**3. Method**

Concerning the basic ORC system, we are using the most straightforward and simplest layout to minimize the installation and maintenance costs for the system. The layout assuming a hypothetical dry working fluid can be seen in Figure 1a. It contains a pump, used to compress the working fluid from the low-pressure liquid state (1) to high pressure one (2). The first heat exchanger is for heating the compressed working fluid. It has two parts, the liquid heater (LH) to heat up the compressed liquid to the saturated state (2→3) and the evaporator (EV), to evaporate the saturated liquid to saturated vapor state (3→4). Cost minimization comes here first; no superheater is included, not even for wet working

fluids. The fluid expands directly from this saturated vapor state (4) through the expander (E) to cold, dry vapor state (5). It will be cooled down in the condenser (C) in two steps, first by pre-cooling the vapor down to the saturated state (5→6), then by condensing it to the saturated liquid state (1). The heat extracted by the pre-cooler is lost here; for the sake of simplicity, recuperative or regenerative heat exchangers are omitted. Corresponding *T*-*s* diagram can be seen in Figure 1b. Using wet working fluid, the expansion line (4→5) would run in the two-phase region (wet vapor) and points 5 and 6 would be identical; therefore, pre-cooling would not be necessary.

**Figure 1.** Schematic layouts and *T-s* diagrams for simple single-source (**<sup>a</sup>**,**b**) and hybrid (**<sup>c</sup>**,**d**) ORC systems. P: pump; LH: liquid heater; EV: evaporator; T: turbine or expander; PC: pre-cooler; C: condenser.

For the hybrid system, we have a second heat source; corresponding figures are Figure 1c,d. In this case, higher initial compression (from point 1 to 2) is used, so the pressure ratio *p*2/*p*1 will be higher. Pressurized working fluid (in liquid state) will be preheated by the primary heat source of the maximal temperature provided by the primary source (from point 2 to point 2\*; this was the point corresponding to maximal temperature in the single source cycle). Then the secondary heat source (for example the solar one) is heating the fluid up to the maximal cycle temperature (from point 2\* to point 3), and also the heat of this source is used to evaporate it (from point 3 to 4). Expansion and cooling steps (the 4-5-6-1 sequence) would be similar to the previous case, although the expansion part (from point 4 to 5) will be more significant and the cooling of the dry vapor (5–6) would start at a higher temperature.

As it has been already stated, we are discussing only the thermodynamic cycle here; therefore, all steps are considered to be ideal. Compression and expansion steps are isentropic, while for the heating and cooling steps, pressure drops are omitted, and strict

isobaric heat exchange processes are assumed. Because for first-law efficiency, the knowledge of the intermediate temperature ( *T*max1) is not needed, only the effect maximal cycle temperature ( *T*max2) will be investigated.

Although wet working fluids are considered as not suitable ones for ORC applications [19], due to their unfavorable behavior (most remarkably the ones related to the appearance of droplets), in this paper we are studying the thermodynamic, rather than technical aspects of ORC. Therefore, both dry and wet working fluids were checked.

Efficiencies are going to be calculated for six different materials (see Table 1), representing a wide range of working fluids from wet ones (almost symmetrical, bell-shaped *T-s* diagrams) to very dry ones (with very skewed *T-s* diagrams), see Figure 2. Four of them might be used for hybrid geothermal/solar ORC or RC while the other two having very low critical temperature might be used in cryogenic cycles [20–22]. Therefore here—when the heat source is solar heat, presumably providing source temperatures exceeding the critical temperatures of these two materials—they are only used for demonstration purpose. Two of them are wet working fluids (using the traditional classification), or class ACZ working fluids [23]. The other four are dry working fluids. Concerning the sequence-based classification [23], dry working fluids can be divided into two classes; ACZM and AZCM types. Letters represent the special point on the *T-s* diagrams; A is the initial point of the *T-s* saturation curve, Z is the final one, C is the critical point and M is the point corresponding to maximal entropy. From these points, four-letter sequences can be built, based on the entropy values of these points. Initial point A always has the smallest entropy, while M has the highest one; therefore, only sequences starting with A and ending with M are possible. Having only two intermediate points, these are the ACZM and AMCZ. Liquid states are located in the A-C part of the curve, while vapor ones are stretching in the C-M-Z part. The crucial difference is that for ACZM, the entropy values for liquid states are always below the entropies of any of the saturated vapor states; while for AZCM, there are liquid states with entropies higher than for some vapor states. This means that for AZCM-type fluids, isentropic expansion starting from some saturated liquid states can reach the fully vaporized dry-vapor region. The ACZM-ones are closer to the wet-ones (ACZ), as it is shown in Ref. [21]. Three of the working fluids represented here are ACZM, and one for AZCM. In several fluids, a local entropy minimum (N) also can be seen; these fluids can also be classified as dry ones, or alternatively as "real isentropic ones" because, for them, it is possible to have an isentropic expansion step from saturated vapor state into another saturated vapor state [21,23,24]. In this paper, this fifth point will not be relevant.


**Table 1.** Working fluids used in this study with some of their relevant properties.

One more piece of information is needed concerning the use of this working fluid classification [23]. The ultimate initial and final points of the saturation curve in *T-s* diagrams are related to the triple points (the last point of the liquid/vapor equilibrium; solid-phase appears at that point). The corresponding entropy values are material properties, and therefore the classification (being the location of C and M points also material-dependent) is also a material property. However, in several cases, triple points are located in technically irrelevant, low temperatures (like for butane, where it is located at 134.6 K, which is −138.5 ◦C). Therefore, it might be better to terminate the *T-s* curve in a technologically more relevant minimal temperature, for example, at the minimum cycle temperature. In ORC applications

(with air cooling) this should be done somewhere around (or exactly at) 20 ◦C (293.15 K). The relevant part of the curve (relevant means that the one above 20 ◦C) might mimic a novel class; for example, the vicinity of the top of all the *T-s* diagrams (above point M) looks like a wet, ACZ type curve. For the sake of clarity, sometimes these "relative" classes are distinguished by marking the related points with an upper-case star (\*), like A\*CZ\*M. Since for hybrid solar/geothermal case, usually, 20 ◦C can be taken as minimum temperature; therefore, both absolute and relative classes are shown in Table 1.

**Figure 2.** Temperature dependencies of ORC (or other Rankine-like) and TFC cycles for various working fluids (Water (**a**), CO2 (**b**), R116 (**c**), Butane (**d**), Neopentane (**e**), Dodecane (**f**)); see text for further explanation.

The efficiency values for various maximal and minimal temperature pairs were calculated with MATLAB software; working fluid data were taken from the NIST Chemistry Webbook [25]. On the schematic *T-s* diagram of a wet working fluid, the blue part represents the saturated liquid state, and the red part represents the saturated vapor state.

The efficiency calculation for ORC and TFC were done in the manner shown in Figure 3. First, a minimal cycle temperature was chosen (represented by the light green line) and used as condenser temperature. For this given condenser temperature, various maximal temperature values were chosen, up to the critical temperature. For each calculation, the temperature interval between the condenser and maximal temperature was divided into 500 equal part; efficiency values were calculated for all of these 500 points, using them as maximal cycle temperatures. Some of them are shown in Figure 3a, corresponding to the evaporation plateau (ORC) and on Figure 3b, corresponding to the topmost point of the "triangle" (TFC). In this way, the efficiency of the first cycle (1-2-3-4-1) were calculated. After finishing one run, a new minimal temperature was chosen slightly above the previous one (by shifting the green base-line a bit higher), and the calculation was repeated for the cycle (1-2a-3a-4a-1) up to the last one, just one step below the critical temperature, to cycle (1-2d-3d-4d-1). In this way, separate curves representing the Tmax dependence of efficiencies were obtained for different condenser temperature.

**Figure 3.** Steps of the efficiency-calculation for ORC (**a**) and TFC (**b**) cycles.

Concerning efficiency, the actual values were calculated by using the following equation:

$$
\eta = \frac{Q\_{in} - Q\_{out}}{Q\_{in}},
\tag{1}
$$

where (concerning the isobaric heat exchanges) the heat added to the system is *Qin* = *h*3 − *h*2 and the heat taken from the system is *Qout* = *h*4 − *h*1; numerical enthalpy values were taken from the NIST Chemistry Webbook [25] as a function of temperature and entropy.
