Investigation of a Novel CO₂ Transcritical Organic Rankine Cycle Driven by Parabolic Trough Solar Collectors

Abstract

The objective of the present study is the detailed investigation and optimization of a transcritical organic Rankine cycle operating with CO₂. The novelty of the present system is that the CO₂ is warmed up inside a solar parabolic trough collector and there is not a secondary circuit between the solar collector and the CO₂. Therefore, the examined configuration presents increased performance due to the higher operating temperatures of the working fluid in the turbine inlet. The system is studied parametrically and it is optimized by investigating different pressure and temperature level in the turbine inlet. The simulation is performed with a validated mathematical model that has been developed in Engineering Equation Solver software. According to the results, the optimum turbine inlet temperature is ranged from 713 up to 847 K, while the higher pressure in the turbine inlet enhances electricity production. In the default scenario (turbine inlet at 800 K and turbine pressure at 200 bar), the system efficiency is found 24.27% with solar irradiation at 800 W/m². A dynamic investigation of the system for Athens (Greece) climate proved that the yearly efficiency of the unit is 19.80%, the simple payback period of the investment is 7.88 years, and the yearly CO₂ emissions avoidance is 48.7 tones.

1. Introduction

Solar energy is a promising renewable energy source that can be used for facing critical problems like the increasing energy needs [1], fossil fuel depletion [2] and the decarbonization of our society [3]. There are various solar collectors like typical flat plate collectors [4], improved flat plate collectors [5], heat pipe solar collectors [6] and solar passive systems in buildings [7]. Solar concentrating technology is an interesting idea for producing thermal energy at high-temperature levels and so to feed various energy systems with high exergy rates [8]. Therefore, there is a possibility for producing different useful outputs like electricity [9], space-heating [10], cooling [11], hydrogen [12], desalination [13], etc. Among them, electricity production is the most usual and attractive output because it can cover the greatest part of human energy needs. Practically, the electricity can be converted into various other energy types and so it has high importance.

Another important aspect of future energy systems is the utilization of new and environmentally friendly working fluids [14]. Among them, CO₂ seems to be an ideal choice because it is a natural refrigerant that is non-toxic, non-flammable and not expensive [15]. CO₂ presents zero ozone depletion potential and a global warming potential equal to one which is an ideal value. However, CO₂ has a low critical temperature at 31 °C that makes the power cycle operates at transcritical or supercritical designs, something that creates the need for investigating more artificial power cycles than the usual [16].

In the literature, there are numerous investigations that investigate the use of CO₂ in various energy systems and especially for power. This working fluid can be used in the Brayton and in the Rankine cycle. In the Brayton cycle, the main devices are a compressor, a burning chamber, a turbine and a gas cooler. In the Rankine cycle, there is a liquid pump, a boiler (or a heat recovery system), a turbine and a condenser. The main difference between these cycles is the existence of a gas compressor in the Brayton cycle, while the Rankine cycle includes a liquid pump. The compressor consumes high amounts of work for the compression, while the pump consumes small amounts of work.

The use of a transcritical Rankine cycle is a very common configuration in the literature, while there are also studies about Brayton systems. Wang et al. [17] studied a simple transcritical ORC with CO₂ as the working fluid. They performed parametric work and they used genetic algorithms and artificial neural networks. According to their results, the thermodynamic efficiency of the cycle is about 29.5% with turbine pressure at 107 bar and temperature in the turbine inlet at 75 °C. Shu et al. [18] studied a waste heat recovery system that includes a transcritical CO₂ ORC system that is fed by the exhaust gases of an internal combustion engine. They compared a simple cycle and an advanced cycle with a regenerator and preheater. They found a 150% electricity production increase with the advanced systems, the fact that proves the need for recovering as much as heat is possible in a transcritical cycle. Kim et al. [19] examined different CO₂ power cycles and they found that the low-temperature transcritical Rankine cycle leads to 12% efficiency while the high temperature to 40%, so they found the great importance of the heat source temperature level on the results. In another work about a complex configuration, Cao et al. [20] studied a unit with an air-based gas turbine as the prime mover with a supercritical CO₂ Brayton cycle as the intermediate cycle and a transcritical CO₂ Rankine cycle as the last bottoming cycle. The overall system electricity efficiency is found to be 52.33% which is a relatively high value.

The next part of the literature regards investigations about the use of solar-fed CO₂ systems. Singh and Mishra [21] examined a solar-driven supercritical CO₂ Brayton thermodynamic cycle with a bottoming ORC with parabolic trough collectors (PTC) as the solar technology. They used an indirect system with thermal oil in the solar field as it is done in the usual system. They found that the use of R407c in the ORC is the best choice which leads to 3740 kW electricity production and total system efficiency (electricity to solar) at 21.8%. Khan and Mishara [22] studied a system with oil-based PTC that drives a supercritical CO₂ Brayton system in an indirect way and also the Brayton cycle drives a bottoming ORC. It is found that the best organic fluid is the R1233zd(E) and in the optimum design the solar collector efficiency is 62.93%, the thermodynamic efficiency is 48.61% and consequently, the system efficiency is 30.59%. In another work, Abdollahpour et al. [23] studied the use of a thermal oil-based nanofluid as the working fluid in a solar field that feeds a transcritical CO₂ Rankine cycle and there is an extra auxiliary system in order to aid the solar energy. Moreover, the Rankine cycle rejects heat to a liquefied natural gas system which includes a turbine for extra electricity production. They calculated the system exergy efficiency at 8.53% and they also indicated that the nanofluid-based system has higher performance than the operation with pure thermal oil in the PTC field. In another interesting work, AlZahrani and Dincer [24] studied an oil-based PTC field that feeds a transcritical CO₂ power system coupled to an absorption chiller. They conducted a parametric analysis and they concluded that the thermodynamic efficiency is 34% and the system efficiency is 20%.

In the literature, the studies that investigate the use of CO₂ inside the solar collector and its direct incorporation in the power cycle are seldom. Practically, there is a group of studies [25,26,27,28] that investigates the use of a simple transcritical Rankine cycle with evacuated tube collectors. More specifically, Zhang et al. [26] found experimentally that the system efficiency is up to 9.45%, while the solar collector efficiency is in the range of 65–70%. At this point, it is useful to refer to some important studies that examine the modeling of the supercritical CO₂ inside the solar concentrating collectors. Qiu et al. [29] investigated a supercritical CO₂-based PTC with a developed model with non-constant heat flux in the absorber tube periphery. They developed a formula for the prediction of the collector thermal efficiency for two pressure levels which are the 90 bar and the 200 bar. They found that the PTC thermal efficiency can reach high values by up to 84%. In another work, Bellos and Tzivanidis [30] studied a supercritical CO₂-based PTC for various pressures and temperatures. They emphasized the thermal and exergy analysis of the solar collector. The global maximum exergy efficiency is found at 45.3% for inlet CO₂ temperature at 750 K. Moreover, Munoz-Anton et al. [31] found the experimental operating feasibility of a supercritical CO₂-based PTC at 500 °C.

The aforementioned investigations indicate that there is a huge interest in CO₂-based units in the last years due to the high efficiency of these systems and because CO₂ is a very environmentally friendly fluid. In this direction, this analysis investigates a solar-driven power cycle that uses a parabolic trough collector and CO₂ in a transcritical Rankine cycle. The main innovation novelty of this investigation is based on the direct pass of the CO₂ through the PTC, something that increases the efficiency of the total system. More specifically, the direct heating of the CO₂ from the PTC makes possible the operation in high temperatures without the limitations of the maximum temperatures of an intermediate fluid (e.g., thermal oil operates up to 400 °C). Moreover, the systems with different fluids in the solar field and the power cycle lead to exergy destruction in the heat transfer process, something that does not exist in the present configuration. Therefore, the use of a supercritical fluid is able to achieve high-temperature levels, something beneficial for the thermodynamic efficiency and also not to have an intermediate fluid between solar field and cycle which creates exergy destruction issues due to the pinch points. The examined system is studied for different values of critical parameters such as the turbine inlet temperature, the turbine inlet pressure, the heat rejection temperature, the solar irradiation level and also the optimization of this configuration is presented. Furthermore, the default scenario is studied on a daily, monthly and yearly basis and a basic financial evaluation is conducted. Moreover, the system is compared with a basic solar-fed ORC in order to prove the superiority of the suggested configuration. The investigation is conducted with a detailed created mathematical model which is programmed in Engineering Equation Solver (EES) [32] that has been validated with data from the literature.

2. Material and Methods

Section 2 includes the examined model description, the mathematical formulation part, the model validation and the explanation of the followed steps of the analysis.

2.1. The Investigated Solar-Driven Power Unit

The objective of this study is the detailed investigation of a solar-fed power cycle. The system is a transcritical Rankine cycle operating with CO₂ as the working fluid. Solar parabolic trough collector is used for giving the heating in the system and its module is given in Figure 1.

It has to be said that in this work, 10 modules of the LS-3 are used in this study, they have been examined in Ref. [30]. The solar field has 10 modules connected in series and the total length is 40.6 m (=10 × 4.06 m). The solar collector has an evacuated tube receiver and its details are summarized in

Table 1. Data about the solar collector and the present simulation [29,30,33].

Parameter	Symbol	Value
Width of the module	(W)	5.6 m
Length of the module	(L)	4.06 m
Number of modules	(MOD)	10
Total length	(Lrow)	40.6 m
Focal distance	(F)	1.71 m
Module area	(Amod)	22.74 m2
Total collecting area	(Aa)	227.4 m2
Inner diameter of the absorber	(Dri)	0.05 m
Outer diameter of the absorber	(Dro)	0.07 m
Inner diameter of the cover	(Dci)	0.108 m
Outer diameter of the cover	(Dco)	0.115 m
Cover emittance	(εc)	0.86
Maximum optical efficiency	(ηopt,max)	0.8419
Absorber absorbance	(α)	0.96
Cover transmittance	(τ)	0.96
Reflector reflectance	(ρr)	0.94
Intercept factor	(γ)	0.972
Concentrating ratio	(C)	25.46
Heat convection coefficient between cover and ambient	(hout)	10 W/m2 K
Ambient temperature	(Tam)	298.15 K
Equivalent sky temperature	(Tsky)	290.15 K
Equivalent sun temperature	(Tsun)	5770 K
Default solar beam irradiation	(Gb)	800 W/m2
Default solar angle	(θ)	0°

[29,30,33]. An important modification of the LS-3 module is the thickness of the absorber tube which is 10 mm in this work in order to withstand the high-pressure levels [29,30].

Regarding the thermodynamic cycle, it is a Rankine cycle that uses CO₂ as the working fluid and it is depicted in Figure 2. The state point “1” after the condenser is saturated liquid and it has the saturation temperature and pressure of the condenser. It is compressed up to the high pressure in the state point “2” which is in a supercritical situation. The compression consumes an important amount of work but it is not as high as in compressors because the CO₂ has liquid characteristics in this region. The fluid of the state point “2” recovers heating from the recuperator by exploiting waste heat after the turbine outlet. The state point “3” has a higher temperature and enthalpy compared to the state point “2” and so lower amounts of heat source energy rate are needed. The PTC is used for providing the heating in the CO₂ which warms up to the high cycle temperature (turbine inlet temperature—TIT) which is a characteristic parameter of the cycle. During the heating process in the PTC, there is an extremely small pressure drop, as it has been calculated, which has no practical effect on the pressure level at the state point “4”. The turbine is used in order to expand the CO₂ of high-pressure to the lower pressure (state point “5”). This situation has high enthalpy and so the recuperator is used for heat recovery. The fluid gives heat to the other stream and so it reaches the state point “6”. The next step is the pass through the condenser in order to reject heat to the ambient and so the cycle closes and the fluid goes again in the state point “1”.

2.2. Mathematical-Formulation Part

Section 2.2 presents the basic mathematical equations of the developed modeling which simulates the studied solar power system. These equations have been inserted properly in the developed model in EES for performing the system simulation.

2.2.1. Modeling of the Solar System

The solar thermal efficiency of the PTC (η_collector) is defined as the ratio of the useful heating (Q_u) to the incident solar direct beam irradiation (Q_sol):

$${\mathsf{\eta }}_{\mathrm{collector}}=\frac{{\mathrm{Q}}_{\mathrm{u}}}{{\mathrm{Q}}_{\mathrm{sol}}}$$

(1)

The useful thermal product (Q_u) is expressed as follows:

$${\mathrm{Q}}_{\mathrm{u}}=\mathrm{m}\cdot \left({\mathrm{h}}_4-{\mathrm{h}}_3\right)$$

(2)

It has to be said that the enthalpies are functions of pressure and temperature as below: h₃ = f(p_l, T_in) and h₄ = f(p_h,T_out). Moreover, an alternative expression of Equation (2) is by using the mean fluid specific heat capacity (c_p,fm):

$${\mathrm{Q}}_{\mathrm{u}}=\mathrm{m}\cdot {\mathrm{c}}_{\mathrm{p},\mathrm{fm}}\cdot \left({\mathrm{T}}_{\mathrm{out}}-{\mathrm{T}}_{\mathrm{in}}\right)$$

(3)

The solar energy input in the system (Q_sol) is defined as:

$${\mathrm{Q}}_{\mathrm{sol}}={\mathrm{A}}_{\mathrm{a}}\cdot {\mathrm{G}}_{\mathrm{b}}$$

(4)

The thermal losses of the absorber (Q_loss,a) are practically radiation thermal losses. It has to be said that the convection thermal losses are neglected due to the vacuum between cover and receiver:

$${\mathrm{Q}}_{\mathrm{loss},\mathrm{a}}={\mathrm{A}}_{\mathrm{ro}}\cdot \mathsf{\sigma }\cdot \frac{\left({\mathrm{T}}_{\mathrm{r}}^4-{\mathrm{T}}_{\mathrm{c}}^4\right)}{\frac{1}{{\mathsf{\epsilon }}_{\mathrm{r}}}+\frac{1-{\mathsf{\epsilon }}_{\mathrm{c}}}{{\mathsf{\epsilon }}_{\mathrm{c}}}\cdot \left(\frac{{\mathrm{D}}_{\mathrm{ro}}}{{\mathrm{D}}_{\mathrm{ci}}}\right)}$$

(5)

The absorber emittance (ε_r) is a function of the mean receiver mean temperature [29]:

$${\mathsf{\epsilon }}_{\mathrm{r}}=0.000327\cdot {\mathrm{T}}_{\mathrm{r}}-0.065971$$

(6)

The thermal losses from the cover to the ambient heat sink (Q_loss,c) are radiation and convection thermal losses.

$${\mathrm{Q}}_{\mathrm{loss},\mathrm{c}}={\mathrm{A}}_{\mathrm{co}}\cdot \mathsf{\sigma }\cdot {\mathsf{\epsilon }}_{\mathrm{c}}\cdot \left({\mathrm{T}}_{\mathrm{c}}^4-{\mathrm{T}}_{\mathrm{sky}}^4\right)+{\mathrm{A}}_{\mathrm{co}}\cdot {\mathrm{h}}_{\mathrm{out}}\cdot \left({\mathrm{T}}_{\mathrm{c}}-{\mathrm{T}}_{\mathrm{am}}\right)$$

(7)

The sky temperature (T_sky) is set to be 8 K lower than the ambient temperature [29]:

$${\mathrm{T}}_{\mathrm{sky}}={\mathrm{T}}_{\mathrm{am}}-8\left[\mathrm{K}\right]$$

(8)

The convection heat transfer coefficient for the cover tube (h_out) can be estimated at 10 W/m²K by applying the following formula [34], for a wind speed (V_w) at 1 m/s:

$${\mathrm{h}}_{\mathrm{out}}=\frac{4\cdot {\mathrm{V}}_{\mathrm{w}}^{0.58}}{{\mathrm{D}}_{\mathrm{co}}^{0.42}}$$

(9)

In steady-state conditions, the energy storage in the glass cover can be neglected and so it can be said [30] about the thermal losses (Q_loss):

$${\mathrm{Q}}_{\mathrm{loss}}={\mathrm{Q}}_{\mathrm{loss},\mathrm{a}}={\mathrm{Q}}_{\mathrm{loss},\mathrm{c}}$$

(10)

The energy balance in the absorber shows that the absorbed solar energy (Q_abs) is separated into useful production (Q_u) and thermal losses (Q_loss):

$${\mathrm{Q}}_{\mathrm{abs}}={\mathrm{Q}}_{\mathrm{u}}+{\mathrm{Q}}_{\mathrm{loss}}$$

(11)

The absorbed solar irradiation is found as the product of the optical efficiency (η_opt) and of solar energy (Q_sol):

$${\mathrm{Q}}_{\mathrm{abs}}={\mathsf{\eta }}_{\mathrm{opt}}\cdot {\mathrm{Q}}_{\mathrm{sol}}$$

(12)

The optical efficiency is a function of the incident angle modifier (K) which is a function of the solar angle (θ). Therefore, the optical efficiency can be written as:

$${\mathsf{\eta }}_{\mathrm{opt}}=\mathrm{K}\left(\mathsf{\theta }\right)\cdot {\mathsf{\eta }}_{\mathrm{opt},\max }$$

(13)

Moreover, the incident angle modifier is calculated by using a reliable model that has been presented by Gaul and Rabl [35] and takes into account the sun position and the PTC end losses:

$$\mathrm{K}\left(\mathsf{\theta }\right)=\cos \left(\mathsf{\theta }\right)-\left(\frac{\mathrm{F}}{{\mathrm{L}}_{\mathrm{row}}}\right)\cdot \left(1+\frac{{\mathrm{W}}^2}{48\cdot {\mathrm{F}}^2}\right)\cdot \sin \left(\mathsf{\theta }\right)$$

(14)

In this study, the PTC axis is located in the south-north direction and there is a tracking mechanism in order to follow the sun–path in the east–west direction. The solar angle can be calculated for this tracking system as below [36]:

$$\cos \left(\mathsf{\theta }\right)=\sqrt{{\cos }^2\left({\mathsf{\theta }}_{\mathrm{z}}\right)+{\cos }^2\left(\mathsf{\delta }\right)\cdot {\sin }^2\left(\mathsf{\omega }\right)}$$

(15)

where the solar time angle (ω), the declination angle (δ) and the zenith angle (θ_z) are used in the previous formula.

The next part of the modeling regards the heat transfer between the absorber and the fluid. Practically, the useful heat can be also calculated by using the fluid heat transfer coefficient (h_fm) as below:

$${\mathrm{Q}}_{\mathrm{u}}={\mathrm{A}}_{\mathrm{ri}}\cdot {\mathrm{h}}_{\mathrm{fm}}\cdot \left({\mathrm{T}}_{\mathrm{r}}-{\mathrm{T}}_{\mathrm{fm}}\right)$$

(16)

The mean fluid temperature (T_fm) can be estimated as follows:

$${\mathrm{T}}_{\mathrm{fm}}=\frac{{\mathrm{T}}_{\mathrm{in}}+{\mathrm{T}}_{\mathrm{out}}}{2}$$

(17)

The Nusselt number (Nu) is used in order to determine the value of the fluid’s heat transfer coefficient:

$${\mathrm{h}}_{\mathrm{fm}}=\frac{\mathrm{Nu}\cdot {\mathrm{k}}_{\mathrm{fm}}}{{\mathrm{D}}_{\mathrm{ri}}}$$

(18)

For turbulent flow inside a tube (Re > 2300), the Nusselt number can be found by using the well-known formula of Dittus-Boelter [37]:

$$\mathrm{Nu}=0.023\cdot {\mathrm{Re}}^{0.8}\cdot {\mathrm{Pr}}^{0.4}$$

(19)

The Reynolds number (Re) is defined as below for the tubular absorber:

$$\mathrm{Re}=\frac{4\cdot \mathrm{m}}{\mathsf{\pi }\cdot {\mathrm{D}}_{\mathrm{ri}}\cdot {\mathsf{\mu }}_{\mathrm{fm}}}$$

(20)

The Prandtl number (Pr) can be found by the next equation:

$$\mathrm{Pr}=\frac{\mathsf{\mu }\cdot {\mathrm{c}}_{\mathrm{p},\mathrm{fm}}}{{\mathrm{k}}_{\mathrm{fm}}}$$

(21)

The solar collector’s pressure drop (ΔP) can be written as:

$$\mathsf{\Delta }\mathrm{P}=\mathrm{f}\cdot \left(\frac{\mathrm{L}}{{\mathrm{D}}_{\mathrm{ri}}}\right)\cdot \left(\frac{1}{2}\cdot {\mathsf{\rho }}_{\mathrm{fm}}\cdot {\mathrm{u}}^2\right)$$

(22)

The friction factor (f) for the turbulent flow regime is found with the next equation [38]:

$$\mathrm{f}=\frac{1}{{\left[0.79\cdot \ln \left(\mathrm{Re}\right)-1.64\right]}^2}$$

(23)

The fluid velocity (u) can be expressed as:

$$\mathrm{u}=\frac{\mathrm{m}}{{\mathsf{\rho }}_{\mathrm{fm}}\cdot \left(\mathsf{\pi }\cdot \frac{{\mathrm{D}}_{\mathrm{ri}}^2}{4}\right)}$$

(24)

The useful exergy product of the PTC (Ex_u) can be found as follows [30]:

$${\mathrm{Ex}}_{\mathrm{u}}={\mathrm{Q}}_{\mathrm{u}}-\mathrm{m}\cdot {\mathrm{c}}_{\mathrm{p}}\cdot {\mathrm{T}}_{\mathrm{am}}\cdot \ln \left[\frac{{\mathrm{T}}_{\mathrm{out}}}{{\mathrm{T}}_{\mathrm{in}}}\right]-\mathrm{m}\cdot {\mathrm{T}}_{\mathrm{am}}\cdot \frac{\mathsf{\Delta }\mathrm{P}}{{\mathsf{\rho }}_{\mathrm{fm}}\cdot {\mathrm{T}}_{\mathrm{fm}}}$$

(25)

The exergy rate of the solar irradiation (Ex_sol) is estimated by using the Petela model [39]:

$${\mathrm{Ex}}_{\mathrm{sol}}={\mathrm{Q}}_{\mathrm{sol}}\cdot \left[1-\frac{4}{3}\cdot \left(\frac{{\mathrm{T}}_{\mathrm{am}}}{{\mathrm{T}}_{\mathrm{sun}}}\right)+\frac{1}{3}\cdot {\left(\frac{{\mathrm{T}}_{\mathrm{am}}}{{\mathrm{T}}_{\mathrm{sun}}}\right)}^4\right]$$

(26)

The temperature of the sun is 5770 K, while the temperature levels of Equations (25) and (26) have to be expressed in Kelvin units.

The definition of the solar collector’s exergy efficiency (η_ex,col) is the following:

$${\mathsf{\eta }}_{\mathrm{ex},\mathrm{col}}=\frac{{\mathrm{Ex}}_{\mathrm{u}}}{{\mathrm{Ex}}_{\mathrm{sol}}}$$

(27)

2.2.2. Modeling of the Transcritical Rankine Cycle

The transcritical Rankine cycle can be modeled by applying the equations of the present section. These formulas regard mainly energy balances and efficiency indexes definitions.

The heat input in the ORC (Q_in) is the same as the useful heat product in the solar collector:

$${\mathrm{Q}}_{\mathrm{in}}={\mathrm{Q}}_{\mathrm{u}}$$

(28)

The turbine’s work production (W_t) is calculated as follows:

$${\mathrm{W}}_{\mathrm{t}}=\mathrm{m}\cdot \left({\mathrm{h}}_4-{\mathrm{h}}_5\right)$$

(29)

The pump’s work consumption in (W_p) is calculated as follows:

$${\mathrm{W}}_{\mathrm{p}}=\mathrm{m}\cdot \left({\mathrm{h}}_2-{\mathrm{h}}_1\right)$$

(30)

The net electricity production of the unit (P_el) is calculated according to the next expression:

$${\mathrm{P}}_{\mathrm{el}}={\mathsf{\eta }}_{\mathrm{mg}}\cdot {\mathrm{W}}_{\mathrm{t}}-\frac{{\mathrm{W}}_{\mathrm{p}}}{{\mathsf{\eta }}_{\mathrm{motor}}}$$

(31)

The expansion in the turbine is modeled with the turbine’s isentropic efficiency (η_is,t):

$${\mathsf{\eta }}_{\mathrm{is},\mathrm{t}}=\frac{{\mathrm{h}}_4-{\mathrm{h}}_5}{{\mathrm{h}}_4-{\mathrm{h}}_{5,\mathrm{is}}}$$

(32)

The compression in the pump is modeled with the pump’s isentropic efficiency (η_is,p):

$${\mathsf{\eta }}_{\mathrm{is},\mathrm{p}}=\frac{{\mathrm{h}}_{2,\mathrm{is}}-{\mathrm{h}}_1}{{\mathrm{h}}_2-{\mathrm{h}}_1}$$

(33)

The heat rejection in the ambient (Q_out) can be calculated as:

$${\mathrm{Q}}_{\mathrm{out}}=\mathrm{m}\cdot \left({\mathrm{h}}_1-{\mathrm{h}}_6\right)$$

(34)

The conduction of the energy balance in the recuperator device leads to the next expression:

$${\mathrm{h}}_3-{\mathrm{h}}_2={\mathrm{h}}_5-{\mathrm{h}}_6$$

(35)

The minimum recuperator temperature difference (ΔΤ_rec) is given as:

$${\mathsf{\Delta }\mathrm{T}}_{\mathrm{rec}}={\mathrm{T}}_6-{\mathrm{T}}_2$$

(36)

The efficiency of the thermodynamic cycle (η_cycle) is defined as:

$${\mathsf{\eta }}_{\mathrm{cycle}}=\frac{{\mathrm{P}}_{\mathrm{el}}}{{\mathrm{Q}}_{\mathrm{in}}}$$

(37)

The overall unit energy efficiency (η_sys) can be found by the next equation:

$${\mathsf{\eta }}_{\mathrm{sys}}=\frac{{\mathrm{P}}_{\mathrm{el}}}{{\mathrm{Q}}_{\mathrm{sol}}}$$

(38)

The overall unit exergy efficiency (η_ex,sys) is defined as:

$${\mathsf{\eta }}_{\mathrm{ex},\mathrm{sys}}=\frac{{\mathrm{P}}_{\mathrm{el}}}{{\mathrm{Ex}}_{\mathrm{sol}}}$$

(39)

The data of the thermodynamic cycle analysis are included in

Table 2. Data of the thermodynamic cycle analysis [20,24].

Parameter	Symbol	Value
Turbine isentropic efficiency	(ηis,t)	85%
Pump isentropic efficiency	(ηis,p)	85%
Electromechanical generator efficiency	(ηmg)	97%
Motor efficiency	(ηmotor)	80%
Recuperator temperature difference	(ΔΤrec)	10 K
Temperature level in the condenser device	(Tcon)	298.15 K

.

2.2.3. Modeling of the Financial and Environmental Analysis

The electricity production during the year (E_el) is calculated by integrating the power rate (P_el) for all the year:

$${\mathrm{E}}_{\mathrm{el}}={{\displaystyle \int }}_0^{8760}{\mathrm{P}}_{\mathrm{el}}\cdot \mathrm{dt}$$

(40)

Similarly, the yearly solar energy (E_sol) is calculated as:

$${\mathrm{E}}_{\mathrm{sol}}={{\displaystyle \int }}_0^{8760}{\mathrm{Q}}_{\mathrm{sol}}\cdot \mathrm{dt}$$

(41)

The system’s capital cost (C₀) is estimated according to the next equation:

$${\mathrm{C}}_0={\mathrm{C}}_{\mathrm{cycle}}\cdot {\mathrm{P}}_{\mathrm{el},\mathrm{ref}}+{\mathrm{C}}_{\mathrm{solar}}\cdot {\mathrm{A}}_{\mathrm{a}}$$

(42)

The yearly cash flow (CF) is calculated as:

$$\mathrm{CF}={\mathrm{K}}_{\mathrm{el}}\cdot {\mathrm{E}}_{\mathrm{el}}-\left(\mathrm{O}\&\mathrm{M}\right)$$

(43)

The yearly operating and maintenance cost (O&M) is set to be 1% of the initial investment cost [39]:

$$\left(\mathrm{O}\&\mathrm{M}\right)=0.01\cdot {\mathrm{C}}_0$$

(44)

The simple payback period (SPP) of the examined configuration is defined as:

$$\mathrm{SPP}=\frac{{\mathrm{C}}_0}{\mathrm{CF}}$$

(45)

The payback period (PP) of the examined configuration is calculated as:

$$\mathrm{PP}=\frac{\ln \left[\frac{\mathrm{CF}}{\mathrm{CF}-{\mathrm{C}}_0\cdot \mathrm{r}}\right]}{\ln \left(1+\mathrm{r}\right)}$$

(46)

The internal rate of return (IRR) can be expressed as follows:

$$\mathrm{IRR}=\mathrm{SPP}\cdot \left[1-\frac{1}{{\left(1+\mathrm{IRR}\right)}^{\mathrm{N}}}\right]$$

(47)

The calculation of the net present value (NPV) is done with the next equation:

$$\mathrm{NPV}=-{\mathrm{C}}_0+\mathrm{R}\cdot \mathrm{CF}$$

(48)

The equivalent project life (R) can be found as follows:

$$\mathrm{R}=\frac{{\left(1+\mathrm{r}\right)}^{\mathrm{N}}-1}{\mathrm{r}\cdot {\left(1+\mathrm{r}\right)}^{\mathrm{N}}}$$

(49)

Table 3. Data of the financial analysis [40,41,42].

Parameters	Symbols	Values
Cost of the electricity	(Cel)	0.2 €/kWhel
Solar Specific	(Csolar)	250 €/m2
Cycle specific cost	(Ccycle)	1400 €/kWel
Operation and maintenance cost	(O&M)	1%·C0
Project lifetime	(N)	25 years
Equivalent project lifetime	(R)	17.41 years
Discount factor	(r)	3%
Nominal electricity	(Pel,ref)	44.14 kWel

gives the data of the economic analysis [40,41,42]. The nominal power of the system is 44.14 kW for this work, as it has been calculated by the results.

Moreover, about the CO₂ emissions avoidance, the Greek energy mix for 2020 leads to approximately 600 kg CO₂ per MWh_el according to the DAPEEP [43]. Therefore, yearly CO₂ emissions avoidance is calculated as:

$${\mathrm{M}}_{\mathrm{CO}2,\mathrm{y}}\left[\mathrm{kg}\right]=600\cdot \frac{{\mathrm{E}}_{\mathrm{el}}\left[\mathrm{kWh}\right]}{1000}$$

(50)

While the total CO₂ emissions avoidance for the life cycle of the project is calculated as:

$${\mathrm{M}}_{\mathrm{CO}2,\mathrm{tot}}\left[\mathrm{kg}\right]=\mathrm{N}\cdot 600\cdot \frac{{\mathrm{E}}_{\mathrm{el}}\left[\mathrm{kWh}\right]}{1000}$$

(51)

2.3. Validation of the Developed Mathematical Models

The developed model has been validated by using literature data. There are two validation procedures; one for the solar collector with supercritical CO₂ and one for the transcritical CO₂ power cycle.

2.3.1. Validation of the Parabolic Trough Collector Model

The PTC model is validated with data from the work of Qiu et al. [29]. The authors of this work had suggested a regression model for efficiency and

Table 4. Validation evidence for the PTC model [29].

Inputs		Collector Efficiency
Gb (W/m2)	Tin (K)	This Study	Literature [29]	Deviation
600	500	81.91%	81.90%	0.011%
600	550	80.24%	80.26%	0.025%
600	600	77.80%	77.84%	0.048%
600	650	74.38%	74.47%	0.120%
600	700	69.73%	69.98%	0.352%
600	750	63.60%	64.19%	0.918%
800	500	82.38%	82.29%	0.105%
800	550	81.07%	81.00%	0.089%
800	600	79.18%	79.14%	0.049%
800	650	76.53%	76.55%	0.024%
800	700	72.95%	73.12%	0.231%
800	750	68.24%	68.72%	0.699%
1000	500	82.65%	82.52%	0.162%
1000	550	81.57%	81.45%	0.144%
1000	600	80.00%	79.91%	0.110%
1000	650	77.82%	77.80%	0.020%
1000	700	74.87%	75.01%	0.182%
1000	750	71.01%	71.43%	0.583%

presents the results of this literature formula and with the developed model of this work. The validation is conducted for different combinations of solar irradiation and inlet temperatures. The other parameters have been selected to have the same values as in Ref. [29]. The mean thermal efficiency deviation is calculated to be 0.215% which is a very low value, while the maximum deviation is 0.918% which is also a low value. Therefore, the found results are in accordance with the literature and so the present model can be adopted as a reliable one.

2.3.2. Validation of the Transcritical Rankine Cycle

The validation of the transcritical Rankine cycle is performed with the study of Wang et al. [17] which is about a simple cycle with a pump, turbine, condenser and heat recovery system. For the validation procedure, the condensation temperature is set at 21 °C, the temperature in the turbine inlet at 79.93 °C, the high pressure at 107.51 bar and the heat input in the cycle at 34,130 kW.

Table 5. Validation evidence for the thermodynamic cycle [17].

Parameters	Literature [17]	This Study	Deviation
Thermodynamic cycle efficiency (-)	0.0475	0.04723	0.568%
CO2 mass flow rate (kg/s)	174.3	176	0.975%
Net cycle work (kW)	1620	1612	0.494%
Turbine work (kW)	3178	3184	0.189%
Pump work (kW)	1558	1572	0.899%
Heat rejection (kW)	32,510	32,518	0.025%
Temperature after the pump (°C)	29.71	29.76	0.168%
Turbine temperature outlet (°C)	35.09	34.73	1.026%
Low cycle pressure (bar)	58.57	58.65	0.137%

summarizes the literature data for various parameters, the calculated values of the present modeling and the deviation values. Generally, the deviations are up to 1% and so it can be said that the present methodology is acceptable.

2.4. Followed Steps

In this investigation, a mathematical model is created by using the equations of Section 2.2 and it is validated with literature data which are presented in Section 2.3. The first step in this analysis is the examination of the unit in the default design scenario and the results are presented in Section 3.1. The next step is the presentation of a simple parametric analysis which is given in Section 3.2. The third step is a detailed parametric analysis with the most interesting parameters; the turbine inlet temperature and pressure levels. These results are presented in Section 3.3 and also an optimization analysis is presented in this subsection. Section 3.4 includes results about the daily, monthly and yearly performance of the examined system. The last part of Section 3 is Section 3.5 which regards the financial analysis, the environmental analysis and further discussion of the found results.

Table 6. The examined parameters in the parametric analysis.

Parameter	Symbol	Default Value	Range
Solar direct beam irradiation [W/m2]	(Gb)	800	[400–1000]
Condenser temperature [°C]	(Tcon)	25	[15–30]
Turbine inlet pressure [bar]	(Ph)	200	[100–220]
Turbine inlet temperature [K]	(TIT)	800	[600–1000]

includes the studied parameters, their default values and the examined range of their values. During the optimization of the cycle, the conjugate directions method is applied with the goal to maximize the system efficiency. The optimization variables are the pressure level in the turbine inlet and the temperature level in the turbine inlet. The optimization is conducted with the EES [32] by using its proper module.

The daily analysis is done for the city of Athens in Greece. The data about the solar direct beam irradiation have been taken by Ref. [36] and 12 mean monthly days are examined. Moreover, for each month, a proper number of sunny days have been used, as they have been presented in Ref. [44]. More specifically, for the months from January to December, the respective sunny days are the following: 13-10-15-18-20-22-29-29-20-18-17-15. Totally, 226 days are assumed that the examined system can operate which is a reasonable number for the climate conditions of Athens.

3. Results and Discussion

Section 3 includes the results of the present analysis and the respective discussion. Results from the parametric analysis, the optimization, the daily performance, as well as financial and environmental data are provided.

3.1. Analysis of the Default Design Scenario

The first part of the results section regards the default operating scenario with the maximum cycle temperature at 800 K and the high pressure at 200 bar. The condenser temperature is equal to 25 °C, the solar direct beam irradiation level 800 W/m² and there is a zero-incident angle.

Table 7. State point thermodynamic properties for the default scenario.

State Points	T (K)	P (bar)	h (kJ/kg)	s (kJ/kgK)
1	298.2	64.34	−231.99	0.510
2	321.3	200.00	−211.08	0.519
3	545.8	200.00	186.41	1.495
4	800.0	200.00	499.66	1.967
5	676.0	64.34	366.29	2.002
6	331.3	64.34	−31.21	1.172

includes the results of the thermodynamic data of the cycle state points.

Figure 3 and Figure 4 show the thermodynamic cycle in usual depictions in order to explain it properly. More specifically, Figure 3 illustrates the temperature—specific entropy, while Figure 4 exhibits the pressure—enthalpy depiction. It can be said that the cycle is a transcritical one with its low pressure to be close to the CO₂ critical pressure. More specifically, the low cycle temperature is 64.34 bar which is close to the critical pressure of 73.8 bar. Figure 4 shows that the enthalpy increase that the PTC gives to the cycle (h₄–h₃) is 313.25 kJ/kg, while the recuperator gives an enthalpy increase (h₃–h₂) of 397.49 kJ/kg. Therefore, it is obvious that the recuperator is responsible for 56% of the heat input in the CO₂, while the PTC gives the rest 44%. Therefore, it can be said that the existence of the recuperator in the system is very important and it is a correct design idea. Moreover, it has to be said that the collector efficiency is 73.62%, the cycle efficiency is 32.96% and the system efficiency is 24.27% for the nominal design point.

3.2. Simple Parametric Investigation

The next step in the present investigation regards the parametric evaluation of the examined unit. Four critical parameters are investigated and more specifically the turbine inlet temperature, the turbine inlet pressure, the cycle low-temperature level in the condenser and the solar irradiation level. Figure 5, Figure 6, Figure 7 and Figure 8 depict the impact of the aforementioned parameters on system efficiency, collector efficiency, cycle efficiency and electricity production.

Figure 5 shows the influence of the TIT on the results. Higher TIT leads to lower collector efficiency because the higher operating temperature in the collector leads to smaller thermal efficiency due to the thermal loss increase. Higher TIT leads to higher cycle efficiency due to the higher potential for work production, something that is in accordance with the Carnot cycle. The unit efficiency is the product of the other efficiencies and so the different impact of the TIT on the collector and the cycle efficiency makes the system efficiency have a combined behavior. Therefore, there is an optimal TIT that maximizes the system efficiency and consequently the electricity production. The next formula shows clearly the aforementioned analysis:

$${\mathsf{\eta }}_{\mathrm{sys}}={\mathsf{\eta }}_{\mathrm{collector}}\cdot {\mathsf{\eta }}_{\mathrm{cycle}}$$

(52)

The optimum TIT for the examined case is found for 800 K. In this case, the collector efficiency is 73.62%, the cycle efficiency is 32.96%, the system efficiency is 24.27% and the electricity production is 44.14 kW.

Figure 6 illustrates the impact of the high pressure on the examined parameters. Higher values of the pressure in the turbine inlet lead to an increase in all the efficiency indexes and electricity production. The increase of the pressure makes higher the cycle efficiency, something that is usual in Brayton cycles. The increase of the pressure level in the collector efficiency is a result that has been also found in Ref. [30]. More specifically, higher pressure leads to higher density and consequently higher mass flow rate inside the solar collector. The higher mass flow rate makes the heat transfer from the absorber to the fluid to be more effective due to the increase of the heat convection coefficient and thus the collector efficiency improves. The increase in the system and electrical efficiency is a result of the increases in the other efficiencies.

Figure 7 depicts the influence of the low cycle temperature on the results. Generally, lower values of this parameter are beneficial for all the examined parameters. The lower condenser temperature makes the system operate to lower temperatures and so the collector operates in lower temperatures and the smaller thermal losses lead to higher collector efficiency. Moreover, lower condenser temperature reduces the low pressure of the system and so the turbine has a higher margin to produce work due to the higher expansion ratio. Therefore, the cycle efficiency, system efficiency and electricity are improved with the lower heat rejection temperature. At this point, it has to be commented that the system can operate during transcritical mode during the summer when there is a geothermal heat sink or a water-cooled condenser (e.g., from a river or from the sea).

Figure 8 includes the results of the influence of solar beam irradiation on collector efficiency. Firstly, it has to be said that the cycle efficiency is not affected by the solar irradiation level. On the other hand, the collector efficiency enhances with the solar potential increase and consequently the system efficiency and the electricity improves. It is remarkable to state that the impact of the solar irradiation on the collector efficiency makes it increase from 63.90% to 75.62%, but the electricity production presents a high increase from 19.16 to 56.66 kW which is a large increase. The curve of electricity production has an approximately linear shape, and this fact is explained by the not so large impact of solar irradiation on the efficiency values. To conclude, solar irradiation has a not too high impact on efficiencies but it has a huge impact on electricity production because solar energy, and the heat input, is directly increased with higher solar irradiation.

3.3. Detailed Parametric Analysis and System Optimization

The detailed parametric analysis is presented in Section 3.3 for different values of TIT and high pressure. Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 show the impact of many values of TIT and high pressure on the evaluation indexes. Figure 9 indicates that the collector thermal efficiency reduces with the TIT increase and increases for higher pressure levels. Generally, the curves are close to each other in low TIT, something that proves that the increase of the pressure is more important in the cases with higher TIT. Figure 10 shows clearly that higher TIT increases the cycle efficiency, as well as the pressure level. However, the impact of the temperature is more important than the pressure level and it is obvious that for a pressure level over 200 bar the enhancement is small.

Figure 11 illustrates the system efficiency for the different combinations of the pressure level in the turbine inlet and the temperature level in the turbine inlet. For every constant pressure curve, there is an optimal temperature level that maximizes the unit’s efficiency. This result is a reasonable one according to Equation (52). The system efficiency is the product of the cycle and the collector efficiency. The different impacts of the TIT on these parameters make their product to be maximized in an intermediate value. An extra interesting result is that the optimal TIT is getting to higher values when there is an increase in the pressure level. Figure 12 shows the electricity production results which are in total accordance with the shape of the curves in Figure 11 about the system efficiency. This is a reasonable result because electricity production is the product of the system efficiency and solar energy input. The solar energy input is the same for all the cases in this parametric investigation, and therefore, the electricity production and the system efficiency are proportional parameters.

Figure 13 illustrates the collector exergy efficiency and Figure 14 illustrates the system exergy efficiency. The collector exergy efficiency is maximized for TIT in the range of 800 to 850 K, as well as the system exergy efficiency which has similar optimum TIT values. Therefore, it is very interesting to state that the collector exergy map is the critical parameter for the determination of the optimal operating range of the system. In other words, the optimum value of the TIT which maximizes the collector exergy efficiency makes the total system to be optimized in this temperature range. The exergetic performance of the system is lower than the collector exergy efficiency which is a reasonable result, and it is explained by the irreversibilities of the examined real thermodynamic cycle. The collector exergy efficiency assumes that the PTC feeds an ideal Carnot cycle and thus the collector exergy efficiency is an upper limit for the system exergy efficiency if the operation of the PTC is taken into account. In any case, the most important conclusion by the analysis of Figure 13 and Figure 14 together is that designing the system with the optimum TIT according to the collector exergy efficiency is able to lead to the overall optimum designs for all the systems. Moreover, these results indicate that the selection of PTC in the present cycle is a proper choice due to the exergetic compatibility of the PTC and transcritical CO₂ Rankine cycle.

In the last part of Section 3.3, two approximating formulas about the system efficiency are given. Equation (53) indicates the optimum values of the TIT which maximizes the system efficiency as a function of the pressure level and the solar beam irradiation. The suggested approximation has a mean absolute percentage error (MAPE) of 0.40% and an R² of 99%. These are acceptable and proper values that show a suitable approximation. Figure 15 shows the results about the validity of Equation (53) by comparing the real values calculated by EES and the respected value calculated by Equation (53).

$${\mathrm{TIT}}_{\mathrm{optim}}=627.32+0.1756\cdot {\mathrm{G}}_{\mathrm{b}}+0.2279\cdot {\mathrm{P}}_{\mathrm{h}}$$

(53)

The optimum results of the TIT lead to an optimum system efficiency which is presented in Equation (54). Equation (54) has the high pressure and the solar beam irradiation levels as inputs, while the TIT has the respective optimum value according to Equation (53). This formula has R² equal to 99.60% and MAPE equal to 0.87% which are acceptable values. In addition, Figure 16 shows the comparison of the real values from EES with the values from the approximation formula.

$${\mathsf{\eta }}_{\mathrm{sys},\mathrm{optim}}=-0.1042+6.99\cdot {10}^{-5}\cdot {\mathrm{G}}_{\mathrm{b}}+0.00274\cdot {\mathrm{P}}_{\mathrm{h}}+1.7516\cdot {10}^{-7}\cdot {\mathrm{G}}_{\mathrm{b}}\cdot {\mathrm{P}}_{\mathrm{h}}-3.4927\cdot {10}^{-8}\cdot {\mathrm{G}}_{\mathrm{b}}^2-6.5263\cdot {10}^{-6}\cdot {\mathrm{P}}_{\mathrm{h}}^2$$

(54)

The previous formulas can give a quick estimation of the system efficiency for different conditions, and they can be used for the design of systems as the studies.

3.4. Daily, Monthly and Yearly Analysis

The investigated unit is studied for the default design scenario on a daily, monthly and yearly basis for the city of Athens (Greece). The simulation was performed by using 12 typical days, one for every month and by assuming that the unit operates only on the sunny days of every month according to the weather data of the studied location. Figure 17 illustrates the daily performance of the unit for the mean day of September which is generally a representative day for all the year as it has been observed by the present analysis. It can be said that solar irradiation presents maximum value at the solar noon while the electricity production has a relatively stable profile from 8:00 up to 16:00. This result can be explained by the impact of the incident angle modifier on the results; this parameter has minimum values close to the solar noon and higher earlier in the morning and later in the afternoon. Practically, the single-axis tracking system with the collector main axis in the south–north direction eliminates the optical losses in the morning and in the afternoon but close at solar noon it is not able to reach very high values due to the sun altitude. Therefore, the optical efficiency is reduced at solar noon but the higher solar irradiation level at this period makes generally electricity production have a relatively stable profile, something important for the usual applications. The system efficiency is influenced by both the electricity and the solar rates and the results show maximization in the morning and the afternoon. The mean system efficiency is found at 20.19%, the maximum at 22.37% and at the solar noon at 18.37%. These deviations are not high, but they are remarkable.

The next step is the monthly and yearly investigation of the system. Figure 18 shows the monthly electricity production and the monthly system efficiency, while

Table 8. Monthly and yearly performance of the system at the default design scenario.

Months	ηsys	Sunny Days	Eel (kWh)		Esol (kWh)
			Day	Month	Day	Month
January	12.90%	13	192	2491	1485	19,305
February	15.96%	10	263	2630	1648	16,485
March	19.17%	15	345	5177	1800	27,003
April	21.65%	18	418	7516	1929	34,723
May	22.63%	20	455	9108	2012	40,246
June	22.75%	22	464	10,202	2038	44,844
July	22.69%	29	456	13,224	2010	58,283
August	22.11%	29	427	12,373	1930	55,971
September	20.19%	20	366	7324	1813	36,268
October	16.96%	18	282	5070	1660	29,888
November	13.65%	17	206	3499	1508	25,643
December	11.95%	15	170	2549	1422	21,332
Year	19.80%	226	-	81,164	-	409,991

includes the respective results. Moreover, Table 8 includes the results for the mean day and for all the months. At this point, it has to take into account that the number of sunny days every month is used for the calculation of the monthly results. This fact makes the calculation more complex and proves that all the mean monthly days have not the same contribution to the yearly results. The maximum monthly production is found for July with 13,224 kWh, the second month is August with 12,373 kWh, while June follows with 10,202 kWh. The lowest yield is found for January with 2491 kWh while December follows with 2549 kWh and February is close with 2630 kWh. On the other hand, the system efficiency is maximized for June at 22.75% with July and May to follow with close values at 22.69% and 22.63% respectively. August has a reduced efficiency at 22.1% and the lowest efficiency is found for December at 11.95%. Generally, the efficiency and the electricity yield are maximized in the summer, and they are minimized in the winter; reasonable and acceptable results are in accordance with the solar potential availability in Athens. The yearly efficiency is found to be 19.80% and the yearly electricity yield for the 226 sunny days in Athens is 81,164 kWh.

3.5. Financial Analysis, Environmental Analysis and Further Discussion

Section 3.5 is devoted to presenting a basic financial and environmental analysis of the examined configuration, as well as discussing the found results in detail. The default design scenario has been assumed and the total results are summarized in

Table 9. Summary of the energy, financial and environmental results of the default design scenario.

Index	Value
Yearly electricity production	81.16 MWh
Total electricity production	2029 MWh
Yearly system efficiency	19.80%
Capital cost (C0)	118.55 €
Yearly cash flow (CF)	15.05 €
Simple payback period (SPP)	7.88 years
Payback period (PP)	9.12 years
Net present value (NPV)	143.5 k€
Internal rate of return (IRR)	11.94%
Yearly CO2 emissions avoidance	48.7 ton
Total CO2 emissions avoidance	1217.5 ton

. In this case, the yearly electricity production is 81.16 MWh and the total production in the 25-year life of the installation is about 2029 MWh, while the mean efficiency is 19.80%. The capital cost of the investment is 118.55 k€, the yearly income cash flow is 15.05 k€ and the NPV is 143.5 k€. The simple payback period and the payback period are found to be 7.88 and 9.12 years respectively, while the IRR is 11.94%. The CO₂ emissions avoidance is found 48.7 tons on a yearly basis and 1217.5 tons during the total life of the project. The aforementioned results indicate that the studied investment is viable, and the financial parameters are promising, especially for a renewable-based energy system. Moreover, significant amounts of CO₂ are avoided, and this fact proves that the system has a positive environmental impact.

At this point, it is essential to compare the present system with other similar systems and other literature studies. Firstly, it has to be said that the present system is more efficient than a respective typical solar-driven ORC with PTC. More specifically, an extra simulation has been done by using a system with toluene as the working fluid and thermal oil in the PTC field, as it has been studied in Ref. [44]. The same parameters have been used in the new evaluation as in the present one and it is found that in steady-state conditions the conventional CO₂ has a system efficiency of 21.69% while the present system with CO₂ has 24.27% system efficiency. Therefore, an efficiency enhancement of 11.89% was found; a significant improvement that proves the importance of the suggested idea. In another work about a solar-driven ORC, Tzivanidis et al. [45] have found system efficiency up to 15.78% which is a very lower value compared to the results of the present analysis. Moreover, another work by Bellos and Tzivanidis [44] proved 15.38% system efficiency which is also lower than the present study. In other studies, relatively low system efficiencies have been found such as 8% with PTC-ORC [46,47] and 12.5% [48].

The financial analysis shows that this work has a payback period of 7.88 years, while Ref. [45] found payback period around 9 years so this work has also beneficial financial evidence. Ref. [46] shows that the payback period is about 8.4 years and Ref. [49] about 12 years, values which are also higher than the values of this work. Therefore, the use of a CO₂-based transcritical cycle is an idea with important energy efficiency gains and also positive financial performance compared to other ORCs.

In the end, it is important to state that the examined unit faces some limitations. There is a need for operation in high-pressure levels which creates the need for designing properly the devices and the tubes in order to withstand these pressure levels. Moreover, the high-temperature levels in the PTC and in the turbine inlet create the need for using suitable materials. Another problem regards the storage technology that can be applied in order to store properly the electricity or the produced heating from the PTC. An idea is to use batteries or chemical storage devices, while it is able also to store heat in a tank by applying heat exchangers after the PTC outlet. These issues can be studied in the future in order to evolve the suggested configuration of this work.

4. Conclusions

The objective of the present analysis is the energy, exergy and financial investigation of a novel CO₂-based transcritical Rankine cycle which incorporates parabolic trough collectors inside the cycle for direct heating of the supercritical CO₂. The system is examined in detail for a default scenario, as well as there is parametric analysis, optimization and daily investigation. The results of this work have been obtained by a validated mathematical model which has been programed in Engineering Equation Solver. Below, the most valuable outcomes are summarized:

In the default scenario with 800 K turbine inlet temperature and 200 bar inlet turbine pressure, the collector efficiency is 73.62%, the cycle efficiency is 32.96%, the system efficiency is 24.27% and the electricity product is 44.14 kW.

The increase of the pressure level enhances the collector, the cycle and the system efficiency. On the other hand, the high-temperature increase is beneficial for the cycle performance but not for the collector performance. Thus, for every high-pressure level, there is an optimal high-temperature level that leads to maximum system performance and consequently electricity production.

The parametric analysis proved that the lower heat rejection temperature and the higher solar irradiation levels lead to increased performance of the examined system.

The monthly performance of the system shows that the maximum monthly production is found in July and the maximum system efficiency in June. On the other hand, the minimum electricity product is found in January and the minimum system efficiency in December. The yearly system efficiency is 19.80% and the yearly electricity yield is 81.16 MWh.

The financial analysis proved that the suggested investment is viable. More specifically, the NPV is 143.5 k€, the IRR is 11.94%, the simple payback period is 7.88 years, and the payback period is 8.12 years. The yearly CO₂ emissions avoidance is calculated to be 48.7 tons and the total CO₂ emissions avoidance in the 25-year period is 1217.5 tons.

The comparison of the found results with other literature studies, in energy and economic terms, shows that the present solar CO₂-based transcritical Rankine cycle is better than other solar-driven conventional ORC, something that proves that the proposed system is a promising idea for future power systems.