Multi-objective optimization of a biomass microCHP-ORC system under supercritical conditions

Abstract

Keywords: ORC; Supercritical; CHP; Biomass; Parametric Analysis; Multi-Objective Optimization; Genetic Algorithm; R124; Isobutane; R245fa

Introduction

Organic rankine cycle is considered as a reliable technology for the efficient conversion of low temperature heat into electrical power [1]. Conventional technologies are unsuitable for this purpose, with Kalina cycle being the only viable alternative [2]. The most common sources of

low-grade heat are solar and geothermal energy, biomass and waste heat from industrial activities [3-4]. The working principle of an ORC cycle is similar to that of the conventional water-steam Rankine cycle. Their most significant difference lies in the working fluid. The conventional Rankine cycle uses water. It is cheap, abundant,

*Corresponding author. *E-mail address: pkladisios@central.ntua.gr This paper was recommended for publication in revised form by Müslüm Arıcı Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

thermochemically stable, non-flammable and non-corrosive. Furthermore, it has high normal boiling point (NBP), high critical temperature (373.95 °C) and low molecular weight. At the same time, water is associated with serious drawbacks. In a steam turbine cycle, a significant amount of superheating is required in order to avoid expansion at the two-phase region and the consequent turbine blade erosion. In addition, steam turbines are generally linked with more complex and expensive installations. ORC systems use organic fluids with higher molecular weights and lower normal boiling points [4-7]. ORC cycle is suitable for the utilization of heat sources with temperatures up to 300 °C. At higher temperatures the conventional Rankine cycle is more efficient [8]. ORC is considered as a proven viable technology for decentralized applications and power from a few kWel up to 1 MWel [1]. Despite their advantages, ORC systems suffer from low thermal efficiencies, due to the low temperature levels of the heat sources [5, 8]. Their optimization is, therefore, of paramount importance. It is indicative that a large portion of the pertinent literature is focused on the development of methods for the selection of suitable working fluids and optimal working conditions [9-10]. In some types of problems, the optimization is a simple process, since the achievement of a single target is sufficient (Single Objective Optimization, SOO). However, in most real applications many criteria (objective functions) have to be met, which may be competitive to each other. This is the subject of multi-objective optimization (MOO) [8]. The single objectives include a great number of thermodynamic, financial and environmental indices such as thermal efficiency, exergy efficiency and net produced power. One of the most common approaches to the multi-objective optimization of ORC systems is the genetic algorithm (Genetic Algorithm, GA) [9-10]. GA is inspired by the biological evolution and specifically from the concepts of natural selection, heredity and mutation. It stands out for its ability to handle complex optimization problems and large numbers of variables. It tracks universal optima, even with the presence of multiple local optima. In comparison with other search methods, it is less dependent on the initial conditions [11-12]. A significant portion of the literature concerning the multi-objective optimization of ORC systems is devoted solely to genetic algorithms [14-16]. In the present paper a steady-state thermodynamic model is developed and 4 candidate working fluids are selected, based on desired properties: R124, isobutene (R600a), R245fa and isopentane (R601a). The multi-optimization with a simple genetic algorithm follows. Thermal efficiency, exergy efficiency and total heat exchanger surface are the single objectives, whose combination formulates the fitness function. The problem was programmed and solved in Python codes, developed by the writers. An essential difference with other approaches pertains to the pinch point of the heat exchangers. The pinch

point of a heat exchanger is the position, where the temperature difference between the two streams receives its lowest value. This difference is called PPTD (Pinch Point Temperature Difference) and is an important design and operating parameter of an exchanger. It affects the efficiency as well as the cost of the system. Lower PPTD values lead to a more efficient exchanger. At the same time, the required surface and consequently the cost are increased. Higher PPTD values are associated with elevated heat losses to the environment. In practice, the pinch point temperature difference is usually the one recommended by the manufacturer. Alternatively, it is the result of an optimization process. In most theoretical cases, a specific PPTD value is assumed for each exchanger [17-19]. This paper includes the pinch point temperature differences in the vector of the decision variables of the genetic algorithm. For a given inlet temperature, the exit temperature and mass flow of the thermal oil and water are adjusted through pinch point analysis, in order to achieve the PPTD value that is required each time. The multi-objective optimization revealed isopentane as the best-performing fluid. In addition, the concepts of cogeneration and recuperation reduce biomass fuel consumption and are beneficial to the overall efficiency of the system.

Organic Rankine Cycle

The basic components of an ORC cycle are: pump, evaporator, expander and condenser. In cases where the temperature at the expander exit is sufficiently high, an internal heat exchanger (IHE) known as the recuperator can be used to increase the thermal efficiency of the cycle. Preheating the cold stream through a recuperator improves the thermal efficiency of the cycle, because the same amount of work is produced using less heat input [20-21]. Depending on the presence of an internal heat exchanger, ORC cycles can be conventional or recuperative [22,23]. In an ideal Rankine cycle there are no pressure loses in the evaporator, condenser and piping. There is also no heat exchange between the components and the environment. Furthermore compression and expansion are considered to be isentropic processes. Thus, a working fluid in an ORC system undergoes the following processes: Process 1-2 (pump): Isentropic compression from condenser to evaporator pressure. Fluid enters the compressed liquid region. Process 2-3 (evaporator): Isobaric heating in the evaporator. Depending on the cycle type, the fluid becomes saturated or superheated vapor or it may enter the supercritical state. Process 3-4 (expander): Isentropic expansion from evaporator to condenser pressure. Fluid may be in the superheated state or in the two-phase liquid-vapor region.

Process 4-1 (condenser): Isobaric cooling in the condenser. Fluid reaches its original saturated liquid state [24]. An Organic Rankine Cycle can be classified into subcritical or supercritical, based on the state of the working fluid at the exit of the evaporator [20]. In Fig. 1 the types of ORC cycle are depicted in temperature versus specific entropy diagrams. In the subcritical ORC the working fluid at the exit of the evaporator may be at a state of saturated or superheated vapor. In the latter case, superheating is involved. Depending on the working fluid properties and the operating conditions, superheating may be linked to higher thermal efficiencies. The permissible degree of superheating, however, is usually small due to the low heat transfer coefficients of the vapor states. Excessive superheating is tantamount to larger and more expensive heat exchangers. Moreover, superheating is commonly used to avoid the two-phase liquid-vapor region during the expansion process. The presence of droplets is responsible for blade erosion and turbine efficiency reduction. Since in the case of most organic fluids the expansion ends at the superheated region, superheating becomes optional [1, 25]. In the case of a supercritical cycle, the fluid is vaporized from the compressed liquid state at the exit of the pump directly to a supercritical state before it is expanded. The fluid does not undergo phase change. In a supercritical ORC the average temperature of heat input is higher which leads to increased thermal efficiencies. In addition there is an improvement in the heat transfer between heat source and working fluid at the external heat exchanger [20]. At the same time, however, supercritical cycle increases the complexity and installation cost due to the high operating pressures [26].

Modeling

In the present paper a 50 kWel biomass-fed microCHP-ORC will be examined. The schematic of the system is displayed in Fig. 2. The combustion of biomass produces warm flue gases. Their heat is transferred to the working fluid indirectly, through a thermal oil loop [4]. This ensures the thermochemical stability of the working fluid. The present model is based on the following assumptions [28-31]: Ø Steady-state operation. Ø Negligible pressure losses in heat exchangers and piping. Ø No heat transfer between components and environment. Ø No changes in kinetic and dynamic energy. Ø Constant isentropic efficiencies in expander and pump. Ø Negligible consumed power by the thermal oil and water pumps. Ø Constant electromechanical losses in the expander-generator coupling. Ø Negligible variations of specific heat under constant pressure versus temperature.

(c) Figure 1. Types of ORC cycles: subcritical without superheating (a), subcritical with superheating (b) and supercritical (c).

(b) Figure 2. Typical biomass CHP-ORC configurations: (a) without internal heat exchanger and (b) with internal heat exchanger.

System Description The equations that describe the components of the system are derived from the conservation of mass and energy and the entropy and exergy balance in a closed volume [24, 32]. The power consumed by the pump is: (1) The pump isentropic efficiency is defined thusly:

(2) The rate of exergy destruction in the pump is: (3) where To represents the temperature in the dead state (temperature of the environment). The heat absorbed by the evaporator is:

The rate of exergy destruction in the evaporator can be calculated by the following equation:

and the corresponding rate of exergy destruction can be expressed as follows:

where To is the average thermal oil temperature. Specifically: (6)

The corresponding isentropic efficiency is defined as follows:

In the case of an internal heat exchanger, states 2 and 4 are replaced by states 2r and 2r respectively. This has to be reflected in the equations for the evaporator and condenser, as well as for the thermal efficiency. The calculation of the working fluid mass flow rate is based on the required power output. Assuming that the expander-generator coupling has efficiency nem, the relationship between power output and net power is: (17) Net power and specific net work are related by the following equation: (18)

(8) The rate of exergy destruction in the expander can be calculated with the following equation:

In a cogeneration system, the electrical efficiency is defined as the ratio of the produced electrical power to the power, which is introduced into the system:

Accordingly, the thermal efficiency of a cogeneration system is defined as the ratio of the heat produced by the system to the heat that is introduced to it:

The heat produced by the CHP system is essentially the heat, which the condenser rejects. The total cogeneration efficiency is defined as the ratio of the total produced power to the power input: (21)

The power input of the cogeneration system is the heat inserted in the form of biomass:

The thermal efficiency of the cycle is defined as the ratio of the net power to the heat input of the cycle:

the mass flow rate and LHVbm the lower where heating value of biomass. By considering a constant boiler efficiency nboiler , the heat that is absorbed by the thermal oil loop, and, consequently, by the evaporator is:

The corresponding specific heat capacity under constant pressure is:

The exergy efficiency is the ratio of the rate of output exergy to the rate of input exergy:

(24) is the total rate of exergy destruction. The where is equal to the heat , that the biomass input exergy provides [33-37]. The PHR index (Power-to-Heat Ratio) is the ratio of the produced electrical power to the produced heat: (25) The PESR index (Primary Energy Saving Ratio) reveals how much energy a cogeneration system conserves, in comparison to the systems that produce electrical power and heat separately. Is it defined as follows:

The total heat exchanged by the two streams is: (29) From Equation (29), the thermal oil mass flow can be calculated: (30) The heat absorbed by the working fluid between state 2 and a random state x is: (31) From the previous equation the thermal oil temperature To,x, which corresponds to position x of the working fluid can be calculated:

(26) (32) where nel,ref and nth,ref are the electrical and thermal efficiency of the reference systems. The typical values of nel,ref = 33% and nth,ref = 90% will be used [38-42]. Pinch Point Analysis Pinch point analysis is used to determine the exit temperature and mass flow rate for the thermal oil and the water in the evaporator and the condenser respectively. In the case of the internal heat exchanger the exit states for both streams are determined. Given are the PPTD values, the inlet temperature and pressure of the thermal oil and the water, as well as the states of the ORC cycle. Each case will be explained in detail in the following sections. Evaporator in supercritical conditions In a supercritical cycle, due to the fact that the curve of the working fluid is non-linear, the location of the pinch point can be anywhere [43-44]. Thus, it has to be iteratively sought. A characteristic example is given in Fig. 3. Given the thermal oil inlet temperature T0,in and the cycle states, suitable exit temperature To,out and mass flow of the thermal oil are sought, in order to achieve a given temperature difference, ΔTpp,evap. By choosing an initial value of To,out near T0,in, the mean thermal oil temperature can be calculated: (27)

This allows the calculation of the local temperature differences and, consequently, the calculation of the minimum. If the minimum temperature difference is greater than the desired value, the process is repeated for a decreased exit temperature, until the following criterion is met: (33) The presence of a recuperator affects the working fluid inlet. The working fluid is entering in state 2r and Equations (29)-(32) are expressed accordingly. Internal heat exchanger In the internal heat exchanger, the heat exchanged between the two streams is: (34) Since no phase change takes place, equation can be expressed as follows: (35) Using the capacity rate definition: (36) Equation (35) can be rewritten as:

Figure 3. Temperature variation as a function of the thermal load and pinch point location for the evaporator under supercritical conditions. (37) Due to the fact that, the hot stream is always vapor and the cold stream always fluid, the following is always true: (38)

A direct conclusion of Equations (37) and (38) is that the temperature drop in the hot stream is always higher than the temperature rise in the cold stream. As a consequence, the minimum temperature difference of the two streams lies invariably on the hot stream exit (state 4r)/cold

Figure 4. Temperature variation as a function of the thermal load and pinch point location for the internal heat exchanger.

stream entry (state 2). The relationship between temperatures T4r, T2 and the pinch point is:

From the previous equation, the water mass flow can be calculated:

(39) Condenser In contrast to the evaporator, the minimum temperature difference lies in a single position: at the start of the phase change of the working fluid (state 4g) [43-44]. Fig. 5 depicts the temperature as a function of thermal load of the condenser. Given the water inlet temperature Tw,in, and the cycle states, the suitable water exit temperature Tw,out and mass are sought, in order to satisfy the desired pinch flow rate point temperature difference, ΔTpp,cond. Similar to the case of the evaporator, the mass flow rate and the exit temperature Tw,out are determined iteratively. An initial value Tw,out close to the inlet temperature Tw,in is selected. The mean water temperature is:

(43) The heat, that the working fluid absorbs between states 4 and 4g, is: (44) Therefore, the water temperature at the pinch point is: (45) If the temperature difference at the pinch point is higher than the desired value, the process is repeated by progressively increasing the exit temperature Tw,out, until the following condition is satisfied:

and the corresponding specific heat capacity under constant pressure is:

Apparently, in the case of the recuperative cycle, state 4 is succeeded by state 4r in Equations (42)-(45).

Heat Exchanger Surface Calculation The heat exchangers are assumed to be Plate Heat Exchangers (PHE). The surface calculation of the plate heat exchanger is directly affected by its geometry. Therefore, a brief description of the basic plate geometry must precede.

Figure 5. Temperature variation as a function of the thermal load and pinch point location for the condenser.

Plate heat exchanger geometry In Fig. 6 the main geometric parameters of a chevron plate are depicted. Plate is characterized by the port diameter Dp, the horizontal distance, Wp and the vertical distance Lp between the ports. The plate section that participates in the heat transfer is defined by the effective length Le and the effective width We. The corrugation geometry is characterized by the chevron angle β, the corrugation wavelength λ, the mean channel spacing b, the corrugation thickness t and the plate pitch p [45].

The presence of corrugations increases the effective surface of the plate. This increase can be expressed by the enlargement factor, which is defined as: (50) The dimensionless parameter X is given by: (51) Typical values of Φ are between 1.15 and 1.25. The effective surface of the plate can be calculated by the following equation: (52) The hydraulic diameter can be calculated as follows: (53) The mean channel spacing p can be estimated using the plate pitch p and corrugation thickness t: (54) Common values for b are within 2 - 5 mm. Despite the fact that, it is important for the heat transfer calculations, it is rarely provided by the manufacturer. In case where the p is unknown, b can be found indirectly, from the length of the compressed plate array, Lc: (55) In the previous equation Nt is the total number of plates. The first and last plate do not contribute in the heat transfer. Therefore, the effective number of plates is:

Figure 6. Geometric parameters of a chevron plate (adjusted from [45]).

The chevron angle β is the most influential and studied parameter. It affects the efficiency and the pressure drop of the exchanger. Typical values lie in the range 22° - 65°. If the port diameter and the distance between the ports are known, the effective dimensions of the plate can be calculated by the following equations: (47)

(56) Channel mass velocity (mass flow rate per unit of cross section surface) is derived from: (57) The number of channels per pass Ncp is: (58)

where Np is the number of passes. A channel is the space between two consecutive plates. Pass is a group of channels that have the same flow direction. The surface of a channel is calculated by:

Table 1. Geometric parameters for the chevron corrugation of a PHE [46-48]. Parameter

Table 2. Geometric parameters and thermal conductivity for the plate of a PHE [46-48]. Parameter

Tables 1-2 contain the geometric parameters of the chevron corrugation and the plate respectively, that are used. The selection was based on the typical ranges of the available literature [46-48].

in the interface between the two streams. By ignoring the thermal resistance due to fouling, the reciprocate of U is expressed thusly:

LMTD method The exchanger surface is calculated with the mean Logarithmic Mean Temperature Difference method (LMTD). According to this method, the exchanged heat can be expressed as follows:

where t is the width and k the thermal conductivity of the interface. ah and ac are the convection coefficients for the hot and cold stream respectively. In a forced convection, the Nusselt number equations are usually of the following form:

The logarithmic mean temperature difference can be calculated by: (61) In the case of a counter-flow exchanger, the temperature differences ΔTa and ΔTb are defined: (62)

(65) where µm is the dynamic viscosity which corresponds to the mean temperature of the flow and µwl is the dynamic viscosity which corresponds to the wall temperature. Coefficients c1, c2 and c3 depend on the plate geometry [52]. The Nusselt number is defined as the ratio of convective to conductive heat transfer at the boundary of a fluid: (66)

(63) With the heat flux and the inlet and outlet temperatures for both streams known, the calculation of the overall heat transfer coefficient U remains [32, 49]. Overall heat transfer coefficient The overall heat transfer coefficient in Plate Heat Exchangers has been an object of studies for decades. A general theory or correlation capable of describing all possible combinations of geometric parameters and flow characteristics is unavailable, due to the complex nature of the problem. Its calculation is based on semi-empirical correlations for the Nusselt number [50-52]. In a plate heat exchanger, the overall heat transfer coefficient includes three discreet phenomena: convection of the hot fluid, convection of the cold stream and conduction

Hydraulic diameter is defined as: (67) where A and P are the surface and perimeter of the cross section. The Reynolds number is the ratio of inertial forces to the viscous forces within a fluid: (68) The Prandtl number is defined as the ratio of momentum diffusivity to thermal conductivity:

(69) At this point, three temperatures, upon which the required calculations are based, must be explicitly defined. The mean or bulk temperature corresponds to the average temperature between the inlet and outlet temperature: (70)

The calculation of the Nusselt number takes place with the correlation proposed by Jackson and Hall, which takes the phenomenon into consideration and includes the pertinent correction [58-59]: (75) The average specific heat capacity,

The plate temperature is the average temperature between the mean temperatures of the two streams: Exponent n is estimated as follows: (71)

Lastly, the wall temperature on the side of a fluid is the average temperature between its mean temperature and the plate temperature [32, 46-47]: (72) Unless declared explicitly by the presence of the indices pl and wl, all properties are calculated based on the mean temperature Tm [53]. When phase change takes place, the temperature of a fluid remains constant. In these cases, properties are calculated based on the mean enthalpy between inlet and outlet [54]:

Condensation In a condenser two zones are observed: precooling or desuperheating and condensing zone. The Nusselt number is calculated with Han, Lee and Kim correlation [52, 60]: (78) The equivalent Reynolds number is defined as:

The following sections describe the Nusselt numbers correlations, that were selected for each heat transfer phenomenon.

Single-phase flow In all cases of single phase-flow, the Kim is used, which is a function of the chevron angle [52, 55]: (74) Supercritical region In the supercritical region two zones are distinguished: subcritical and supercritical. In the supercritical zone, the typical equations cannot be used. The properties of a working fluid change rapidly, especially near the pseudocritical temperature. For a given supercritical pressure, the pseudocritical temperature Tpcr is the temperature that corresponds to the maximum specific heat capacity under constant pressure cp [56-59].

(80) The geometric parameters Ge1 and Ge2 are calculated by the following equations: (81)

Number of plates For the calculation of the total number of plates, and therefore of the required surface, the method described by Buonopane et al. is followed [46, 61]:

7. Estimation of the total number of effective plates:

of multi-objective optimization with conflicting objectives do not have a solution that satisfies all the criteria simultaneously. The success of the algorithm depends strongly on the control parameters, i.e. the size of the population, the crossover and mutation probabilities and the number of parents and generations. Typical values for the population size and generations are 50 - 100 and 50 - 500 respectively. The crossover probability receives values usually 0.5 - 0.90, while the mutation probability in the range 0.001 - 0.1 [62-64]. Table 3 contains the values of the control variables that are used.

This methodology is appropriate when the inlet and outlet temperatures, the mass flow rates and the plate geometry are readily available.

Genetic Algorithm Most variations of the genetic algorithm involve the following three operators: selection, crossover and mutation. Selection applies pressure to the population analogous to that of natural selection. The weaker individuals are gradually discarded, while the fitter have a higher chance of passing their genetic information on. Through crossover, solutions exchange information, similar to the way organisms reproduce. The offsprings have a combination of the genetic material of the parents. Selection and crossover are followed by mutation. Through mutation, new genetic information is introduced and thusly the diversity of the population is ensured. Mutation assists in avoiding local optima. Every time, the population undergoes these processes, the number of generations increases by 1. In the present paper the simple genetic algorithm was implemented. The population of solutions has a real number representation and the parents were chosen through Tournament selection with a subgroup size of 2. The crossover method is the single-point crossover. The children replace the worst performing individuals at every generation. The algorithm ends after a predefined number of generations. The steps of the algorithm follow:

4. Crossover results in the creation of offsprings, which

replace the individuals of the population with the lowest fitness values.

6. The algorithm ends when the termination criterion is

met. Otherwise steps 2-5 are repeated. It should be clarified that, the genetic algorithm does not provide a single optimal solution, but a set of non-dominated solutions that lie on the Pareto front. Problems

Multi-objective function A typical multi-objective problem is stated thusly: (83)

where is the multi-objective function and deciare the objectives, sion variables vector. Functions and express the equality and while functions inequality types of constraints respectively [65-66]. In the field of ORC optimization a plethora of thermodynamic, environmental and economic criteria are used. In the present paper, the thermal efficiency, the exergy efficiency and the total heat exchanger surface are selected as single objectives. The multi-objective function is the weighted sum of these single objectives (weighted sum method). Consequently, the aim of the optimization is the maximization of the following function: (84) where f1 is the thermal efficiency nth, f2 is the exergy efficiency nex and f3 the total exchanger surface At. Due to the fact the single objectives have different units and value ranges, a normalization is in order. The transformation is achieved through the upper-lower-bound approach:

Table 7. Working fluid selection criteria (85) This approach does not face numerical issues, in cases where the denominator is 0 and receives values between 0 and 1 [65-66]. The decision variable vector for the conventional supercritical cycle is: (86)

Table 4. Examined range of the decision variables for the supercritical conditions Tevap = [(Τ_cr+1), 275] °C phigh = [3.7,4.0] MPa ΔTpp,evap = [3,25] °C ΔTpp,cond = [3,25] °C ΔTpp,rec = [3,10] °C

Table 5. Weight coefficients for the multi-objective function w1 = 0.4 w2 = 0.3 w3 = -0.3

Tables 4 and 5 summarize the examined ranges of the decision variables for the subcritical and the supercritical conditions respectively. The weight coefficients for the objectives f1,f2 and f3 receive the arbitrary values of 0.4, 0.3 and -0.3 respectively. Working Conditions The working conditions are presented in Table 6. The power output of the system is set at 50 kWel. The expander machine is assumed to be a radial turbine with maximum working temperature of 275 °C and a constant isentropic efficiency of 0.80. The maximum working pressure is assumed 4 MPa and the maximum pressure ratio 30. The pump has also a constant isentropic efficiency of 0.80. The thermal oil enters the evaporator at temperature of 300 °C, while the water has an entry temperature of 20 °C. Both have a pressure of 2 MPa. The biomass fuel is assumed to have an average LHV of 16 MJ/kg [67]. Table contains the working conditions for both cycles. It should be noted that, the properties of a fluid undergo rapid changes during phase change or supercritical flow. Therefore, the surface calculation must be divided into subsections. The total surface is the sum of these subsections [88-90]. It was found that the discretization into 100 subsections suffices for a low relative error (below 0.1%). Lastly, the working fluids are selected using the constraints of Table 7 and their basic properties are depicted on Table 8. Their thermodynamic properties are calculated with CoolProp with NBP reference state (h = 0 kJ/kg, s = 0 kJ/kgK) for saturated liquid under pressure of 1 atm) [6870]. As thermal oil Dowtherm Q was selected.

Results And Discussion

The optimal solutions for the basic and recuperative cycle are depicted in Fig. 7 and 9 and Tables 9 and 11. The corresponding values of the examined parameters can be found in Figures 11-16 and Tables 10 and 12. Fig. 8 and 10 provide the average and best fitness values versus the generations. Considering that the maximum pinch point temperature difference is 25 °C, it can be deduced that the maximum

Table 8. Properties of the selected working fluids by order of increasing critical temperature [68-70] Working fluid Chemical formula

Figure 7. Optimal solutions for the conventional supercritical cycle.

operating temperature cannot exceed 275 °C. The evaporation temperatures of the recuperative cycle are located near that boundary. In contrast, in the case of conventional cycle an important deviation from the maximum allowed temperature is noticed. Furthermore, it is observed that the optimal evaporation pressure is located near the maximum allowed value of 4 MPa for all working fluids, regardless of the cycle type. The pinch point temperature difference receives low values for the condenser and the recuperator. In the first case the value range is 4 - 7 °C, while in the latter it is 4 - 6 °C. The corresponding values for the evaporator are remarkably higher. Generally, the lowest temperature difference is higher than 15 °C, with one exception. For R245fa in the recuperative cycle this value is below 10 °C. Fitness diagrams show that isopentane generally performs better, with R245, isobutane and R124 following. R124 and isobutane display the lowest values of thermal efficiency, with 0.0985 and 0.1072 respectively. R245fa

Figure 8. Average and best fitness versus generations for the conventional supercritical cycle.

Table 9. Optimal solutions for the conventional supercritical cycle Working fluid

Table 10. Parameter values corresponding to the optimal solution for the conventional supercritical cycle Working fluid

Figure 9. Optimal solutions for the recuperative supercritical cycle.

follows with 0.1258. The maximum achieved value of 0.1442 corresponds to isopentane. Internal recuperation is beneficial to the thermal efficiency. The percentage increase can surpass 80%. Even though isopentane shows the smallest gain, it achieves the highest nth, 0.2328. The values of exergy efficiency are higher than 0.80 for all working fluids. The highest value of 0.8547 is exhibited by isopentane. The use of recuperation improves slightly nex (below 1%). With internal preheating the required exergetic input is reduced in a manner, that practically counterbalances the improvement in terms of rate of exergy destruction. In the conventional cycle, the lowest total exchanger surface is 3.4407m2 and is attributed to isopentane. Isobutane, R245fa and R124 follow with 3.68832, 3.85102 and 3.86342 respectively. The presence of an internal heat exchanger leads to an increase of At. Isobutane and isopentane require 12.74% and 12.55% more surface respectively. R124 requires 5.00% and R245fa 0.6117% more surface.

Figure 10. Average and best fitness versus generations for the recuperative supercritical cycle.

Table 11. Optimal solutions for the recuperative supercritical cycle Working fluid

In terms of specific net work and working fluid mass flow rate, isopentane displays superior performance. In the conventional cycle the values of 81.8093 kJ/kg and 0.6791 kg/s are achieved. A noteworthy gain results from the use of a recuperator. The improvement for wnet is 25 - 45%, while it is 20 - 30% across all fluids. Thusly, isopentane for reaches specific net work 103.3 kJ/kg and required mass flow rate 0.5378 kg/s. The large deviation is attributed to the fact that the optinal evaporation temperature between the two cycles differs greatly. R245fa and isopentane exhibit the highest values for the PHR and PESR indices. In the case of the conventional cycle, R245fa achieves 0.1295 for PHR and 0.1992 for PESR. Internal recuperation improves these values to 0.2390 and 0.2742 respectively. Similarly, for isopentane PHR is 0.1516 without recuperation and 0.28 with recuperation. The presence of a recuperator improves PESR from 0.2169 to 0.2957.

The overall cogeneration efficiency receives value in the region 0.92 - 0.95, without noticeable differences between cycles and working fluids. In the conventional cycle R124 displays at the same time the maximum thermal and minimum electrical cogeneration efficiency. In contrast, isopentane achieves the maximum electrical and the minimum thermal cogeneration efficiency. The addition of an internal heat exchanger leads to noticeable improvements. For instance, in the case of isopentane nel,chp increased from 0.1235 to 0.2029 and nth,chp from 0.7246 to 0.8130. The required biomass mass flow rate in the conventional cycle lies in the range 0.02 - 0.04 kg/s, with isopentane and R124 exhibiting the minimum and maximum values respectively. Biomass fuel consumption can be reduced greatly through internal preheating. The reduction can be more than 45% (R124 and isobutane).

Table 12. Parameter values corresponding to the optimal solution for the recuperative supercritical cycle Working fluid

Figure 11. Thermal and exergetic efficiency corresponding to the optimal solution for the supercritical cycle. The lowest thermal oil exit temperature is 76.6785 °C in the basic cycle and 156.7209 °C in the recuperative cycle with R124. The lowest thermal oil mass flow rate is 0.8015 kg/s for isopentane without recuperation. R245fa with recuperation achieves 0.8527 kg/s.

In the conventional cycle, the maximum water exit temperature is 74.2279 °C and is achieved by R124, while the lowest water mass flow rate is 1.5242 kg/s and is attributed to isopentane. In the recuperative cycle, R245fa achieves exit temperature 56.6073 °C and mass flow rate 1.3675 kg/s. Generally, the use of recuperation lowers Tw,out significantly, by 15-20 °C.

Figure 12. Specific net work and working fluid mass flow rate corresponding to the optimal solution for the supercritical cycle.

Figure 13. Electrical, thermal and overall cogeneration efficiency corresponding to the optimal solution for the supercritical cycle.

Figure 14. PHR, PESR indices and biomass mass flow rate corresponding to the optimal solution for the supercritical cycle.

Figure 15. Mass flow rate and exit temperature for the thermal oil and water corresponding to the optimal solution for the supercritical cycle.

Figure 16. Heat exchanger surface histogram corresponding to the optimal solution for the supercritical cycle.

Conclusions

In the present paper, a steady-state thermodynamic model for a 50 kWel biomass CHP-ORC was developed and four candidate fluids were selected: R124, isobutane, R245fa and isopentane. The genetic algorithm was used for the optimization of the system under supercritical conditions. The fitness function was a linear combination of the thermal efficiency, the exergy efficiency and the total heat exchanger surface. The evaporation temperature and pressure and the pinch point temperature differences at the heat exchangers were selected as decision variables. The comparison of the results revealed a series of interesting observations. Firstly, there is no single working fluid that is superior in every aspect. However, isopentane stands out. It has also been noted that, working fluids with higher critical temperature display higher thermal efficiency. The use of an internal heat exchanger leads to an increase in total surface, despite the reduction in evaporator and condenser surface. In addition, a recuperator restricts the water exit temperature at lower levels. Therefore, its use is advisable only when the increase in thermal efficiency offsets the increase in cost and the water exit temperature can be maintained at sufficient levels. Lastly, it is evident that, optimization alone is not enough. The system designer must make the final decision based on subjective criteria. Another two noteworthy observations are related to the optimal values of the decision variables. The optimal

pinch point temperature difference at the recuperator tends towards the lower values 4 - 6 °C. In addition, the evaporation temperature of the supercritical cycle receives values in the region 3.95 - 4 MPa, for all working fluids, for both conventional and recuperative cycle. These observations can be exploited, in order to reduce the search space and lead to a faster convergence of the genetic algorithm. No systematic tendencies were observed for the remaining decision variables. An important aspect of the CHP-ORC system is the utilization of the heat, which the condenser would otherwise reject. In this case, the total cogeneration efficiency surpasses 90%, while simultaneously saving 20 - 30% biomass fuel. The paper concludes with suggestions for future research. The evaporator is significant source of exergy destruction. During the phase change, the temperature of a pure substance remains constant, which leads to an increased temperature difference between the two streams. This is tantamount to an increased rate of exergy destruction. One of the most effective ways to mitigate the evaporator losses is the use of zeotropic mixtures. Their main attribute is the non-isothermal phase change. Compared to pure substances, zeotropic mixtures have not been investigated at the same degree. The heat transfer on the interior of plate heat exchanger is a complex phenomenon. It depends on the characteristics of the flow, the geometry of the plates and the configuration

of the flow. Consequently there is no general theory or correlation, which describes all possible flow conditions and combinations of geometric parameters. The calculation of the Nusselt number relies on semi-empirical correlations, that are developed for specific heat exchangers, geometries and flow regiments. Unlike the single-phase region, the evaporation, condensation and supercritical flow have not been investigated sufficiently. Also, apart from the chevron angel, the influence of the other geometric parameters on the heat transfer characteristics has yet to be delineated. The success and efficiency of the genetic algorithm is greatly affected by its control parameters. The dynamic adjustment variables such as the crossover and mutation probability could offer faster convergence. This area has plenty of room for research activity. A relative attempt has been performed by A. Hassanat et al [64]. Lastly, the relationship between critical temperature and thermal efficiency has been verified before by J. Xu and C. Yu [8]. Working fluids with higher critical temperature are expected to exhibit higher values of nth. This conclusion could be expanded to more parameters. A similar tendency is observed for other indicators such as the thermal and electrical cogeneration efficiencies (nel,chp, nth,chp) and the PHR and PESR ratios. This requires further systematic investigation with more fluids, in order to verify it.

Acknowledgements

The present work was financially supported by National Technical University’s Research Committee scholarship.

Nomenclature

Latin characters [m2] Ach [m2] Ae [m2] Ap b [m] c [J ⁄ kgK] C [J ⁄ sK] [m] Dh [m] Dp E [J] [W] f G F(X) Ge h k Lc

channel surface plate effective surface plate projected surface mean channel spacing specific heat capacity heat capacity rate hydraulic diameter port diameter energy exergy rate

objective function mass velocity multi-objective function geometric parameter specific enthalpy thermal conductivity compressed PHE length

[m] [m] [MJ ⁄ Nm3] [kg ⁄ s] [-] [-] [-] [-] [-] [-] [-] [-] [-] [-] [Pa or atm] [m] [-] [-]

effective plate length vertical distance between ports lower heating value mass flow rate efficiency number of chromosomes number of genes number of generations number of parents number of channels per pass number of effective plates number of passes total number of plates Nusselt number pressure plate pitch crossover probability mutation probability

perimeter primary energy saving ratio power-to-heat ratio Prandtl number specific heat energy heat flux heat transfer rate

[-] [J ⁄ kgK] [m] [K or °C ] [W ⁄ m2 K] [m2 ⁄ s] [J ⁄ kg] [-] [W] [m] [m]

Reynolds number specific entropy corrugation thickness temperature overall heat transfer coefficient velocity specific work weight factor power effective plate width horizontal distance between plate ports plate dimensionless parameter

convection coefficient chevron angle difference corrugation pitch or wavelength dynamic viscosity kinematic viscosity

Indices 0 bm boiler c ch chp cond cr d e eq el em evap f fg g h ign in is lm m max min net norm o orc out p pl preheat pp ps r ref rec s sub super

dead state biomass boiler cold stream channel cogeneration condenser critical destruction expander equivalent electric electromechanical evaporator or evaporating section saturated liquid state liquid-vapor two-phase region saturated vapor state hot stream auto-ignition input isentropic logarithmic mean of bulk maximum minimum net normalized thermal oil ORC output pump plate preheating section pinch point pseudocritical recuperative reference recuperator saturation subcritical section supercritical section

superheating section total triple vertical cooling water working fluid wall

Abbreviations

Latin characters CAS CHP GA GWP IHE LHV MOO NBP ODP ORC PHE SOO WHR

Chemical Abstracts Service Combined Heat and Power Genetic Algorithm Global Warming Potential Internal Heat Exchanger Lower Heating Value Multi-Objective Optimization Normal Boiling Point Ozone Depletion Potential Organic Rankine Cycle Plate Heat Exchanger Single-Objective Optimization Waste Heat Recovery

ORC states Point Description 1 Condenser exit Pump exit, isentropic 2is compression 2 Pump exit, real compression Recuperator, cold 2r stream exit Evaporator 2f Evaporator 3g 3 Evaporator exit 4is 4 4r 4g

Working fluid state Saturated liquid Compressed liquid Compressed liquid Compressed liquid Saturated liquid Saturated vapor Saturated vapor, superheated Vapor or supercritical state Superheated vapor

Turbine exit, isentropic expansion Turbine exit, real Superheated vapor expansion Recuperator, hot stream Superheated vapor exit Evaporator Saturated vapor

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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