# Dynamic Simulation and Exergo-Economic Optimization of a Hybrid Solar–Geothermal Cogeneration Plant

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## Abstract

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## 1. Introduction

^{−1}for each net MW produced. Second Law efficiency of the plants ranges from 20% to 45% depending on the brine inlet temperature and on the selection of the energy conversion cycle. Astolfi et al. [19,20] investigated binary ORC power plants for the exploitation of low-medium temperature geothermal sources in the temperature range of 120–180 °C and defining the optimal combination of fluid, cycle configuration and cycle parameters. Results showed that supercritical cycles, employing fluids with a critical temperature slightly lower than the temperature of the geothermal source, lead to the lowest electricity cost. El-Emam and Dincer [22] presented thermodynamic and economic analyses for a geothermal regenerative ORC. The authors assessed that the ORC evaporator and condenser showed the highest exergy destruction rates. Habka and Ajib [23] investigated some hybrid integrations of cogeneration plants based on ORC driven by low-temperature geothermal water. The results showed excellent performance in terms of power production and exergy efficiency.

_{2}O/LiBr absorption chiller; a geothermal well (at 95 °C); a solar field obtained by new prototypal flat-plate evacuated solar collectors. Geothermal energy is obtained by a combination of a downhole heat exchanger and by fluid extraction. Solar energy is used only to boost ORC power production. The system can produce simultaneously electrical energy, cooling energy and thermal energy for a hotel located in Ischia (South of Italy). The system showed excellent energy performance indexes. From the economic point of view, good results are also obtained. In fact, in the worst operating conditions the Simple Payback Period is 7.6 years, decreasing to 2.5 years in the most convenient considered scenario (public funding and full utilisation of the produced thermal energy).

- 1-year dynamic simulation: this paper presents the development of a dynamic simulation model of an Organic Rankine Cycle (ORC) fed by solar and medium-enthalpy geothermal energy. Zero-dimensional energy and mass balances simulate the ORC model with Engineering Equation Solver (EES), which allows one to evaluate the off-design performance of the plant setting the geometrical features of ORC heat exchangers and the design conditions of the turbine. Moreover, the simulation model has been implemented in a more complex TRNSYS project in order to consider both geothermal and solar sources, simulating solar collectors, geothermal well, heat exchangers, pumps, tank and controllers. Therefore, a dynamic simulation of the hybrid plant has been obtained. In the authors’ knowledge, none of the papers available in literature includes a 1-year dynamic simulation of a hybrid solar geothermal ORC system, which allows one to vary the ORC driving temperature as a function of the available solar radiation.
- Exergy analysis: a detailed exergetic analysis is performed, aiming at detecting the sources and the magnitude of irreversibilities within the components of the plant and within the plant as a whole. Such analysis allows one to calculate the magnitude of the irreversibilities and to analyze possible actions to reduce such losses in the system. An exergetic optimization has been also performed with the scope to calculate the set of design parameters, minimizing the overall system exergy destruction.
- Thermo–economic optimization: a rigorous optimization has been implemented varying the main design parameters (such as collectors area and tank volume). Detailed heuristic/deterministic algorithms have been implemented in order to calculate the set of the design parameters minimizing the payback period of the system.

## 2. System Layout

^{2}Parabolic Trough Collector (PTC) solar field raises the temperature of the diathermic oil. Considering the possible temperature achieved, n-pentane is selected as organic fluid, in accordance with the results shown in reference [6]. In fact, according to the results found in reference [6] and to the findings of the majority of the papers cited in the previous section, n-pentane shows one of the most suitable (T, s) diagrams, for the selected inlet hot fluid temperatures. A possible further improvement of system energetic and exergetic efficiencies could be achieved using mixtures from alkanes (e.g., butane/pentane mixtures). In fact, for such mixtures evaporation and condensation temperatures show temperature glides which better fit with hot fluid and cooling water temperature profiles. This will reduce exergy losses. Unfortunately butane/pentane mixtures are not available in conventional thermodynamic databases. Therefore, the implementation of such fluid will require a different work aiming at evaluating all its correlations.

## 3. TRNSYS Simulation Model

#### 3.1. Exergetic Model

_{sun}to 3/4 of the black body temperature, i.e., 4,350 K [43].

- PTC$${\dot{Ex}}_{des,PTC}={\dot{Ex}}_{sun}+{\dot{Ex}}_{2D}-{\dot{Ex}}_{3D}$$$${\text{\eta}}_{ex,PTC}=\frac{{\dot{Ex}}_{3D}-{\dot{Ex}}_{2D}}{{\dot{Ex}}_{sun}}$$The previous equations represent the most common formulation of exergy efficiency for solar thermal collectors. Equations (3)–(5) typically lead to low exergetic efficiency values due to the significant difference between Tsun and the average temperature of the fluid flowing in the collectors. To overcome this issue, some researchers have proposed a new approach which consists in referring the exergetic efficiency to the absorber plate temperature, rather than Tsun.. Using this approach, an different definition of the exergetic efficiency of the solar collectors can be obtained (${\text{\eta '}}_{ex,PTC}$) [45]:$$\text{\eta}{\prime}_{ex,PTC}=\frac{{\dot{Ex}}_{3D}-{\dot{Ex}}_{2D}}{{\dot{Ex}}_{in,PTC}}$$$${\dot{Ex}}_{in,PTC}=A\xb7I\xb7\left(1-\frac{{T}_{a}}{{T}_{pl}}\right)$$$${T}_{pl}=\frac{{T}_{2D}+{T}_{3D}}{2}$$
- ORCExergetic analysis has been conducted for each component of the ORC subsystem. None of the ORC components involves chemical reactions. Therefore, only physical exergy related to the material and energy flows for each component have been considered for the formulation of exergy balances. Physical exergies for all the state points of the system are calculated as shown in ref. [43].The overall ORC plant exergetic performance has been calculated as follows:$${\text{\eta}}_{ex}=\frac{\dot{{W}_{net}}+{\dot{m}}_{w}\left(e{x}_{w,out}-e{x}_{w,in}\right)}{{\dot{m}}_{oil}\left(e{x}_{1p}-e{x}_{3p}\right)}=1-\frac{{\dot{Ex}}_{des}}{{\dot{m}}_{oil}\left(e{x}_{1p}-e{x}_{3p}\right)}=1-{\displaystyle \sum}_{i}^{n}{\text{\delta}}_{i}$$
- Geothermal HE$${\dot{Ex}}_{des,GeothermalHE}={\dot{Ex}}_{2G}+{\dot{Ex}}_{6D}-{\dot{Ex}}_{3G}-{\dot{Ex}}_{7D}$$$${\text{\eta}}_{ex,GeothermalHE}=\frac{{\dot{Ex}}_{7D}-{\dot{Ex}}_{6D}}{{\dot{Ex}}_{2G}-{\dot{Ex}}_{3G}}$$
- Recuperator$${\dot{Ex}}_{des,Recuperator}={\dot{Ex}}_{3G}-{\dot{Ex}}_{4G}-{\dot{\u2206Ex}}_{rec}$$$${\text{\eta}}_{\text{ex},Recuperator}=\frac{{\dot{\u2206Ex}}_{rec}}{{\dot{\text{Ex}}}_{3\text{G}}-{\dot{\text{Ex}}}_{4\text{G}}}$$$${\dot{\u2206Ex}}_{rec}={\dot{\text{Ex}}}_{4\text{W}}-{\dot{\text{Ex}}}_{3\text{W}}$$
- Total$${\dot{Ex}}_{des,Total}={\dot{Ex}}_{sun}+{\dot{\u2206Ex}}_{wells}+{\dot{W}}_{aux}-{\dot{\u2206Ex}}_{cond}-{\dot{\u2206Ex}}_{rec}-{\dot{W}}_{net,ORC}$$$${\text{\eta}}_{ex,Total}=1-\frac{{\dot{Ex}}_{des,Total}}{{\dot{Ex}}_{sun}+{\dot{\u2206Ex}}_{wells}}$$$${\dot{W}}_{aux}={\dot{W}}_{P1}+{\dot{W}}_{P2}+{\dot{W}}_{P3}$$$${\dot{\u2206Ex}}_{cond}={\dot{Ex}}_{2W}-{\dot{Ex}}_{1W}$$
- Balance of the Plant (BOP)$${\dot{Ex}}_{des,BOP}={\dot{Ex}}_{des,Total}-{\dot{Ex}}_{des,PTC}-{\dot{Ex}}_{des,GeothermalHE}-{\dot{Ex}}_{des,Recuperator}-{\dot{Ex}}_{des,ORC}$$

#### 3.2. Economic Model

- PTC [46]C
_{PTC}= 600 · A_{PTC} - Geothermal wells [47]C
_{Geothermal wells}= 1000 · DP_{wells}· N_{wells} - Geothermal HE [48]C
_{Geothermal HE}= 17,500 + 699 · (A^{0.93}) - Recuperator [48]$${C}_{\text{Recuperator}}=\text{}20\xb7{\dot{Q}}_{rec}$$
- BOP (including pipes, tanks, valves, etc.) [24]C
_{BOP}= 0.10 · (C_{PTC}+ C_{Geothermal HE}+ C_{Recuperator}) - Management and maintenanceC
_{M&M}= 0.02 · (C_{PTC}+ C_{Geothermal wells}+ C_{Geothermal HE}+ C_{Recuperator}+ C_{ORC})

Parameter | Value |
---|---|

Generated electricity from solar energy revenue | 0.34 €/kWh |

Generated electricity from geothermal source revenue | 0.165 €/kWh |

Thermal energy produced revenue | 0.052 €/kWh |

## 4. Results and Discussion

^{2}. The daily radiation varies from 0.52 kWh/(m

^{2}·day) on a cloudy winter day up to 8.33 kWh/(m

^{2}·day) on a sunny summer day. Annual radiation is 1.56 MWh/(m

^{2}·year). The minimum monthly radiation is achieved in December, 54.95 kWh/(m

^{2}·month), whereas the maximum one occurs in June, 203.58 kWh/(m

^{2}·month). The case study is performed using the set of system design parameters shown from Table 2 to 8. Here, an iterative procedure has been implemented in order to achieve a suitable design configuration of the ORC system, as discussed in references [6,18,24,36]. In particular, for the selected operating fluid and design inlet temperatures and mass flow rates (hot and cooling stream), heat exchangers are appropriately designed selecting tubes and shells diameters on the basis of a maximum flow velocity (1–2 m/s). Such parameters (diameters, pitch, thickness, etc.) are fixed using TEMA standards [24]. Then, number of pipes, pipe lengths and shell lengths are iteratively varied in order to achieve the desired temperature profiles and heat transfer rates. Finally, such values are subject to a parametric optimization in order to maximize the energetic efficiency of the ORC.

Parameter | Value |
---|---|

Inlet temperature ${T}_{in}$ | 135 °C |

Expansion ratio ${\text{\tau}}_{design}$ | 7 |

Isentropic efficiency ${\text{\eta}}_{design}$ | 0.80 |

Mass flow rate ${\dot{m}}_{design}$ | 12 kg/s |

Parameter | i = 1 | i = 2 | i = 3 | i = 4 |
---|---|---|---|---|

a_{i} | −0.2679 | 0.3338 | −1.308 | 84.59 |

b_{i} | 0.6987 | −10.03 | 43.44 | – |

Parameter | Value |
---|---|

Shell Length | 11 m |

Tubes Number | 626 |

Shell Diameter | 1482 mm |

Tube Diameter Out | 19.1 mm |

Tube Pitch | 23.75 mm |

Tube Thickness | 0.71 mm |

Enhancement parameter for the finned tube area | 3.5 |

Parameter | Value |
---|---|

Shell Length | 9 m |

Tubes Number | 400 |

Shell Diameter | 800 mm |

Tube Diameter Out | 19.05 mm |

Tube Pitch | 25.4 mm |

Tube Thickness | 1.5 mm |

Parameter | Value |
---|---|

Shell Length | 15 m |

Tubes Number | 781 |

Shell Diameter | 1460 mm |

Tube Diameter Out | 19.1 mm |

Tube Pitch | 25.4 mm |

Tube Thickness | 0.71 mm |

Row number | 8 |

Enhancement parameter for the finned tube area | 3.5 |

Parameter | Value |
---|---|

Collectors area | 10,000 m^{2} |

Mass flow solar cycle | 40 kg/s |

Mass flow geothermic fluid | 40 kg/s |

Mass flow diathermic oil primary heat exchanger input | 50 kg/s |

Tank volume | 0.010 · A_{PTC} m^{3} |

Geothermal source temperature | 160 °C |

Power of the Pump 1 | 7.5 kW |

Power of the Pump 2 | 7.5 kW |

Power of the Pump 3 | 60 kW |

Reinjection temperature | 70 °C |

Water inlet recuperator temperature | 55 °C |

Water outlet recuperator temperature | 67 °C |

Parameter | Value |
---|---|

Tube Length | 15 m |

Shell Diameter | 0.8382 m |

Tubes Number | 600 |

Tube Passages | 2 |

Tube Thickness | 0.0015 m |

Tubes Diameter Out | 0.01905 m |

Tube Pitch | 0.0254 m |

#### 4.1. Daily Analysis

_{out,PTC}) becomes higher than the tank bottom temperature (T

_{cold,tank}). Then, the PTC loop continues to be activated until the late evening when it is stopped in order to prevent heat dissipation. Note that the solar loop is equipped with a fixed speed pump. Therefore, the PTC outlet temperature (T

_{out,PTC}) varies as a function of solar radiation availability. The maximum temperature reached by PTC (T

_{out,PTC}) on this day is 220 °C, while the one at the top of the tank (T

_{hot,tank}) is 207 °C. Note that T

_{hot,tank}is also the temperature of the diathermic oil to be supplied to the ORC. As a consequence, during daytime the ORC driving temperature varies significantly. Therefore, solar radiation also determines a general increase of all the temperatures within the ORC cycle, as shown Figure 4. The maximum evaporation temperature (T

_{eva}equal to 150 °C) is achieved around midday. Such a figure also shows a significant variation of the evaporation temperature (T

_{eva}) whereas the condensing temperature (T

_{1}) shows only a slight increase. As a consequence, the evaporation pressure (P

_{eva}) increases up to 16 bar, while the condensation one (P

_{cond}) does not significantly vary (not shown in Figures for brevity).

_{ORC}) (Figure 5), which reaches a maximum of 0.146. Figure 5 also shows that during the central hours of the day a significant increase of ORC mass flow rate is detected. In fact, as expected, the higher the temperature of the fluid entering the turbine, the higher the mass flow rate. As mentioned above, the net power produced by the plant is related both to the expansion ratio and mass flow rate. Therefore, the simultaneous increase of both expansion ratio and mass flow rate determines the increase in net power production, shown in Figure 6, where a maximum net power of the whole plant (${\dot{W}}_{net}$) of 1.03 MW is obtained. It is worth noting that the net power also includes the electrical demands of Pump 1, Pump 2 and Pump 3. In Figure 6, the higher contribution of solar energy is also clearly shown. PTC thermal power produced (${\dot{Q}}_{PTC}$) achieves a peak of 5.14 MW determines a corresponding increase of thermal power supplied to the ORC (${\dot{Q}}_{in,ORC}$). Consequently, the higher the ORC input thermal power, the higher the electrical and cogenerative (${\dot{Q}}_{cond}$) powers. During the night, thermal power supplied by the geothermal heat exchanger (${\dot{Q}}_{geothermal,HE}$) is almost constant since the overall system operates very close to the steady state. Conversely, during daylight thermal capacity of geothermal HE (${\dot{Q}}_{geothermal,HE}$) decreases (due to the higher oil inlet temperature), determining a corresponding increase of the heat recovered by the recuperator (${\dot{Q}}_{rec}$).

_{sun}) and the fluid to be heated decreases, showing a maximum of 0.24. Conversely, when the PTC exergetic efficiency is defined by Equations (6) and (7), a much higher reading of such parameter is obtained (even higher than 60%) due to the lower temperature difference between the source (the plate of the collector) and the fluid to be heated. It is worth noting that for the calculation of exergy efficiency of the solar collector, although the approach based on Equations (3) and (5) is the most common selection in literature, the second alternative approach—Equations (6) and (7)—must be considered as the most appropriate one for the system under investigation which also includes geothermal energy. For that energy, input exergy is calculated using the exergy of the extracted geothermal brine, rather than the exergy related to the magma temperature. Therefore, for a suitable comparison, solar collector exergetic efficiency must be calculated with respect to the plate temperature rather than the solar surface temperature. As for the geothermal HE, the exergetic efficiency slightly increases during daytime, achieving a peak of 0.96, due to the lower temperature difference between the hot stream (geothermal brine) and the cold one (oil exiting from the ORC). Conversely, ORC and recuperator exergetic efficiencies slightly decrease during daytime, reaching a minimum of 0.64 and 0.54, respectively. Finally, the total exergetic efficiency of the system is the same shown in winter during the night. Conversely, the increase of solar radiation determines a decrease of the overall system exergetic efficiency, reaching a minimum (0.26) slightly lower than in the winter case.

#### 4.2. Weekly Analysis

_{wells}) is dominant during the year. It is almost constant during the year around 628 MWh/week. The geothermal contribution is about from 63% to 91% of the overall exergetic fuel of the plant. Exergy related to the solar energy (Ex

_{sun}) varies during the year, increasing during the summer. It ranges from 50 MWh/week obtained in the second week, to 350 MWh/week of the thirty-seventh week. The exergy associated to the auxiliary components (Ex

_{aux}) is basically constant and negligible, being about 12 MWh/week.

_{rec}), ranging from 230 to 240 MWh/week, corresponding to a variation from 53% to 67% of the total exergy product. The exergy product related to the heat recovered by the condenser (ΔEx

_{cond}) varies from 46 to 53 MWh/week, while the net exergy obtained in the ORC (Ex

_{net,ORC}) ranges from 82 MWh/week of the first week to 100 MWh/week of the thirty-fifth week. This result shows that heat recovery from recuperator and condenser is crucial in order to reduce exergetic dissipations in the system.

#### 4.3. Annual Energetic, Exergetic and Economic Analyses

^{3}MWh compared to 35.7 × 10

^{3}MWh exchanged in the geothermal HE. In terms of exergy, instead, exergetic flow received from the Sun is equal to 9.4 × 10

^{3}MWh, compared to 32.7 × 10

^{3}MWh of geothermal exergy input. It is worth noting that the exergy destroyed in PTC is an order of magnitude higher than that destroyed in the geothermal HE, being respectively 8.0 × 10

^{3}MWh and 825 MWh. In fact, the solar thermal collectors operate subject to large temperature differences, (between the Sun temperature and the one of the fluid to be heated), causing large irreversibilities. Conversely, in geothermal HE, the temperature differences between the geothermal brine and diathermic oil are much lower, allowing one to achieve a better exergetic performance. Therefore, it is reasonable to expect that the values of exergetic efficiency in these components are significantly different. In fact, it is 14.94 for the PTC ( ${\text{\eta}}_{ex,PTC}$) and 92.40% for the geothermal HE (${\text{\eta}}_{ex,GeothermalHE}$). In addition, as expected, the exergetic efficiency calculated referring to the absorber plate temperatures rather than Tsun ( $\text{\eta}{\prime}_{ex,PTC}$) is much higher than the previous one (${\text{\eta}}_{ex,PTC}$), being 41.53%. As shown by the previous figures, the system is dominated by the geothermal source. As a consequence, the calculated solar fraction is very low (10%).

Parameter | Value |
---|---|

PTC thermal energy | 4126 MWh |

Geothermal HE thermal energy | 35,736 MWh |

Recuperator thermal energy | 99,894 MWh |

ORC inlet thermal energy | 34,150 MWh |

ORC net electric energy | 4,634 MWh |

Condenser thermal energy | 32,475 MWh |

Solar fraction | 0.1035 |

Exergy associated to the solar source | 9,422 MWh |

Exergy associated to the geothermal source | 32,727 MWh |

Exergy associated to the recuperator | 12,247 MWh |

Exergy associated to the condenser | 2,549 MWh |

Exergy associated to the auxiliary components | 612 MWh |

Exergy destruction in the PTC | 8,014 MWh |

Exergy destruction in the geothermal HE | 825 MWh |

Exergy destruction in the recuperator | 9,631 MWh |

Exergy destruction in the ORC | 3,466 MWh |

Exergy destruction in the BOP | 1,394 MWh |

Total exergy destruction in the plant | 23,330 MWh |

${\text{\eta}}_{ORC}$ | 0.1357 |

${\text{\eta}}_{ex,ORC}$ | 0.6959 |

${\text{\eta}}_{ex,PTC}$ | 0.1494 |

$\text{\eta}{\prime}_{ex,PTC}$ | 0.4153 |

${\text{\eta}}_{ex,GeothermalHE}$ | 0.9240 |

${\text{\eta}}_{ex,Recuperator}$ | 0.5598 |

${\text{\eta}}_{ex,Total}$ | 0.4544 |

PTC cost | 6,000,000 € |

Wells cost | 800,000 € |

Geothermal HE cost | 259,809 € |

Recuperator cost | 270,000 € |

ORC cost | 4,800,000 € |

BOP cost | 652,981 € |

Management and maintenance cost | 255,656 € |

Electric energy produced revenue | 736,540 €/year |

Thermal energy produced revenue | 6,883,192 €/year |

SPB | 1.74 years |

^{3}MWh) and exergy (12.2 × 10

^{3}MWh), the main contribution is due to the recuperator. The electric energy produced in a year is equal to 4.6 × 10

^{3}MWh. It is important to notice that the value of the ORC energetic (${\text{\eta}}_{ORC}$) and exergetic ${\text{\eta}}_{ex,ORC}$ efficiencies, respectively 13.57% and 69.59%, and the value of the exergetic efficiency of the whole system (${\text{\eta}}_{ex,Total}$) equal to 45.44%.

#### 4.4. Parametric Analysis

#### Collector Area

^{2}. In such a large range, the rated solar thermal capacity varies approximately from 2.5 to 7.5 MW. Lower or higher PTC areas are not considered since in those cases the solar fraction would be dramatically low or high, respectively Annual thermal energies are shown in Figure 14. Obviously, the PTC thermal energy increases in case of large solar fields, from 2,100 MWh (5,000 m

^{2}) up to 6,200 MWh (15,000 m

^{2}). For all the configurations, the PTC thermal energy is an order of magnitude lower than the one exchanged in geothermal HE, which decreases in the case of larger solar fields, as discussed in the previous sections. In fact, geothermal energy used for ORC varies from 36,500 MWh (5,000 m

^{2}) to 35,000 MWh (15,000 m

^{2}). The ORC inlet thermal energy ranges from 32,700 to 35,800 MWh, in case of larger solar fields. This increase involves also a growth of the net electric energy obtained by the ORC, ranging from 4,400 MWh up to 4,900 MWh. In addition, thermal energy recovered in the condenser and the recuperator increase, ranging, respectively, from 31,400 MWh up to 33,600 MWh and from 99,100 MWh up to 101,000 MWh.

_{ex,TOT}) which ranges from 0.500 for the minimum collector area to 0.418 for the maximum one. Therefore, for a larger PTC area, exergy fuels increase more than exergy products. This is due to the fact that the larger the solar field, the higher the PTC temperature, determining a lower efficiency and therefore higher irreversibilities. Hence, the exergy destruction in the whole system (Ex

_{des,TOT}) increases significantly, varying from 55.6 to 86.0 MWh, as shown in Figure 15. This figure also shows that the ORC efficiency (η

_{ORC}) is almost constant, ranging from 0.135 to 0.137. Therefore, in terms of energy, a slight increase of the products compared to the fuels has been obtained by the increasing collector area.

_{sol}) and SPB increase for larger solar fields. As discussed above, an increase of the PTC area involves an increase of the energy produced by the PTC and a decrease of the energy exchanged in the geothermal HE. Therefore, the solar fraction increases for larger solar fields. For the considered range of variation of the PTC area, the solar fraction varies approximately from 5% to 15%. Therefore, smaller solar fields are not taken into account since their contribution to the overall energy production would be negligible. Conversely, larger solar fields are not feasible, since in those cases the system’s economic profitability decreases dramatically. In fact, the trend of the SPB shows a linear increase for larger solar fields. It is worth noting that, for the selected range of variation, a monotonic SPB trend is found and therefore, the minimum coincides with the lower limit. This is quite surprising since, from the economic point of view, two opposite trends must be considered in the case of solar fields: a higher capital cost and a higher electrical energy production. Such opposite trends should determine a non-monotonic trend of the SPB. Unfortunately, the minimum occurs very close to zero, corresponding to an extremely low solar fraction that suggests that, in those conditions, the hybridization does not make sense. As a consequence, this result suggests a better economic profitability of the geothermal source compared with the solar one.

#### 4.5. Exergetic and Thermo-Economic Optimizations

_{PTC}), the unitary volume tank and the diathermic oil mass flow rate circulating in the solar loop (${\dot{m}}_{solarloop}$). The first variable varies in the same range used in the parametric analysis, the second between 0.005 and 0.015 m

^{3}/m

^{2}, and the last one between 20 and 60 kg/s. The initial conditions are the design ones discussed above. The objective functions are two: the total exergy destruction (Ex

_{des,TOT}) and the SPB. Obviously, the target is to minimize such functions. The results are shown in Figure 17 and Figure 18. These figures show, for each iteration of the optimization procedures, the values of the objective functions and the ones of the design variables. As a consequence, the optimum is found for the last iteration of the optimization procedure. Here, the optimal values of the design variable, and the corresponding readings of the objective functions can be read.

^{2}, improving the exergetic efficiency from 0.45 to 0.50.

^{3}/m

^{2}is considered the smallest size in order to achieve a suitable hydraulic separation between ORC and solar loops. For lower tank capacities, severe turbulences in pump operation could be achieved. However, the variation of tank volume only slightly affects the objective functions under consideration, as discussed above. However, the optimal configuration is found for the minimum value volume of the tank. In fact, its increase leads to an increment of the thermal losses to the environment. Conversely, the relative amount of thermal losses decreases due to the smaller surface to volume ratio achieved in the case of larger tanks. Furthermore, it is worth noting that thermal losses (and exergy losses) can be reduced by increasing tank insulation. Therefore, the total exergy destroyed increases with the increase of the tank volume. From the economic point of view, using larger tanks is never profitable. In fact, the capital cost is higher and the overall energy production is lower. Therefore, SPB increases for larger tank volumes. Thus, it can be concluded that the tank must act only as hydraulic separator between the solar loop and the ORC loop.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Appendix A: ORC Model

#### Heat Exchangers

#### Primary Heat Exchanger

#### Preheater

#### Evaporator

_{nb}, and the natural convection heat transfer coefficient h

_{natural}:

_{natural}= 250 (W/m

^{2}·K) and the bundle boiling factor F

_{b}= 1.5. The mixture boiling correlation factor F

_{c}= 1 for pure fluid and azeotropic mixtures [61]. A reduced pressure formulation without a surface–fluid parameter or fluid physical properties has been developed by Mostinski [62], which is valid for a wide range of reduced pressures and several fluids.

#### Regenerator

#### Condenser

_{s,t,}has been evaluated. In addition, considering the laminar film condensation of a quiescent vapour on an isothermal horizontal tube, the Nusselt theory [63] has been used:

^{3}is obtained.

_{f}is the condensate retention angle. It is the angle between the top of the tube and the point where the tube begins to flood with condensate. Its evaluation, Equation (A.15), has been proposed by Honda et al. [67]:

#### Expander

#### Pump

_{eq}is set to the design value of 2,900 rpm:

#### Diathermic Oil

^{®}has been selected as the diathermic oil. It can be used both in liquid and vapour phase, and its range applications are, respectively, from 15 °C up to 400 °C and from 257 °C up to 400 °C. It is an eutectic mixture of two very stable compounds, biphenyl (C

_{12}H

_{10}) and diphenyl oxide (C

_{12}H

_{10}O), which have almost equal vapour pressures so the mixture can be treated as if it were a single compound. As a consequence, for the calculation of its thermo-physical properties, it can be treated as a single fluid. In this model it is used in liquid phase. The fluid properties have been calculated using analytical equations obtained from the product datasheet [80].

#### Simulation Algorithm

## Nomenclature

$A$ | area (m ^{2}) |

b | fin spacing (m) |

BOP | Balance Of Plant |

$\dot{C}$ | heat capacity rate (kJ·kg ^{−1}·K^{−1}) |

$c$ | fluid specific heat (kJ·kg ^{−1}·K^{−1}) |

C | cost (€) |

CC | capital cost (€) |

CF | cash flow (€) |

$Cp$ | specific heat at constant pressure (kJ·kg ^{−1}·K^{−1}) |

$D$ | diameter (m) |

DP_{wells} | deep geothermal wells (m) |

$e$ | fin height (m) |

$ex$ | specific exergy (kJ·kg ^{−1}) |

$En$ | energy (kJ) |

$\dot{Ex}$ | exergy flow rate (kW) |

$Ex$ | exergy flow (kJ) |

${F}_{b}$ | boundle boiling factor |

${F}_{c}$ | mixture boiling correlation factor |

${F}_{sol}$ | solar fraction |

$G$ | corrected mass flow rate |

$g$ | gravitational acceleration (m·s ^{−2}) |

$h$ | heat transfer coefficient (W·m ^{−2}·K^{−1}) |

H | Hydraulic Head (m) |

HE | Heat Exchanger |

I | Irradiation solar (kW·m ^{−2}) |

$i$ | specific enthalpy (kJ·kg ^{−1}) |

${i}_{lv}$ | latent heat of evaporation (kJ·kg ^{−1}) |

$k$ | thermal conductivity (W·m ^{−1}·K^{−1}) |

$\dot{m}$ | mass flow rate (kg·s ^{−1}) |

${N}_{row}$ | row number |

N_{wells} | quantity of well |

$n$ | rotating speed |

$Nu$ | Nusselt number |

$NTU$ | number of thermal unit |

OC | operating cost (€) |

ORC | Organic Rankine Cycle |

P | pressure (kPa) |

$Pr$ | Prandtl number |

$PTC$ | Parabolic Trough Collectors |

$\dot{Q}$ | thermal power (kW) |

$\ddot{q}$ | heat flux (W·m ^{−2}) |

$Re$ | Reynolds number |

s | entropy (kJ·kg ^{−1}·K^{−1}) |

SPB | Simple Pay Back (years) |

$T$ | temperature (°C) |

t | fin thickness (m) |

${T}_{1}$ | condensation temperature (°C) |

${T}_{5}$ | organic fluid temperature outlet primary heat exchanger (°C) |

$U$ | overall heat transfer coefficient (W·m ^{−2}·K^{−1}) |

$\dot{V}$ | volume flow rate (m ^{3}·s^{−1}) |

v | specific volume (m ^{3}·kg^{−1}) |

$\dot{W}$ | Power (kW) |

## Greek Symbols

$\text{\beta}$ | angle at the fin tip |

$\text{\delta}$ | defect efficiency |

$\text{\epsilon}$ | heat exchanger efficiency |

${\text{\Phi}}_{f}$ | condensate retention angle |

$\text{\eta}$ | efficiency |

$\text{\rho}$ | density (kg·m ^{−3}) |

µ | dynamic viscosity (Pa·s) |

σ | surface tension (N·m ^{−1}) |

$\text{\tau}$ | expansion ratio |

$\text{\omega}$ | ratio between maximum and minimum heat capacity rate |

## Subscripts

$a$ | ambient condition |

$aux$ | auxiliary |

b | boiling |

$c$ | cold fluid |

$cond$ | condenser |

$crit$ | critical |

$des$ | destruction |

$design$ | design |

$eq$ | equivalent |

$est$ | finned tube outer |

$eva$ | evaporator |

ex | exergetic |

$fin$ | fin |

$geo$ | Geothermal |

$h$ | inlet hot fluid |

hot,tank | upper part of the tank |

$id$ | ideal |

$in$ | inlet |

$l$ | liquid |

$natural$ | natural convection |

$nb$ | nucleate boiling |

M&M | Management and Maintenance |

$min$ | minimum |

$max$ | maximum |

net | net |

$o$ | outer |

$off$ | off design |

$out$ | exit |

$p$ | primary circuit |

pl | absorber plate |

P1 | Pump1 |

P2 | Pump2 |

P3 | Pump3 |

$real$ | real |

$rec$ | Recuperator |

$s$ | isentropic |

$s,t$ | single tube |

$sat$ | saturated |

$shell$ | shell |

solar loop | circulating in the solar loop |

$sun$ | solar |

$T$ | iso-thermal wall |

$tot$ | total |

tur | turbine |

$v$ | vapor |

$w$ | wall |

wells | geothermal fluid |

1-5R | organic fluid points |

1D-7D | diathermic oil points |

1G-4G | geothermal fluid points |

1W-4W | water points |

## Conflicts of Interest

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**MDPI and ACS Style**

Calise, F.; Capuano, D.; Vanoli, L.
Dynamic Simulation and Exergo-Economic Optimization of a Hybrid Solar–Geothermal Cogeneration Plant. *Energies* **2015**, *8*, 2606-2646.
https://doi.org/10.3390/en8042606

**AMA Style**

Calise F, Capuano D, Vanoli L.
Dynamic Simulation and Exergo-Economic Optimization of a Hybrid Solar–Geothermal Cogeneration Plant. *Energies*. 2015; 8(4):2606-2646.
https://doi.org/10.3390/en8042606

**Chicago/Turabian Style**

Calise, Francesco, Davide Capuano, and Laura Vanoli.
2015. "Dynamic Simulation and Exergo-Economic Optimization of a Hybrid Solar–Geothermal Cogeneration Plant" *Energies* 8, no. 4: 2606-2646.
https://doi.org/10.3390/en8042606