**4. Results**

On Figure 4, one can see two efficiency lines for CO2; upper one (blue) represents the efficiency of the Rankine-like subcritical CO2 power cycle, while the lower one (green) represents its TFC counterpart. The minimal temperature (condenser temperature) were chosen as the triple-point temperature of the CO2 (216.6 K), and the efficiencies were calculated up to the critical point (304.13 K). On Figure 4, the whole curve can be seen, while on Figure 4b, the high-temperature region can be seen, showing the well-developed maximum on the upper curve, 4 degrees below the critical point.

**Figure 4.** The efficiency of a CO2 Rankine-like Power Cycle (upper lines) and Trilateral Flash Cycle (lower lines) with fixed condenser temperature (216.6 K) and with increasing evaporator temperature. (**a**) full temperature range; (**b**) the vicinity of the critical temperature.

On Figure 5, a set of these curves (approximately 20 for each material) can be seen for all materials listed in Table 1. The first sign of this maximum was found earlier in butane [26]. For water (5a) and carbon dioxide (5b) (two typical wet working fluids), the maxima on all ORC curves are very characteristic. For CO2, red dots are marking these maxima for better visualization; these points are omitted for the other materials to avoid them to distort the shape of the curves. The increase of the maximal cycle temperature (with a fixed minimal cycle temperature) causes an increase of the efficiency for both cycles (the Rankine and the flash-types, represented by the upper and lower curves). Reaching higher maximal temperatures, the speed of this increase will be lower and lower, and finally a maximum, then a decreasing part can be seen. It means that with a fixed condenser temperature, the increase of maximal temperature has an initially positive effect of the first-law efficiency, but when it is reaching the vicinity of the critical point, this positive effect disappears, and the further increase would be contra-productive. On the contrary, for the TFC-like cycles, the increase holds up to the critical point, i.e., having a TFC-like cycle with a wet working fluid, the increase of maximal cycle temperature always increases the efficiency. In this way, the two curves are forming an elongated rain-drop form; the pointed end is located at the condenser temperature (with *η* = 0). Increasing the condenser temperature, one can obtain smaller efficiencies; both the upper and lower curves are shifting down, while the location of the maximum (with smaller and smaller value) shift closer to the critical temperature (see the location of the red dots for CO2), but always remains below that value. The rain-drop shape formed by the upper and lower curves remains, but it will be smaller and smaller. For the sake of the better visibility, the location of the maxima are marked by red dots for the CO2 curves (Figure 5b); its movement can be clearly seen.

Hexafluoroethane (R116) is originally an almost isentropic working fluid. The *T-s* diagram has an inverse S-shaped saturated vapor part, which is very narrow, i.e., the entropy value for the vapor phase (except in the vicinity of the critical temperature) is almost constant. In the novel classification system, it is a type ACNMZ. From the mapping of various working fluid types [21], this is an almost wet isentropic fluid. The point M located at 253.5 K, i.e., choosing condenser temperature above this value, R116 acts like a wet working fluid. Point N is located at 220.9 K, i.e., choosing condenser temperature between 220.9 K and 253.5 K, this fluid can mimic a moderately dry working fluid. Efficiency-curves can be seen in Figure 5c. The curves are very similar to the ones seen for water and CO2; the maxima are still well-developed (although these points are closer to the critical point than for the two wet ones), and the joint ORC-TFC curves are also raindrop-shaped. For this material, one can also say that increasing the maximal cycle temperature has a positive effect on the first-law efficiency, but in the vicinity of the critical point, this effect disappears and changes to negative.

**Figure 5.** Temperature dependencies of ORC (or other Rankine-like cycles) and TFC cycles for various working fluids (Water (**a**), CO2 (**b**), R116 (**c**), Butane (**d**), Neopentane (**e**), Dodecane (**f**)); see text for further explanation.

Butane is a moderately dry working fluid; it means that although the *T-s* diagram is not bell-shaped, but tilted, the titling is not very strong. ORC-TFC efficiency curves can be seen in Figure 5d. Although maxima can be still noticed, they are remarkably less developed than for the previous three materials.

For neopentane (Figure 5e), which has a more tilted *T-s* diagram that butane (i.e., it is drier), the rain-drop shape is already very distorted, and even the flat end is almost pointed (although the transition from ORC to TFC is still smooth). Very small, nearly diminished maxima can be seen, located very close to the critical temperature. In this case, one can assume, that increase of the maximal cycle temperature is always good or almost neutral for the efficiency. For the uppermost curve, the location of the maximum is only 1.97 K from the critical temperature, and even above this point, the decrease of efficiency is very small, so, one can consider it almost temperature-independent.

Finally, a very dry (on *T-s* diagram, represented by a very tilted *T-s* curve) material, dodecane (Figure 5f) were analyzed. One can see, that at low evaporation temperatures, ORC efficiency exceed TFC one, as it happened before, but increasing this temperature further, the difference will be smaller and smaller, and at a given temperature (which depend on the condenser temperature, i.e., different for each pair of curves) it disappears at the crossing of the two curves. Passing that crossing temperature, the efficiency of TFC is higher than for ORC; finally, they will join smoothly at the critical point. In this case, the efficiencies of TFC can form a shallow maximum, close to the critical curve, while the efficiency of ORC increases continuously. Therefore one can conclude, that by using very dry working fluid for an ideal (strictly isobaric heat exchanges and strictly isentropic expansion/compression) basic-layout (recuperator and superheater-free) ORC-system, the increase of maximal cycle temperature can always increase cycle efficiency, even in the vicinity of the critical temperature; additionally, small maxima might be expected for the efficiency of TFC.

3-D version of these lines for water (Figure 6a–d) and butane (Figure 6e–h) are also shown Rotating animated gif version of these figures are provided as Supplementary Material. The distortion of the initial rain-drop shape can be clearly seen.

**Figure 6.** *Cont*.

**Figure 6.** 3-D representation of the efficiency-curves for water (**<sup>a</sup>**–**d**) and butane (**<sup>e</sup>**–**h**). Rotating version of these figures is provided in the Supplementary Material as Figure S1a (water) and Figure S1b (butane) in animated gif-format.

#### **5. Theoretical Reason for the Existence and Disappearance of the Maximum**

A simple geometrical explanation can be given for the appearance of efficiency maximum in wet working fluids. It can be seen in Figure 7 that the ORC or any other Rankinelike cycle (steam Rankine or CO2 power cycles) can be seen as a superposition of a TFC-like and a Carnot-like cycles, although this superposition is not simply a summing of efficiencies. The TFC-like part is the one at lower entropies, resembling a triangle, while the Carnot-part (located on higher entropies) is the rectangular one. Considering that the Carnot-cycle is the best potential cycle between the given maximal and minimal temperature, adding it to a TFC, the Carnot-like part always has a positive contribution to efficiency. In this way, for wet working fluids, *η*ORC is always bigger, than *η*TFC. One can see, that by increasing the maximal cycle temperature (compare Figure 7a,b), the rectangular part will be smaller

and smaller, almost diminishing in the vicinity of the critical temperature (Figure 7c), and when *T*max = *T*cr, it disappears, and the TFC and ORC cycles will be indistinguishable. Therefore, at *T*max = *T*cr, *η*ORC = *η*TFC, and the transition will be smooth and continuous.

**Figure 7.** Rankine-cycle (black) in a wet working fluid as a superposition (but not the simple sum) of a TFC-like (red) and Carnot-like (green) parts with same condenser temperature with a medium (**a**) high (**b**) and almost critical (**c**) evaporator temperature.

This smoothness and continuity affect the shape of the efficiency-temperature curve. Previously we used *T-η* curves, here, for a short time, the inverse should be used, the *η-T* one (see Figure 8). Being *η*ORC > *η*TFC for wet working fluids at every temperature, except *T*cr, the low-entropy part (left side) represents the TFC, and the high-entropy part (right side) represents the ORC. In this representation, the two curves have to have end-points at the maximal temperature (*T*cr). Continuity causes the equality of the two efficiencies at *T*cr, i.e., the two curves have to join in this point, i.e., instead of two end-points, they will have one common end-point. Due to the smoothness, the derivative d*T*/d*η* will be zero in this point; therefore, this will be a smooth maximum with positive d*T*/d*η* on the left side and negative d*T*/d*η* on the right one. It should be recalled, that the left side represents the efficiency of ORC; being d*T*/d*η* positive on the original, *T*-*η* representation, this part would show negative slope. Therefore, in the vicinity of *T*cr, the smooth and continuous transition between ORC and TFC cycles would cause a part, where efficiency decreases with increasing temperature. Contrary, at lower temperatures, one can see an increasing part; this can be possible only by assuming a maximum on the *η*(*T*) curve of the ORC.

One should remember that a strong point of this reasoning was to handle ORC as a superposition of a TFC-like and a Carnot-like part. While this is true for the wet working fluid (Figure 9a), in a dry one, a third part, resembling a distorted, upside-down, inverse TFC, can also be defined, see Figure 9b–d. This part is also less efficient, that a Carnot-like one. Increasing the "dryness", i.e., having more and more tilted *T-s* diagrams and going from ACZM to AZCM class [23], the weight of the middle Carnot-like part will be smaller and smaller. Finally, this part can completely disappear.

**Figure 8.** Schematic temperature-efficiency (**a**) and its inverse, efficiency-temperature (**b**) curves of an ORC-like and TFC-like system, explaining the necessity of the maximum. Further explanation can be found in the text.

**Figure 9.** Different structure of an ORC cycle (black) with a wet, moderately dry, a medium dry and a very dry working fluid, divided into TFC-like (red), Carnot-like (green) and inverse TFC-like (grey) sub-cycles. The diminishment of the Carnot-like part (green) with increasing dryness can be clearly seen, as well as the loss of triangularity of TFC (red) and the appearance of the inverse TFC-like type (grey). The four figures are representing wet (**a**), moderately dry (**b**), medium dry (**c**) and very dry (**d**) working fluids.

This process can be seen in detail in Figure 9a–c. With wet working fluid (a), the ORC can be seen as a superposition of an almost perfectly triangular TFC (red) and an ideal Carnot (green) cycles, as it has been shown before (Figure 7). For moderately dry one (b), a third part (grey) appears, located at higher entropies, resembling a distorted upsidedown TFC. The appearance of this part makes the Carnot-like part narrower, decreasing

its superficial contribution to net first-law efficiency. For medium dry one (c) the isobaric, but not isothermal part of the heat removal is already so significant, that the Carnot-like part totally disappears and the cycle will be the superposition of an almost triangular TFClike and an upside-down TFC-like part; because the Carnot-part disappeared, one might expect that net efficiency might be smaller than previously. Finally, with a very dry working fluid, the isobaric heat removal part preceding the condensation will be so huge, that it will reach the initial TFC-like part, cutting down the corner and terminating the "triangularity" (Figure 9d).

Because this transition happens with increasing dryness, one can expect to see the similar process by going from wet (water, CO2) through isentropic (R116), moderately dry (butane) and dry (neopentane) working fluids to very dry one (dodecane). Upon this transition, the maximum will be less and less significant, and finally, it can disappear, as it is shown for dodecane.

#### **6. Discussion and Conclusions**

The increase of the maximal cycle temperature (which is equal to the evaporator temperature in ideal, simple, superheater-free systems) is thought to be a good way to increase the first-law efficiency in simple cycles, like steam Rankine, Organic Rankine and CO2 power cycles, although there are signs, that problems might arise when this increase reaches the vicinity of the critical point. Some of the problems related to the properties of the materials in this region; there are significant density fluctuations, causing strong local anomalies; also, the accuracy of the Equation of States can be reduced in this region [27,28]; this can also produce problems during design and in the operational phases. Also, from the technical point of view, high evaporation temperatures, especially for dry working fluids, are also not very favorable [29,30]. For example, choosing the evaporation temperature above the temperature of local entropy maximum (the so-called point M), expansion starts in the wet region, even in dry fluids; therefore, droplets can be formed during the early stage of the expansion, causing troubles in the special "dry" expanders.

Here, we approached the problem of the increase of maximal cycle temperature from a purely thermodynamic side, by determining its effect (with fixed condenser temperature) on the first-law efficiency. Efficiencies for six different materials with increasing dryness (water, CO2, R116, butane, neopentane and dodecane) were evaluated. Although anomalies were expected to be seen for dry working fluids [29,30], surprisingly, a very strange effect was seen for wet working fluids, namely the appearance of an efficiency maximum on the efficiency of the Rankine-like cycle (first seen for butane [26]), located close to the critical temperature. The location of the maximum depends on the condenser temperature as well as on the material. For water, taking the smallest possible condenser temperature (the triple point temperature, 273.16 K), the maximum appears 24.6 K below the critical point, while increasing the condenser temperature, it can approach the critical point. For CO2, R116 and butane, the biggest distance between the location of the maximum from the critical point is in the order of 5–10 K (8.68 K for butane, 4.01 K for CO2 and 4.53 K for R116); for neopentane, it is 1.97 K, and finally, for dodecane, no maximum can be seen. Concerning reduced temperatures, the biggest distances between the entropy maximum and the critical points are 0.038 (water), 0.013 (CO2), 0.015 (R116), 0.020 (butane) and 0.005 (neopentane). Since for most real ORC and RC power plants, the operational temperature is not very near to the critical temperature; therefore, technologically, the existence of the maxima are relevant only for two cases (for water, where the maximal distance between the critical temperature end efficiency maximum is almost 25 K, and for butane, where this difference is almost 9 K). For the rest of the presented working fluids, the relevance is rather theoretical than technical. However, one might assume that the temperature differences will be sufficient to have technological importance for some of the not presented wet working fluids.

The presence of these maxima can be clearly shown for several systems by comparing the efficiency of a Rankine-like cycle with a Trilateral Flash-like Cycle (TFC). Some theoretical explanation is also given to see, that for very dry working fluids (like dodecane), the disappearance of this maximum can always be expected for simple, Rankine-like cycles, using only the basic Rankine-layout (pump, liquid heater with evaporator, expander and condenser), without recuperative unit.

The existence of the efficiency maximum for various working fluids is a disproof of the common misconception that the increase of the maximal cycle temperature always has a positive effect on first law efficiency for simple thermodynamic cycles. This disproof is interesting itself, as a strange thermodynamic fact, but also, it can give a new tool for engineers to find optimal cycle temperature for an ORC (or similar) power plant, by designating a temperature range, where a further increase of the evaporation temperature would be contra-productive.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/1996-1 073/14/2/307/s1, Figure S1a,b: Rotating 3-D representation of the efficiency-curves for water and butane, respectively (in animated gif-format).

**Author Contributions:** Conceptualization, A.R.I.; analysis: A.M.A., preliminary analysis: L.K., writing: A.R.I., A.M.A. and L.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was performed in the frame of the FIEK\_16-1-2016-0007 project, implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the FIEK\_16 funding scheme. Part of the research reported in this paper and carried out at BME has been supported by the NRDI Fund (TKP2020 NC, Grant No. BME-NC) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author, [A.R.I.], upon request.

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

#### **Appendix A. Hybrid Solar Systems**

Renewable energy is environmentally friendly; therefore, it is the most viable future option to dispose of some of the environmental problems. Solar is one of the best renewable energy sources, due to availability and high temperature, wherefore solar thermal power, or concentrated solar power (CSP), is an optimal technology to hybridize with other energy technologies for power generation. The hybrid systems using geothermal and solar energy is one of the options with good potentials. This hybrid technology has a promising future, especially in countries that have abundant solar and geothermal energy. The stand-alone geothermal energy can be classified as medium- or low-enthalpy heat source for power plants, therefore, adding solar energy is the best choice, due to increasing the temperature of a geofluid, which increases the first-law efficiency to convert heat to work (and then to electricity) [15,16,31].

In general, a hybrid system can be categorized based on the kind of used energy, such as merging the solar source with one of the renewable energy sources like geothermal, wind, and biomass (these are the so-called high-renewable hybrids); while merging it with natural gas is called medium-renewable hybrids. Finally, merging it with traditionally fueled Brayton or/and Rankine cycle, the systems are called as a low-renewable hybrid.

Reducing CO2 emission is a crucial factor for the power plant. The lowest CO2 emission is with high-renewable hybrids, and the rate increases with medium-renewable, and it is at its highest rate with the low-renewable hybrids; therefore, the increase of the solar part reduces the CO2 emissions [32]. The efficiency of hybrid systems can be increased by adding thermal energy storage that uses the surplus of solar thermal energy at the night time when the temperature is low. Furthermore, the increase of the solar collector field increases the efficiency in all case with or without using thermal energy storage [33]. The second-law efficiency for the hybrid geothermal-solar system is higher than for separate geothermal and solar system at all ambient temperatures [34]. Concerning the increase of maximal temperature above the critical point, the hybrid supercritical ORC (Organic Rankine cycle) outperforms the hybrid subcritical ORC thermodynamically and economically. Therefore, the hybrid supercritical ORC has higher thermal efficiency, and the ability to produce power two to nine times more than a stand-alone subcritical geothermal plant. The geothermal-solar plant is characterized by produce relative constant power during its period operation, concerned to stand-alone solar ones [18,35].

Concerning working fluids, some preliminary result was achieved, showing that that isobutene (also called isobutylene) is a suitable working fluid in the hybrid geothermalsolar system; it has the best performance with the lower GWP, when compared to other working fluids as isobutene, *n*-butane, and 2-butene [36]. The hybrid solar-geothermal is less vulnerable to the seasonal changes in ambient temperatures than stand-alone solar one, as it was observed in an Australian geothermal-solar hybrid system case study, demonstrating that a properly designed hybrid plant can outperform a stand-alone solar thermal plant in terms of the cost of electricity production [37].

As a final issue, the thermal stability of the working fluid and the capacity of the system components might limit the solar superheating in the hybrid geothermal-solar so, to overcome this problem, water can be used as the heat transfer fluid for ORC that operates under 175 ◦C [38].
