}}_{{\rm{h}}}({h}_{\mathrm{in}}-{h}_{\mathrm{out}})}{{h}_{3}-{h}_{2}}$$	$${\dot{m}}_{{\rm{f}}}=\frac{{\dot{m}}_{{\rm{h}}}({h}_{\mathrm{in}}-{h}_{\mathrm{out}})}{{h}_{3}-{h}_{7}}$$

Abbreviation: TRC, transcritical organic Rankine cycle.

To consider the energy flow in the system and its components, we used the continuity and energy equations for a control volume in the steady state. The corresponding equations are as follows: [Image Omitted. See PDF] [Image Omitted. See PDF]

The power consumed in the pumping process between points 1 and 2 is [Image Omitted. See PDF] [Image Omitted. See PDF]

The shaft work of the expander between points 3 and 4 are defined as [Image Omitted. See PDF] [Image Omitted. See PDF]

The mass flow rate of the cooling source in the condenser is calculated by [Image Omitted. See PDF]where the mass flowrates of the working fluid and heat transfer rates of the condenser for two TRC configurations are listed in Table 2.

The outlet temperature of a recuperator at a given condensing pressure is determined by [Image Omitted. See PDF]

The temperature of stat 6 can be determined by using h₆ at a given condensing pressure. [Image Omitted. See PDF]

Generally, the outlet temperature of the heat source in a TRC evaporator cannot decrease to the reference temperature, and only a fraction of the exergy from the heat source can be transferred into the TRC system; that transferred exergy can be expressed as [Image Omitted. See PDF]

The second law efficiency of the system can be determined by [Image Omitted. See PDF]

To consider the specific power, we express the net power output per unit mass flow rate of the heat source as follows⁴⁰: [Image Omitted. See PDF]

where $${\dot{P}}_{\mathrm{net}}$$ is [Image Omitted. See PDF]

To validate the thermodynamic analysis model developed in the present study, this paper presents a model solution with the same operating conditions and working fluid as those in Maraver and colleagues.^27,41,42 A basic TRC with R134a as a pure working fluid is computed to compare the net shaft power of the expander and η_I, η_II, and $${\dot{{\rm{I}}}}_{\text{tot}}$$ of the TRC model at a heat source outlet temperature of 70.3°C. According to the validation results, the relative deviation in the net shaft work, first-law efficiency, second-law efficiency, and irreversibility are 1.48%, 1.08%, 0.28%, and 0.44%, respectively, as listed in Table 3. Additionally, the basic model was validated in our previous study.⁴¹ The first-law efficiency of R218 in the present study and our previous experimental study are respectively 2.5% and 2.63%, for which the relative deviation is 4.93%. The relative deviation is higher than that of other analyses because our experimental thermal efficiency uncertainty was ± 4.5%. The recuperative TRC model can also be compared with that in Le et al.⁴²; the comparison results are shown in Table 4.

Table 3 Validation of the basic TRC model for pure working fluid R134a.

	$${\dot{W}}_{\mathrm{net}}$$ (kW)	ηI (%)	ηII (%)	$${\dot{I}}_{\mathrm{tot}}$$ (kW)
Le et al.42	4.7	14.0	57.7 (ηex)	3.4
Present study	4.77	14.15	57.53	3.415
Relative deviation	1.48%	1.08%	0.28%	0.44%

Abbreviation: TRC, transcritical organic Rankine cycle.

Table 4 Validation of the recuperative TRC model for pure working fluid R245fa.

	ηI (%)	ηII (%)
Maraver et al.27	21.7	56
Present study	20.98	57.25
Relative deviation	3.37	2.23

Abbreviation: TRC, transcritical organic Rankine cycle. RESULTS AND DISCUSSION

We developed a thermodynamic TRC model in MATLAB software to investigate the effects of a recuperator on working fluids at various expander inlet temperatures and pressures. With this aim, we analyzed the pure fluids R245fa, R134a, R290, R600a, and R1234yf as well as the mixed fluids R245fa/R134a, R245fa/R1234yf, R245fa/R290, R600a/R134a, R600a/R1234yf, and R600a/R290.

The optimal mole fractions of the mixtures corresponding to the maximal specific power for basic and recuperative TRC at various T_exp,in values at a given P_exp,in are listed in Tables 5 and 6, respectively. When T_exp,in is low, the proportions of R245fa and R600a samples with optimal mole fractions are lower than those of R134a, R1234yf, and R290. This is because the maximal specific power occurs for the range of temperature difference between T_exp,in and T_cri ranging from 30°C to 80°C under the effects of condensing temperature glide. Therefore, the optimal mole fraction of R245fa- and R600a-base mixtures must change with T_exp,in. Additionally, the optimal mole fraction for recuperative TRC is the same as that for basic TRC. The optimal mole fraction corresponding to maximal specific power is not substantially affected by P_exp,in. However, as shown in Figure 3, the efficiencies and specific power are considerably improved by P_exp,in because the pressure minimizes the exergy destruction of the evaporator and condenser. Except for R245fa/R290 and R600a/R1234yf, the peak values of the second law efficiency and specific power are observed at a specific mole fraction of the mixtures. However, as shown in Figure 3, the maximal first-law efficiency of the thermodynamics of the mixtures only occurs at X_R245fa and X_R600a = 1.

Table 5 Mole fractions of the mixtures corresponding to the maximal specific power at various T_exp,in values in basic TRCs (P_exp,in = 5.6 MPa).

	Texp,in
Fluids	150°C	160°C	170°C	180°C	190°C	200°C
R245fa/R134a
ζ (kJ/kg)	40.53	47.97	55.02	61.72	71.06	80.49
$${\eta }_{|}\unicode{x02007}( \% )$$	11.67	12.00	12.20	12.33	14.65	14.96
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	57.21	57.29	56.70	55.74	59.33	59.33
X	0.17/0.83	0.17/0.83	0.17/0.83	0.17/0.83	0.95/0.05	0.95/0.05
ΔTexp_cri (°C)	41.80	51.80	61.80	28.74	38.74	48.74
R245fa/R1234yf
ζ (kJ/kg)	40.12	46.64	52.86	58.78	71.20	80.37
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	10.95	11.29	11.48	11.59	14.64	14.98
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	55.91	55.52	54.57	53.31	59.40	59.32
X	0.2/0.8	0.2/0.8	0.2/0.8	0.2/0.8	0.97/0.03	0.97/0.03
ΔTexp_cri (°C)	44.90	54.90	64.90	74.90	38.76	48.76
R245fa/R290
ζ (kJ/kg)	38.87	45.06	50.94	57.83	69.69	78.48
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	10.93	11.13	11.26	14.00	14.85	15.24
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	54.65	53.94	52.82	60.27	59.12	58.96
X	0.42/0.58	0.42/0.58	0.42/0.58	1/0	1/0	1/0
ΔTexp_cri (°C)	42.24	52.24	62.24	25.99	35.99	45.99
R600a/R134a
ζ (kJ/kg)	39.53	45.65	51.45	60.07	67.82	75.3
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	10.83	11.06	11.20	13.23	13.44	13.54
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	55.10	54.36	53.18	56.57	56.06	55.22
X	0.35/0.65	0.35/0.65	0.35/0.65	0.94/0.06	0.94/0.06	0.94/0.06
ΔTexp_cri (°C)	50.85	60.85	70.85	47.27	57.27	67.27
R600a/R1234yf
ζ (kJ/kg)	38.67	45.52	52.63	59.78	66.93	74.05
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	11.10	11.78	12.31	12.78	13.15	13.55
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	55.42	55.69	55.81	55.55	55.18	54.67
X	0.45/0.55	0.29/0.71	0.37/0.63	0.48/0.52	0.59/0.41	0.78/0.22
ΔTexp_cri (°C)	30.68	47.52	53.91	59.55	65.79	70.36
R600a/R290
ζ (kJ/kg)	38.23	45.12	52.32	60.26	67.90	75.18
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	11.36	11.63	12.95	13.23	13.38	13.46
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	55.11	54.90	56.80	56.63	55.98	55.01
X	0.26/0.74	0.26/0.74	0.81/0.19	0.81/0.19	0.81/0.19	0.81/0.19
ΔTexp_cri (°C)	41.38	51.38	40.85	50.85	60.85	70.85

Abbreviation: TRC, transcritical organic Rankine cycle.

Table 6 Mole fractions of the mixtures corresponding to the maximal specific power at various T_exp,in values in recuperative TRCs (P_exp,in = 5.6 MPa).

Fluids	Texp,in
	150°C	160°C	170°C	180°C	190°C	200°C
R245fa/R134a
ζ (kJ/kg)	39.87	46.87	53.46	59.70	69.09	77.64
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	12.96	13.94	14.73	15.41	16.93	18.05
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	59.94	60.97	61.07	60.65	63.20	64.29
X	0.17/0.83	0.17/0.83	0.17/0.83	0.17/0.83	0.95/0.05	0.95/0.05
ΔTexp_cri (°C)	41.80	51.80	61.80	28.74	38.74	48.74
R245fa/R1234yf
ζ (kJ/kg)	39.26	45.28	51.01	56.43	69.26	77.53
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	12.71	13.81	14.68	15.40	16.88	18.07
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	59.33	60.13	60.09	59.56	63.18	64.30
X	0.2/0.8	0.2/0.8	0.2/0.8	0.2/0.8	0.97/0.03	0.97/0.03
ΔTexp_cri (°C)	44.90	54.90	64.90	74.90	38.76	48.76
R245fa/R290
ζ (kJ/kg)	37.99	43.79	49.26	57.00	67.96	75.85
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	12.87	13.70	14.39	14.95	17.01	18.33
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	45.77	46.41	46.64	61.69	63.02	64.38
X	0.42/0.58	0.42/0.58	0.42/0.58	1/0	1/0	1/0
ΔTexp_cri (°C)	42.24	52.24	62.24	25.99	35.99	45.99
R600a/R134a
ζ (kJ/kg)	38.61	44.3	49.67	58.12	65.16	71.95
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	12.81	13.72	14.47	16.02	17.01	17.82
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	59.10	59.34	58.96	61.70	62.27	62.26
X	0.35/0.65	0.35/0.65	0.35/0.65	0.94/0.06	0.94/0.06	0.94/0.06
ΔTexp_cri (°C)	50.85	60.85	70.85	47.27	57.27	67.27
R600a/R1234yf
ζ (kJ/kg)	38.39	44.99	51.05	57.67	64.26	70.82
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	11.62	12.65	15.02	16.08	17.08	17.83
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	56.98	55.59	58.16	56.66	55.66	56.70
X	0.45/0.55	0.87/0.13	0.41/0.59	0.5/0.5	0.63/0.37	0.78/0.22
ΔTexp_cri (°C)	30.68	28.17	52.25	58.83	64.55	70.36
R600a/R290
ζ (kJ/kg)	37.47	43.93	50.96	58.24	65.23	71.87
$${\eta }_{{\rm{|}}}\unicode{x02007}( \% )$$	12.99	13.94	15.11	16.21	17.07	17.79
$${\eta }_{{\rm{\Vert }}}\unicode{x02007}( \% )$$	57.23	57.35	56.64	57.48	57.63	57.37
X	0.26/0.74	0.26/0.74	0.81/0.19	0.81/0.19	0.81/0.19	0.81/0.19
ΔTexp_cri (°C)	41.38	51.38	40.85	50.85	60.85	70.85

Abbreviation: TRC, transcritical organic Rankine cycle.

Figure 3. First- and second-law efficiencies, and specific power of the mixtures at various Pexp,in values and a Texp,in of 200°C.

Figure 4 shows that the heat transfer rate of the recuperator for R245fa/R1234yf and R600a/R290 varied with mole fraction at a given recuperator effectiveness. The results indicate that the temperature glide varies parabolically with the mole fraction of the mixtures. However, the heat transfer rate and condensing pressure linearly decrease with an increase in the mole fraction of a high-critical-temperature working fluid (namely R245fa and R600a). The condensing pressure is slightly affected by the temperature glide. The heat transfer rate may not be relevant to the temperature glide. In addition, the heat transfer rate can be adjusted by controlling the mole fraction of the mixtures in relation to change in condensing pressure. The heat transfer rate of X_{R245fa, R600a} = 1 is smaller than that of X_{R1234yf, R290} = 1. As previously discussed, the condensing pressure of the working fluid is related to the heat transfer rate of the recuperator.

Figure 4. Heat transfer rate of the recuperator, condensing temperature glide, and condensing pressure of the mixtures (A) R245fa/R134a and (B) R600a/R290 at a Texp,in of 200°C under various mole fractions.

The comparison of heat transfer rate between the pure and mixtures at various P_exp,in values is shown in Figure 5. For all of the working fluids, the heat transfer rate is considerably decreased with an increase of P_exp,in. This is because the enthalpy difference between the expander outlet and dew point in the condenser is reduced by P_exp,in. The heat transfer rate of the mixtures shows a similar trend as that with pure fluids, as shown in Figure 5B. Owing to a condensing pressure effect, R245fa and R1234yf respectively have the lowest and peak value heat transfer rate at a given P_exp,in. Because the optimal mole fraction of R245fa/R290 at T_exp,in of 200°C is 1/0, which involves pure R245fa rather than a mixed fluid, it is replaced by 0.42/0.58, which is the optimal mole fraction at T_exp,in of 170°C. The ratio of change (ROC) is a useful performance indicator for recuperators. To understand the effects of recuperators for different working fluids at various T_exp,in values, one can calculate the ROC for the specific power, first- and second-law efficiencies as follows: [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF]

Figure 5. Recuperator heat transfer rate of (A) pure fluids and (B) mixtures at Texp,in = 200°C and Tc,in = 32°C.

ROC_I and ROC_II decrease with an increase of P_exp,in. The trends of these values resemble the aforementioned heat transfer rate of the recuperator, as graphed in Figures 6 and 7. However, ROC_SP demonstrates a trend opposite to that of ROC_I,II. This is because the working fluid's mass flow rate of the TRC is reduced by the recuperator, especially for a constant mass flow rate of any heat source. Therefore, the recuperator exerts a negative influence on ROC_SP; this result agrees with those in Algieri and colleagues.^25,27 According to the discussion in Algieri and Morrone,²⁵ although the output power is reduced by modifying the recuperator, the specific power is independent of the cycle configuration at a fixed thermal power input. In the present study's model, the working fluid's mass flow rate is reduced at a constant mass heat source flow rate obtained through a high exhaust hot fluid temperature in the evaporator. This results from the working fluid being preheated by the recuperator. Additionally, the ROC_SP is increased with an increase in P_exp,in because the working fluid's mass flow rate increases (due to a decrease in the enthalpy difference between the inlet and outlet in evaporator) at a given T_exp,in. In addition, for both pure and mixed fluids, ROC_I,II values are increased by over 30% when T_exp,in is increased from 150°C to 200°C because heat input from the heat source in the evaporator increases with T_exp,in at a given ΔT_pp,eva. However, the ROC_I,II of R245fa/R1234yf have the highest value at a T_exp,in of 180°C. This is because the optimal mole fractions of R245fa/R1234yf corresponding to the maximal specific power output are 0.2/0.8 at 180°C and 0.97/0.03 at 200°C. This maintains a temperature difference—the expander inlet temperature minus the critical temperature—between 30°C and 80°C, as presented in Tables 5 and 6. As previously discussed, the heat transfer rate of the recuperator is affected by the mole fraction of the mixture. Therefore, ROC_I,II for R245fa/R1234yf have a negative trend at high T_exp,in values because of increases in the proportion of R245fa.

Figure 6. Effect of inlet expander temperature on ROCI, ROCII, and ROCSP of pure fluid (A) R134a and (B) R600a.

Figure 7. Effect of inlet expander temperature on ROCI, ROCII, and ROCSP of mixtures (A) R245fa/R1234yf and (B) R600a/R134a.

To further elucidate the effect of the recuperator at various condensing temperatures, we present the ROC of specific power, first- and second-law efficiencies for R245fa/R1234yf at optimal mole fractions in Figure 8. The condensing temperature is defined as follows: [Image Omitted. See PDF]where $${\unicode{x02206}T}_{c}$$ represents the temperature difference between the inlet and outlet temperatures of a cooling source fixed at 5 K.

Figure 8. Effect of the cooling source temperatures on (A) first-law efficiency and ROCI, (B) second-law efficiency and ROCII, and (C) the specific power and ROCSP of R245fa/R1234yf (0.97/0.03) at Texp,in = 200°C.

The results indicate that with an increase in T_c,in, the system performance decreases significantly but ROC_I,II,SP slightly increases. Although a recuperator is most suitable for high condensing temperatures, the effect of condensing temperature on the recuperator is small, especially when P_exp,in is ≥6.3 MPa. This is because the heat transfer rate of recuperator is low at high P_exp,in values, which has a negligible effect on performance. However, the specific power and ROC_SP linearly increase with an increase in P_exp,in. In fact, the Δh_3-2 and Δh_4-5 decrease with an increase in P_exp,in. As a result, the working fluid mass flow rate increases, and the heat transfer rate of the recuperator declines.

To demonstrate the relationship between the thermal properties of the working fluid and the effect of the recuperator, comparisons of ROC_I,II with the specific volume ratio of pure and mixed fluids are graphed in Figures 9 and 10. ROC_I,II and ν₃/ν₄ decrease and increase, respectively, with an increase in P_exp,in. Among pure and mixed fluids, ROC_I,II values for R245fa and R245fa-based mixtures are the lowest; however, such fluids have the highest ν₃/ν₄ values. For high values of ν₃/ν₄, which are affected by P_exp,in and working fluid, the recuperator has a relatively small effect on system efficiency. Moreover, for R134a, R1234yf, and R290, the ν₃/ν₄ is low and approaches the same value; thus, it is difficult to observe the effects of their specific volume ratios on the recuperator (as shown in Figure 9). Figure 10 shows that the ROC_I,II and ν₃/ν₄ of the optimal mole fraction mixtures under the effect of condensing temperature glide. The optimal mole fraction of R245fa/R290 (1/0) at a T_exp,in of 200°C is replaced by the optimal mole fraction of R245fa/R290 (0.42/0.58) at a T_exp,in of 170°C. The results indicate that ROC_I,II decrease with an increase in P_exp,in. In the present study, R245fa and R600a (the high values of ν₃/ν₄) were mixed with R134a, R1234yf and R290 (the low values of ν₃/ν₄) to cause obvious changes of ROC and ν₃/ν₄. If one compares Figure 9 with Figure 10, it can be observed that R245fa/R1234yf of the mixture exhibits a 30% decrease in ROC_I and a 12% decrease in ROC_II relative to pure R1234yf. Notably, the ROC_I,II values of R245fa/R1234yf are close to that of pure R245fa because of the notably low proportion of R1234yf in the mixture.

Figure 9. (A) ROCI, (B) ROCII, and (C) specific volume ratio of pure fluids at various inlet expander pressure values (Texp,in = 200°C).

Figure 10. (A) ROCI, (B) ROCII and (C) specific volume ratio of mixed fluids at various inlet expander pressure values (Texp,in = 200°C).

On the basis of the aforementioned results, η_I,II, ROC_I,II, and ν₃/ν₄ are affected considerably by T_exp,in and P_exp,in. Therefore, the first- and second-law efficiencies for pure fluids and optimal mole fraction of mixtures at basic and recuperative TRCs can be expressed together as a single function of the specific volume, inlet temperature, and inlet pressure of the expander as follows: [Image Omitted. See PDF]where η represents the η_I and η_II of pure and mixed fluids. Equation (18) was derived using PS Imago Pro 6.0 (IBM SPSS Statistics).⁴³ The constants and powers of Equation (18) are listed in Tables 7 and 8. Meanwhile, the standard and maximal deviations calculated with the thermodynamic analysis data and correlation equation is shown in Tables 7 and 8. A comparison of the aforementioned deviations proves that the predicted η_II is more accurate than that of the predicted η_I. In addition, the ranges of P_exp,in and T_exp,in of Equation (18) are listed in Tables 9 and 10.

Table 7 Constants, coefficients, and deviations of the empirical correlation equation for pure fluids

	a	b	c	d	e	f	g	m	R2	Std. dev. (%)	Max. dev. (%)
$${\eta }_{\mathrm{I\_Basic}}$$	4.23	−0.02	−4.6	−9.46	−10.37	−9.13	22.56	0.74	0.91	4.5	9.4
$${\eta }_{\mathrm{II\_Basic}}$$	73.14	0.35	−12.15	−55.06	−27.63	−49.5	80.5	0.768	0.92	2.84	7.1
$${\eta }_{\mathrm{I\_REC}}$$	−7.12	−0.12	−1.59	−13.42	−25.67	−11.52	33.1	0.74	0.83	4.8	15.4
$${\eta }_{\mathrm{II\_REC}}$$	38.2	0.22	−3.98	−54.42	−47.95	−44.01	98.15	0.67	0.89	2.0	7.0

Table 8 Constants, coefficients, and deviations of the empirical correlation equation for the six mixtures

	a	b	c	d	e	f	g	m	R2	Std. dev. (%)	Max. dev. (%)
$${\eta }_{\mathrm{I\_Basic}}$$	32.71	−0.19	−12.73	−17.62	−12.65	−20.46	22.32	1.09	0.92	3.14	10.1
$${\eta }_{\mathrm{II\_Basic}}$$	81.6	0.47	−15.54	−41.55	−9.31	−43.06	63.45	0.77	0.85	2.05	5.8
$${\eta }_{\mathrm{I\_REC}}$$	57.25	−0.4	−17.96	−44.4	−42.1	−45.33	48.19	1.09	0.88	4.23	14.2
$${\eta }_{\mathrm{II\_REC}}$$	153.26	−0.08	−31.48	−89.22	−54.63	−88.26	95.15	1	0.85	1.65	8.4

Table 9 Operating parameters of the empirical correlation equation for five pure fluids

Fluids	Pexp,in (MPa)	Texp,in (°C)
R245fa	3.9–4.5	170a
	3.9–5.5	180a
	3.9–6.9	190–200
R600a	3.7–4.4	150a
	3.7–5.3	160a
	3.7–6.8	170a
	3.7–6.9	180–200
R1234yf	3.7–6.9	150–200
R134a	4.1–6.9	150–200
R290	5–6.9	150–200

a

At a given T_exp,in the P_exp,in cannot exceed a certain level to avoid liquid droplet formation of the working fluid in the expander. The condition at the expander outlet is superheat vapor.

Table 10 Operating parameters of the empirical correlation equation for the mixtures

Fluid	Pexp,in (MPa)	Texp,in (℃)
R245fa/R290	5–6.9	150–200
R245fa/R1234yf	3.9–6.9	150–200
R245fa/R134a	4.1–6.9	150–200
R600a/R290	5–6.9	150–200
R600a/R1234yf	3.7–6.9	150–200
R600a/R134a	4.1–6.9	150–200

Figures 11 and 12 illustrate that η_I and η_II calculated by the empirical correlation equation in basic and recuperative TRCs can be contrasted with thermodynamic analysis data under various mole fractions. To evaluate the accuracy of the predicted equation, the error and standard deviation between the correlation and analysis data were considered (standard deviation is listed in Table 11). The error of η_I and η_II can be expressed as follows: [Image Omitted. See PDF]

Figure 11. Effect of the mixed fluids' mole fraction on predicted correlation, analysis data, and error of ηI at Pexp,in = 6 MPa and Texp,in = 200°C.

Figure 12. Effect of the mixed fluids' mole fraction on predicted correlation, analysis data, and error of ηII at Pexp,in = 6 MPa and Texp,in = 200°C.

Table 11 Standard deviation of error between the correlation and analysis data

	R245fa/R290	R245fa/R1234yf	R245fa/R134a	R600a/R290	R600a/R1234yf	R600a/R134a
$${\eta }_{\mathrm{I\_Basic}}$$	7.72	5.27	3.3	2.2	1.87	5.2
$${\eta }_{\mathrm{II\_Basic}}$$	7.08	1.75	1.91	0.85	1.14	2.8
$${\eta }_{\mathrm{I\_Rec}}$$	11.97	10.96	6	4.65	5.32	8.7
$${\eta }_{\mathrm{II\_Rec}}$$	10.32	5.61	3.4	1.6	1.98	5.8

Figure 11 shows that for η_I, the recuperative TRC errors vary abruptly with mole fractions that have any degree of variation higher than that of basic TRCs. Table 11 also shows the standard deviation for the recuperative TRC is augmented by the recuperator. A comparison of Figures 11 and 12 shows that the errors of η_II for basic and recuperative TRC are lower with respect to that of η_I, except for R245fa/R290 and R245fa/R1234yf. The errors of η_I and η_II at half mole fraction (namely, X_{R245fa, R600a} = 0.5) approach 0%, except for R245fa/R290 and R245fa/R1234yf of recuperative TRC. This is because of abrupt variation of the system performance at low and high proportions of the X_R245fa,R600a under the effects of condensing temperature glide (T_glide). Furthermore, the error of η_I is highly related to the T_glide of the mixtures because they have similar trends with various mole fractions, as shown in Figure 13. When the T_glide value is increased, the error varies dramatically, especially when T_glide is ≥16 K. Figure 13 and Table 11 also show that the variation of error associated with a recuperator is higher than that of a basic TRC; this results in a high standard deviation. According to the analysis results, the error of the predicted equation is considerably affected by condensing temperature glide and the recuperator. Additionally, although the correlation is developed from the data of the optimal mole fraction corresponding to the maximal specific power, it can directly and accurately predict the system efficiency of an arbitrary mole fraction, especially that for which η_II is of basic TRC.

Figure 13. Error of ηI and condensing temperature glide of mixtures at Pexp,in = 6 MPa and Texp,in = 200°C.

Figures 14–16 indicate the effects of T_exp,in on the errors of η_I and η_II for pure and mixed fluids at various P_exp,in. Results reveal that the errors of η_I,II are varied dramatically by P_exp,in at a T_exp,in of 150°C and 200°C, except for R600a/R134a. At a T_exp,in of 160°C and 170°C, the fluctuation of the error is small because of the insignificant effect of P_exp,in. A comparison between Figures 15 and 16 reveals that the fluctuation in errors of η_II for recuperative TRC is lower than that of η_I, with fluctuation in the errors of η_II as intervals between −4% and 6%, except for R600a/R134a and R600a/R1234yf. Figures 14–16 illustrate that the η_I,II errors of pure and mixed fluids under different T_exp,in values intersect at a point in which the intersections of η_II are in a P_exp,in the range of 4.9–5.8 MPa and the error values of the intersection are close to 0%, except for R600a/R134a. Furthermore, the P_exp,in for the intersection of the η_I error are varied by working fluid. Fluctuations with T_exp,in in the error of η_I,II for pure and mixed fluids have small values at a low P_exp,in, particularly when P_exp,in approaches the intersection.

Figure 14. Effect of the expander inlet temperature on the error of ηI and ηII for pure fluids with a recuperator

Figure 15. Effect of the expander inlet temperature on the error of ηI for mixed fluids with a recuperator at mole fraction 0.2/0.8

Figure 16. Effect of the expander inlet temperature on the error of ηII for mixed fluids with a recuperator at mole fraction 0.2/0.8

CONCLUSIONS

In the present study, the specific power, first- and second-law efficiencies of the basic and recuperative TRC with the use of the pure and zeotropic fluids as working fluids were examined using a thermodynamic method at various cooling temperatures, inlet expander pressures, and temperatures to investigate the effects of a recuperator on system efficiency. The optimal mole fractions of the mixtures corresponding to specific powers were explored. An empirical correlation equation was developed through an analysis of data of optimal mole fractions. The conclusions are summarized as follows:

• 1.

  The heat transfer rate of the recuperator is reduced with an increase in X_{R245fa, R600a} and P_exp,in. However, with an increase in T_exp,in, the associated heat transfer rate can be substantially improved. Moreover, the aforementioned heat transfer rate is negligibly affected by T_glide. Additionally, the recuperating performance indicator (ROC) associated with the first- and second-law efficiencies was defined to evaluate the effect of the operating parameters and working fluids on the recuperator. The effect of the heat transfer rate is also reflected by ROC_I,II; they have similar trends under various operating parameters and working fluids. However, ROC_SP exhibits a trend opposite to that of ROC_I,II because the recuperator reduces the working fluid's flow rate, especially for a constant heat source flowrate. Additionally, ROC_I,II and ROC_SP slightly increase with an increase in T_c. However, the effect of T_c on ROC_I,II is small, especially when P_exp,in is higher than 6.3 MPa.

• 2.

  Among pure and mixed fluids, ROC_I,II for R245fa and R245fa-based mixtures have the lowest; however, they have the highest ν₃/ν₄ values. Moreover, for R134a, R1234yf, and R290, the ν₃/ν₄ values are low, which resulted in their ROC_I,II values being higher than those of R245fa, R600a, R245fa-, and R600a-based mixtures. The ν₃/ν₄ and ROC_I,II increase and decrease, respectively, with an increase in P_exp,in. According to the aforementioned discussion, ν₃/ν₄ and ROC_I,II have opposite trends for P_exp,in. In fact, the ROC_I,II values in optimal mole fractions of the mixtures are smaller than those for R1234yf, R134a, and R290; however, ROC_I,II values of mixtures are close to those of pure R245fa and R600a, except R245fa/R290. This is because of the notably low proportions of R1234yf, R134a, and R290 in optimal mole fractions of the mixtures at a certain high T_exp,in.

• 3.

  On the basis of the analysis of data regarding pure and optimal mole fractions in mixtures, a universal empirical correlation equation of system efficiency is proposed. The values of η_I and η_II can be directly and accurately predicted by the equation, especially in the case with basic TRC and low T_glide of the mixture. The predicted accuracy values of the empirical correlation equation range from high to low as follows: η_{II_basic}, η_{I_basic}, η_{II_Rec}, and η_{I_Rec}. Additionally, R245fa/R134a and R600a-based mixtures have low standard deviations because their condensing temperature glides are lower than 16 K.

• 4.

  Errors of the η_I and η_II are significantly affected by the P_exp,in value, particularly for T_exp,in at 150°C, 180°C, 190°C, and 200°C. Furthermore, the fluctuation in errors of η_II for recuperative TRC is lower than that of η_I, with fluctuation in the errors of η_II as intervals between −4% and 6%. Therefore, the empirical equation is highly accurate in predicting η_II at a T_exp,in of 160°C and 170°C. In addition, the error of η_I and η_II at different T_exp,in values intersects at a point; however, the η_II error in the intersection is close to 0% at a P_exp,in range of 4.9–5.8 MPa.

• 5.

  Although the equation is correlated at fixed values of 0.8 and 0.65 of the expander and pump isentropic efficiencies, respectively, they are widely used in the thermodynamic analysis. However, in future work, the effects of isentropic efficiency in the expander and pump will be considered in the equation to improve the applied range.

NOMENCLATURE

• $$h$$
• specific enthalpy (kJ/kg)
• $$\dot{I}$$
• irreversibility (kW)
• $$\dot{m}$$
• mass flowrate (kg/s)
• $$P$$
• pressure (MPa)
• $$\dot{P}$$
• electrical power (kW)
• $$\dot{Q}$$, Q
• heat transfer rate (kW)
• ROC
• ratio of change
• $${\rm{s}}$$
• specific entropy (kJ/kg K)
• $$T$$
• temperature (°C, K)
• $$\dot{W}$$
• power (kW)
• $${\rm{X}}$$
• mole fraction of mixture

GREEK LETTERS
• $$\varepsilon ,\eta $$
• effectiveness and efficiency (%)
• $${\rm{\zeta }}$$
• specific power (kJ/kg)
• ν
• specific volume (m³/kg)
• $$\unicode{x02206}\dot{E}$$
• available exergy (kW)
• $$\unicode{x02206}T$$
• temperature difference (K)

SUBSCRIPT
• 0
• environment state
• 1–7
• state point
• Basic
• Basic
• c
• cooling source
• in/out
• inlet/outlet
• cond
• condenser/condensation
• cri
• critical point
• eva
• evaporator/evaporation
• exp
• expander
• f
• working fluid
• g
• generator
• glide
• glide
• h
• heat source
• h,in/h,out
• heat source inlet/outlet
• I/II
• first/second-law of thermodynamics
• is
• isentropic
• in/out
• inlet/outlet
• m
• motor
• net
• net
• pp
• pinch point
• pump
• pump
• rec, REC, Rec
• recuperator
• tot
• total

ACKNOWLEDGMENTS

The financial support provided to this study by the Ministry of Science and Technology of the Republic of China under MOST 108-2221-E-167-007-MY3 and MOST 110-2622-E-006-001-CC1 is gratefully acknowledged.