Multi-objective analysis of an influence of a brine mineralization on an optimal evaporation temperature in ORC power plant

The fact that Organic Rankine cycle system is very promising technology in terms of electricity production using low grade heat sources, necessitates constant research in order to determine the best cycle configuration or choose the most suitable working fluid for certain application. In this paper, multi-objective optimization (MOO) approach has been applied in order to conduct an analysis that is to resolve if there is an influence of a mineralization of a geothermal water on an optimal evaporation temperature in ORC power plant with R1234yf as the working fluid.


Introduction
Organic Rankine Cycle system (ORC) is constantly developing technology in past several decades. The main advantage of such systems is the availability of effective conversion low grade heat into power. Moreover, they are characterised by simple construction, low negative environmental influence and relatively high effectiveness. In view of the above advantages, a lot of research is still being conducted in order to ensure the most efficient work of designed ORC system. These studies are taking into account different technical and ecomomic performance indicators, called objective functions, which are to be minimized or maximized. Discussed criteria include: energy efficiency η I (first law efficiency), exergy efficiency η II (second law efficiency), net power output W net , heat transfer area A, overall heat transfer coefficient k, overall cost of the system C, total exergy destruction rate δB tot [1] or size parameter SP, which allows to initially evaluate the size of a turbine [2]. Some authors also used different combinations of these quantities. Shengjun et. al [3] concucted thermoeconomic optimization of subcritical ORC and transcritical power cycle system for low temperature geothermal power generation using several objective functions: ratio of heat transfer area and net power output, also known as APR parameter, or levelized energy cost LEC. Roy et. al [4] carried out performance analysis of an ORC with superheating under different heat source temperature conditions using so called availability ratio Ф, which is defined as the ratio of the available energy (difference between heat obtained from the thermal source and total exergy destruction in the system) to the total energy obtained from the thermal source. Guo et. al [5] conducted thermodynamic analysis of waste heat power generation system. One of the applied objective function was so called sustainability index SI, which is also used in this paper.
Most of the conducted studies converge singleobjective analysis, which are focused on optimizing one of the above performance indicators. Much less attention has been paid to multi-objective optimization (MOO). This approach allows to carry out more comprehensive analysis, using more than one objective function, which means that ORC system can be optimized from both thermodynamic and ecomonic perspectives. Moreover, the selection process of the working fluid can be also optimised including several criteria [6].
Wang et. al [7] carried out multi-objective optimization of an ORC for low grade waste heat recovery using evolutionary algorithm. The authors used exergy efficiency and capital cost as performance indicators of the system. Xiao et. al [8] conducted multiobjective optimization of evaporation and condensation temperatures for subcritical ORC, using sustainability index SI and the ratio of the cost of the system C to net power output as objective functions. The authors solved optimization problem using linear weighted evaluation function. Toffolo et. al [9] made the optimal selection of working fluid and design parameters in ORC system, using multi-criteria approach. The objective functions that the authors applied, was levelized cost of electricity LCOE, which considers the annual energy production of the system or specific investment cost SIC defined as the ratio between total investment cost and net power output.
In several cases, it is very important to not only seeking to improve all thermodynamic and economic indexes, but also include some inconveniences of the system. Therefore, in case of geothermal water, which is the energy source in the system analysed in this paper, the level of mineralization M of a brine should be considered. In order to solve this problem, multiobjective optimization has been carried out using weighted sum method [10] with global function G, which is to be minimized.

System description
The energy source of the system has a significant impact on the overall performance and effectiveness of ORC power plant and it is important to carry out complex analysis in order to determine the optimal working parameters of the ORC. In case of considered system, geothermal water (brine) has been used to supply ORC power plant. One of the crucial property of such fluid is mineralization M. It is worth to emphasise that this parameter can quite considerably vary depending on local hydrogeothermal conditions. The level of salinity of the brine affects its values of basic thermophysical properties, such as: specific heat c p1 and density ρ 1 , so in calculation procedure, these parameters should be considered as function of either temperature and mineralization.
The ORC system, as shown in figure 1 and 2, works as classical power plant. Slightly superheated (point 1) vapour flows into the expander and its enthalpy is converted to useful work of the turbine. The low pressure vapour (point 2) is cooled (to the state corresponding to point 2") and phase change occurs in condenser (cold water is a coolant). Then the saturated liquid (3) is pumped to higher pressure and goes to the inlet of vapour generator, where it is heated up, evaporated and, optionally, superheated.
The choice of the working fluid affects substantially the overall performance of ORCs and a lot of analysis have been done in order to choose the best one for certain configuration of the system [11], [12], [13]. In this paper, R1234yf has been chosen as one of the most promising working fluid, because of its beneficial impact on an effectiveness of ORC, as well for a favourable environmental indicators, such as: ODP=0 and GWP=4 [14].
As more attention in calculation procedure has been devoted to the evaporation process of the working fluid, in figure 3 only vapour generator (VG) and its T-ΔH diagram are presented. The VG is considered as countercurrent plate heat exchanger (PHE). It is worth mentioning that the evaporator section of the VG is divided into finite small elements in order to assume constant thermophysical properties of evaporating fluid for each element. The values of the parameters shown in figure 3 are provided in table 1.

Calculation model
The proposed model is created in order to reflect if there exists significant influence of the salination of the geothermal water on the optimum evaporation temperature of the working fluid in ORC power plant. However, it is difficult to make a comprehensive optimization using only single objective functions, since, during designing process of ORCs, it is impornant to include several factors, such as: thermodynamic, economic or environmental. In that model, multi-objective optimization (MOO) approach has been proposed with multi-objective function G(X), which consists of the quantities that are most substantially affected by decision variables, i.e.: the evaporation temperature t ev and the mineralization M of the brine. Component quantities are the following: heat exchanger area of the vapour generator A VG , net power output of ORC system N out , overall exergy destruction rate δB tot in the system and exergy drop ΔB w of the geothermal water at the inlet and outlet .
Calculations have been made using MATLAB 2017b software [15] with REFPROP 9.1 [16] as database of the thermophysical properties of R1234yf fluid.

Optimization model
The optimization model has been solved using weighted sum method with aforementioned G(X) as objective function, which, in this case, is the composition of two single objective functions: Single objective functions, which are to be minimised, can be written as follows: out Function f 1 is an economic indicator, while function f 2 is called sustainability index SI, which determines the influence on the environment. However, in equation (1), normilized forms of the functions (2) and (3)

Heat transfer area of vapour generator A VG
A VG in this model is considered to be the sum of preheater, evaporator and superheater, which, using heat transfer formula, can be written in the following form: It is worth to remind that the evaporator section (middle part of the (5) equation) is divided into finite small elements (see figure 3), so each quantity is determined with respect to i-th fininte element.
Heat fluxes for preheater, evaporator and superheater sections can be obtained from energy balance equation. On the assumption that ORC power plant is in steady state as well as there is no heat loss, corresponding heat fluxes can be calculated from the following formulas: where enthalpies corresponding to states of working fluid (see figure 1 or 2) are determined using Refprop database. Enthalpy h 4 is obtained using equation (23). Mass flow rate of the R1234yf can be calculated from the following equation: As has been mentioned before, density ρ 1 and specific heat c p1 of the geothermal water should be considered as function of temperature as well as mineralization of the brine. Equations of these properties provided in [17] are as follows: The logarithmic mean temperature difference ΔT log can be calculated using equation : where superscripts: ' and " indicates inlet and outlet temperatures respectively. Overall heat transfer coefficient k can be obtained from the general formula: The so called chevron angle is assumed to be equal to: β=60°. The Reynolds number Re is defined as follows:

Superheater section
The following Nusselt number correlation [19] for organic working fluid side has been used:

Evaporator section
Yan and Lin [20] proposed the following correlation for two-phase flow boiling process: where boiling number is defined as: Detailed description of the equation is provided in aforementioned article.

Preheater section
In case of preheating process of the R1234yf, the same correlation (14) for Nusselt number was used as for water side.

Net power output N out
The general formula for N out is the following: where the enthalpies h 2 and h 4 are calculated using definitions of isentropic efficiencies (equations (22) and (23)), which assumed values for a turbine and a pump are equal to: η iT =80% and η iP =75%.

Overall exergy destruction rate δB tot
The total exergy destruction in analysed ORC system can be presented as the sum: The components of equation (24) correspond to all elements of ORC system are shown in figure 2 and are explained below.

Vapour generator
The exergy destruction in every component can be calculated using Gouy-Stodola equation, which in case of VG has the following form: where subscript r means reference state, which level is characterized by temperature T r =298.15K and pressure p r =101325Pa. The sum of the entropy fluxes can be found from the entropy balance equation:

Exergy drop ΔB w of the geothermal water
Exergy drop of the brine between inlet and the outlet of vapour generator can be calculated from: Using the definition of exergy, b 1w and b 2w are calculated as: ( )

Results and discussion
In figures 4-10 overall results of thermodynamic and optimization analysis are presented. As can be seen in figure 4, heat transfer area A VG of the vapour generator increases with the increase of evaporation temperature, which is, mostly, due to the fact that with the increase of t ev , heat flux ̇ (see equation (7)) of the working fluid R1234yf also increases, which clearly involves A VG (see equation (5)). However, with the increase of salination of a brine, there can be seen the opposite dependence. This fact is mainly influenced by the variability of the geothermal water heat flux ̇1 , as it is the function of mineralization M. Decreasing values of the heat flux ̇1 with increasing mineralization M stems from an influence of the M on the thermophysical properties of the brine, i.e.: density ρ 1 and specific heat c p1 . It turns out that the density ρ 1 is increasing while the specific heat c p1 is decreasing with the increase of mineralization M, however -the overall effect, which comes from the product of ρ 1 and c p1 (see equation (9)), leads to the decrease of ̇1 . Moreover, the changes in heat flux ̇1 affect heat fluxes transfered in preaheater, evaporator and superheater (see equations (6),(7),(8)), since the enegy balance equation must be conserved, thus decreasing values of ̇1 lead to the less values of A VG .
Net power output N out, as can be seen in figure 5, is increasing with the evaporation temperature, since the mass flow rate ṁ 2 , which affects N out , depends on previously discussed ̇1 (see equations (9) and (21)). For an analogical reason, N out is decreasing with the mineralization M, as higher values of M correspond to the lower values of ṁ 2 . Figure 6 presents the results for the first objective function f 1 , i.e. ratio of heat transfer area of the vapour generator A VG and net power output N out .   It is worth noticing that there exists optimum evaporation temperature of the function f 1 and it is located approximetaly at the temperature of 54°C. The influence of mineralization is relatively small, but negative -increasing values of M lead to increasing values of f 1 which is to be minimised.
Overall exergy destruction rate δB tot is decreasing with the increase of evaporation temperature, as can be seen in figure 7. It is worth mentioning that the greatest exergy destruction occurred in vapour generator. Moreover, these losses are changing most rapidly with respect to evaporation temperature t ev and mineralization M. Therefore, it should come as no surprise that exergy destruction in the system is decreasing , as higher values of t ev lead to smaller temperature differences between geothermal water and working fluid and therefore -to less entropy generation (see for instance equations (25) and (26)). The exergy destruction also decreases with respect to mineralization, which can be explained using equation (26). As has been mentioned before, mass flow rate of the working fluid ṁ 2 is decreasing with the increase of mineralization. Simultaneously, mass flow rate of the geothermal water ṁ 1 is increasing, because of the increase of the density ρ 1 . The entropy drops of the water and working fluid are negative and positive respectively, thus the overall effect leads to the decrease of exergy destruction. In figure 8, exergy drop of the geothermal water is changing with respect to only mineralization M, as quantities in equation (33) are independent on the evaporation temperature t ev of the working fluid. The less values of ΔB w for higher levels of mineralization can result from decreasing values of specific heat c p1 , which in turn affect the enthalpies that are the components in equations (34) and (35). Figure 9 presents the results for the second objective function f 2 , which is the ratio of total exergy destruction δB tot to exergy drop of the geothermal water ΔB w . This function is also known as sustainability index SI. Lower values of that indicator correspond to lower negative influence on an environment. The drop in of the values of the objective function with respect to evaporation temperature t ev is obvious, since the exergy destruction from figure 7 has the same tendency and exergy drop of water from figure 8 is not changing. The mineralization M of the brine affect the objective function f 2 analogically as on its components. Although, that function does not indicate optimum evaporation temperature t ev of working fluid, however, there can be seen that higer saturation temperatures would be more beneficial in this case.  Finally, figure 10 presents the results for multiobjective function G. On the assumptions that have been made, i.e. equivalent importance (w 1 =w 2 =0.5) of both objective functions f 1 and f 2 and using evaporation temperature t ev and mineralization M as decision variables, it turns out that optimum evaporation temperature equals to approximately 61°C, but more importantly, it is the same for all analysed levels of mineralization of the brine. This means that mineralization M has no influence on the choice of optimum evaporation temperature. Moreover, it can be seen that minimum values of the function G have been obtained for mineralization M=0.00. However, each of three curves almost coincide with each other, which means that the overall influence of the mineralization M is almost negligible, which can be explained by the fact that single-objective functions f 1 and f 1 have presented the opposite relationship with respect to salination -f 1 was increasing and f 2 was decreasing for certain evaporation temperatures.