Development of a SAFT-γ Mie group-contribution approach to modelling working fluids and its application in the computer-aided molecular and process design of ORCs

UNIVERSIDAD DE VALLADOLID ESCUELA DE INGENIERIAS INDUSTRIALES Máster en Ingeniería Química Development of a SAFT-γ Mie groupcontribution approach to modelling working fluids and its application in the computeraided molecular and process design of ORCs Autor: Santiuste Martínez, Jaime Responsable de Intercambio en la UVa Mato Chaín, Rafael Universidad de destino Imperial College of Science, Technology and Medicine of London Valladolid, Julio 2020. TFM REALIZADO EN PROGRAMA DE INTERCAMBIO TÍTULO: Development of a SAFT-γ Mie group-contribution approach to modelling working fluids and its application in the computeraided molecular and process design of ORCs ALUMNO: Santiuste Martínez, Jaime FECHA: Julio 2020 CENTRO: Chemical Engineering Department, Molecular Systems Engineering (MSE) UNIVERSIDAD: Imperial College of Science, Technology and Medicine of London TUTOR: Haslam, Andrew J. Resumen Los ciclos de Rankine orgánicos (Organic Rankine Cycles, ORCs) se presentan como una buena alternativa para recuperar el calor residual. De entre los fluidos de trabajo posibles, los hidrofluorocarbonos (HFCs) destacan por su potencial para aumentar el rendimiento de estos ciclos. Dentro del método SAFT-γ Mie, se han desarrollado los grupos funcionales pertenecientes a los HFCs, obteniendo una estimación de la presión de vapor y de la densidad de líquido saturado con una desviación media absoluta, respecto a los datos experimentales, de 3,17% y 1,77%, respectivamente. Por último, una vez desarrollado el modelo termodinámico con la ecuación SAFT-γ Mie y analizadas las propiedades de transporte, se propone un marco para el diseño molecular y de proceso asistido por ordenador (Computer-Aided Molecular and Process Design, CAMPD), con el objetivo de maximizar la potencia obtenida con el ciclo, modificando para ello tanto el fluido de trabajo como las condiciones de operación. Palabras clave SAFT-γ Mie, hidrofluorocarbonos (HFCs), estimación de parametros, ciclo de Rankine orgánico (ORC), diseño molecular y de proceso asistido por ordenador (CAMPD) Abstract Organic Rankine Cycles (ORCs) are known to be a good alternative for the recovery pf waste heat. Among the wide variety of working fluids, refrigerants, in particular fluorinated hydrocarbons, have a great potential to increase the thermodynamic performance of these cycles. Within the SAFT-γ Mie approach, functional groups present in HFCs have been developed, obtaining an estimation of the vapour pressure and the saturated-liquid density with an absolute average deviation (AAD), with respect to experimental data, of 3.17% and 1.77%, respectively. Finally, once the thermodynamic model with the SAFT-γ Mie approach is developed and the transport properties are analysed, a computer-aided molecular and process design (CAMPD) framework is proposed, with the objective of maxims the power output, modifying both the working fluid and the operating conditions. Keywords SAFT-γ Mie, hydrofluorocarbons (HFCs), parameter estimation, Organic Rankine Cycle (ORC), Computer-Aided Molecular and Process Design (CAMPD) UNIVERSITY OF VALLADOLID SCHOOL OF INDUSTRIAL ENGINEERING Master in Chemical Engineering IMPERIAL COLLEGE LONDON CHEMICAL ENGINEERING DEPARTMENT Molecular Systems Engineering MASTER’S THESIS Development of a SAFT-γ Mie group-contribution approach to modelling working fluids and its application in the computer-aided molecular and process design of ORCs Author: Jaime Santiuste Martínez Supervisors: Andrew J. Haslam Amparo Galindo Rafael Mato Chaín London, July 2020 2 Abstract 3 Abstract Organic Rankine Cycles (ORCs) are known to be a good alternative for the recovery of waste heat. Among the wide variety of working fluids, refrigerants, in particular fluorinated hydrocarbons, have a great potential to increase the thermodynamic performance of these cycles. Group-contribution (GC) approaches present a great opportunity to screen a wide variety of molecules. Hydrofluorocarbon functional groups are developed for use within the SAFT-γ Mie approach, leading to absolute average deviations for the vapour pressure and the saturated liquid density of 3.17% and 1.77%, respectively, across a wide range of fluorinated hydrocarbon fluids collectively incorporating all of these groups. The model prediction of calorific and thermodynamic properties is analysed, obtaining a suitable description of the groups present in the molecules of interest to model thermodynamic cycles, in particular ORCs. To complete the working-fluid modelling, transport-property GC correlations are analysed for use within the ORC model developed. Finally, an outer approximation (OA) algorithm is proposed to solve the mixed integer nonlinear programming (MINLP) optimisation of the ORC, in order to obtain the optimal working fluid. The proposed parameters for the SAFT-γ Mie groups, along with the transport-property methods, allow a complete description of the working fluid, as used in the ORC model. A computer-aided molecular and process design (CAMPD) framework is proposed, in which the solver selects the groups present in the working fluid, and then optimises the process variables in order to maximise the power output. 4 Index 5 Index Abstract ......................................................................................................................................... 3 Resumen ..........................................................................................Error! Bookmark not defined. Index .............................................................................................................................................. 5 Summary/Objectives ..................................................................................................................... 6 1. Introduction............................................................................................................................... 7 2. Development of new functional groups.................................................................................. 12 2.1. SAFT-γ Mie ........................................................................................................................ 12 2.1.1. Molecular model ....................................................................................................... 12 2.1.2. Helmholtz free energy ............................................................................................... 14 2.1.3. Combining rules......................................................................................................... 17 2.1.4. Functional groups ...................................................................................................... 17 2.1.5. Parameter estimation ............................................................................................... 19 2.2. Results and discussion ...................................................................................................... 22 2.2.1. Parameter estimation ............................................................................................... 22 2.2.2. Properties prediction ................................................................................................ 30 3. Organic Rankine Cycle analysis ............................................................................................... 40 3.1. Methods/Theory .............................................................................................................. 40 3.1.1. Thermophysical properties for ORC modelling ......................................................... 40 3.1.2. Organic Rankine Cycle (ORC) model .......................................................................... 45 3.1.3. ORC optimisation ...................................................................................................... 49 3.2. Results and discussion ...................................................................................................... 52 3.2.1. Thermophysical properties for ORC modelling ......................................................... 52 3.2.2. ORC – CAMPD ............................................................................................................ 59 4. Conclusions.............................................................................................................................. 64 Bibliography ................................................................................................................................ 67 Symbols and abbreviations ......................................................................................................... 71 Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 12 2. Development of new functional groups 2.1. SAFT-γ Mie The SAFT-γ Mie3 EoS is a group contribution-approach that allows us to obtain thermodynamic properties and the phase behaviour of fluids and mixtures. This method is based on a heteronuclear molecular model, in which each molecule is defined by the functional groups present and their multiplicity. The interaction between the functional groups is modelled with Mie potentials of variable range. 2.1.1. Molecular model As other group-contribution (GC) approaches, SAFT-γ Mie can be used to estimate a wide range of molecular properties by representing each molecule as the addition of all the functional groups and their multiplicity3. In this approach, molecules are divided as in Fig. 3, usually grouping together each carbon atom with the groups bounded with them. In the example shown in Fig. 3, the fused heteronuclear model for a molecule formed by four different functional groups is represented. These groups, CHF2, CHF, CF2 and CH2F, have their own parameters, the combination of which leads to the final molecule characterization. Fig. 3. Representation of a heteronuclear molecule (1,1,2,2,3,4-hexafluorobutane) within the SAFT-γ Mie approach. Cyan: carbon, black: hydrogen and Magenta: fluorine Additionally, SAFT-γ Mie theory includes the description of strong interaction forces, such as hydrogen bonding, by adding a number of short-range association sites placed on the segments where these forces take place. Intermolecular potential The intermolecular interaction between two segments, k and l, is assumed to have the form of the Mie potential25, a generalised form of the Lennard-Jones potential. The potential of the interaction between two segments is calculated with eq. [1] as a function of the intersegment distance rkl: ∅𝑘𝑘𝑘𝑘 𝑀𝑀𝑀𝑀𝑀𝑀(𝑟𝑟𝑘𝑘𝑘𝑘)=𝐶𝐶𝑘𝑘𝑘𝑘·𝜀𝜀𝑘𝑘𝑘𝑘·��𝜎𝜎𝑘𝑘𝑘𝑘 𝑟𝑟𝑘𝑘𝑘𝑘�𝜆𝜆𝑘𝑘𝑘𝑘 r−�𝜎𝜎𝑘𝑘𝑘𝑘 𝑟𝑟𝑘𝑘𝑘𝑘�𝜆𝜆𝑘𝑘𝑘𝑘 a� [1] where ∅𝑘𝑘𝑘𝑘 𝑀𝑀𝑀𝑀𝑀𝑀is the intersegment potential, 𝜀𝜀𝑘𝑘𝑘𝑘 the depth of potential well between k and l segments, 𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎 and 𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟 are the intermolecular attractive and repulsive ranges of potential and 𝜎𝜎𝑘𝑘𝑘𝑘 the segment diameter. The term 𝐶𝐶𝑘𝑘𝑘𝑘 the (eq.[2]) is a term to ensure a minimum potential of −𝜀𝜀𝑘𝑘𝑘𝑘. 2. Development of new functional groups 13 𝐶𝐶𝑘𝑘𝑘𝑘=𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟 𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟−𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎·�𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟 𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎�𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎 𝜆𝜆 𝑘𝑘𝑘𝑘 𝑟𝑟−𝜆𝜆 𝑘𝑘𝑘𝑘 𝑎𝑎 [2] The Mie potential is represented in Fig. 3 for the interaction of two equal segments of type k. The parameters described above are shown in the figure and its physical sense. The potential shape is similar in the case of the unlike interaction between two segments, k and l. Fig. 4. Mie potentials. ΦMIE(r): intermolecular potential. 𝜀𝜀𝑘𝑘𝑘𝑘 : dispersive energy. 𝜎𝜎𝑘𝑘𝑘𝑘 : segment diameter. Sk: shape factor. 𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎 and 𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟 : attractive and repulsive intermolecular potentials. r: intersegment distance. 𝜈𝜈𝑘𝑘∗: number of segments For the association sites, like hydrogen bonding or other short-range interaction, the interaction is modelled with a square-well potential between a site of type a place on a segment of type k and a site of type b placed on a segment of type l, as described in eq. [3]: ∅𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 �𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎�=�−𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 𝑖𝑖𝑖𝑖 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎≤𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑐𝑐, 0 𝑖𝑖𝑖𝑖 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎>𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑐𝑐, [3] where 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎is the centre-centre distance between both sites, 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑐𝑐 the cut-off range of the association interaction and 𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 the depth of the association-energy well. The distance between the segment centre and the site centre is represented by 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑑𝑑. To sum up, a functional group is represented by the number of identical segments, 𝜈𝜈𝑘𝑘∗, the shape factor, 𝑆𝑆𝑘𝑘, that represents the extent to which the segment contribute to the overall molecule, the segment diameter, 𝜎𝜎𝑘𝑘𝑘𝑘, the attractive and repulsive ranges of potential, 𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎 and 𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟 and the energy of interaction, 𝜀𝜀𝑘𝑘𝑘𝑘 between segments of type k. The association interaction between groups is calculated with the number of different site types, 𝑁𝑁𝑆𝑆𝑆𝑆,𝑘𝑘, the number of sites of a given type, 𝑛𝑛𝑘𝑘,𝑎𝑎,𝑛𝑛𝑘𝑘,𝑎𝑎…𝑛𝑛𝑘𝑘,𝑁𝑁𝑆𝑆𝑆𝑆,𝑘𝑘, the distance between the group and the site, 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑑𝑑 , the association energy, 𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 , the bonding volume parameter, 𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎, and the cut-off range, 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑐𝑐. In the current version of SAFT used, 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑑𝑑 and 𝑟𝑟𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑐𝑐 are both fixed to a value of 0.4·𝜎𝜎𝑘𝑘𝑘𝑘. The interaction between the groups k and l is characterised by the parameters 𝜎𝜎𝑘𝑘𝑘𝑘, 𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎, 𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟 and 𝜀𝜀𝑘𝑘𝑘𝑘, and the association parameters 𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 and 𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎. A system is represented in the current theory by the set 𝜈𝜈𝑘𝑘,𝑀𝑀, that collects the number of groups of type k in the compound 𝑖𝑖. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 14 2.1.2. Helmholtz free energy As in other formal statistical-mechanical approaches, SAFT-γ Mie EoS is used to provide the Helmholtz free energy, employed to obtain the macroscopic thermodynamic properties from the molecular model, SAFT-γ Mie in the current work1. The expression [4] collects all the terms used to obtain the overall Helmholtz free energy A: 𝐴𝐴 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇=𝐴𝐴𝑀𝑀𝑑𝑑𝑀𝑀𝑎𝑎𝑘𝑘 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇+𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇+𝐴𝐴𝑐𝑐ℎ𝑎𝑎𝑀𝑀𝑚𝑚 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇+𝐴𝐴𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑐𝑐 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇 [4] where 𝐴𝐴𝑀𝑀𝑑𝑑𝑀𝑀𝑎𝑎𝑘𝑘 is the ideal contribution to the free energy, 𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 is the attractive and repulsive Mie-segments interactions free energy, 𝐴𝐴𝑐𝑐ℎ𝑎𝑎𝑀𝑀𝑚𝑚 is the contribution to the free energy derived from the molecule formation and 𝐴𝐴𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑐𝑐accounts the free energy from the association interaction. N represents the total number of molecules, kB the Boltzmann constant and T the absolute temperature. These terms are explained in detail below. Ideal term The ideal contribution to the Helmholtz free energy is calculated with eq. [2]: 𝐴𝐴𝑀𝑀𝑑𝑑𝑀𝑀𝑎𝑎𝑘𝑘 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇=��𝑥𝑥 𝑀𝑀 ln (𝜌𝜌 𝑀𝑀 Λ 𝑀𝑀 3) 𝑁𝑁𝐶𝐶 𝑀𝑀=1 �−1 [5] where 𝑥𝑥𝑀𝑀 is the mole fraction of component 𝑖𝑖, 𝜌𝜌𝑀𝑀=𝑁𝑁𝑀𝑀/𝑉𝑉 the number density, 𝑁𝑁𝑀𝑀 the number of molecules of component 𝑖𝑖 and 𝑉𝑉 the total volume of the system. Λ𝑀𝑀3 represents the thermal de Broglie volume and incorporates the effects of translational, rotational and vibrational contributions to the kinetic energy. 𝑁𝑁𝐶𝐶 refers to the total number of the components. Monomer term The monomer contribution to the free energy refers to the attractive and repulsive interactions characterised by the Mie potential. It is obtained using a Barker-Henderson26 high-temperature perturbation up to third order [6], 𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇=𝐴𝐴𝐻𝐻𝑆𝑆 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇+𝐴𝐴1 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇+𝐴𝐴2 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇+𝐴𝐴3 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇 [6] where the hard-sphere free energy contribution, 𝐴𝐴𝐻𝐻𝑆𝑆, calculated with eq. [7] and the rest of terms, 𝐴𝐴𝑞𝑞, are obtained with eq. [10]. 𝐴𝐴𝐻𝐻𝑆𝑆 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇=��𝑥𝑥 𝑀𝑀 𝑁𝑁𝐶𝐶 𝑀𝑀=1 �𝜈𝜈 𝑘𝑘,𝑀𝑀 𝜈𝜈 𝑘𝑘 ∗𝑆𝑆 𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 �𝑎𝑎𝐻𝐻𝑆𝑆 [7] 𝜈𝜈𝑘𝑘,𝑀𝑀 is the number of groups of the type k in the compound 𝑖𝑖 and 𝑎𝑎𝐻𝐻𝑆𝑆 is the dimensionless contribution to the hard-sphere free energy. This term is a function of the segment density, 𝜌𝜌𝑎𝑎, the mole fractions and the diameters of the sements1 𝜌𝜌 𝑎𝑎 =𝜌𝜌��𝑥𝑥 𝑀𝑀 𝑁𝑁𝐶𝐶 𝑀𝑀=1 �𝜈𝜈 𝑘𝑘,𝑀𝑀 𝜈𝜈 𝑘𝑘 ∗𝑆𝑆 𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 � [8] 𝑥𝑥𝑎𝑎,𝑘𝑘 is the fraction of segments of a group of type k in the mixture, calculated with eq. [9]: 2. Development of new functional groups 15 𝑥𝑥𝑎𝑎,𝑘𝑘=∑𝑥𝑥𝑀𝑀𝜈𝜈𝑘𝑘,𝑀𝑀𝜈𝜈𝑘𝑘∗𝑆𝑆𝑘𝑘 𝑁𝑁𝐶𝐶 𝑀𝑀=1 ∑𝑥𝑥𝑗𝑗 𝑁𝑁𝐶𝐶 𝑗𝑗=1 ∑𝜈𝜈𝑘𝑘,𝑗𝑗𝜈𝜈𝑘𝑘∗𝑆𝑆𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [9] The rest of the terms from eq. [6], 𝐴𝐴𝑞𝑞, are obtained from eq. [10]: 𝐴𝐴𝑞𝑞 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇=�1 𝑘𝑘𝐻𝐻𝑇𝑇���𝑥𝑥 𝑀𝑀 𝑁𝑁𝐶𝐶 𝑀𝑀=1 �𝜈𝜈 𝑘𝑘,𝑀𝑀 𝜈𝜈 𝑘𝑘 ∗𝑆𝑆 𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 �𝑎𝑎 𝑞𝑞 𝑞𝑞= 1, 2, 3 [10] where 𝑎𝑎𝑞𝑞 is the dimensionless contribution to the free energy, 𝑎𝑎𝑞𝑞=��𝑥𝑥𝑎𝑎,𝑘𝑘𝑥𝑥𝑎𝑎,𝑘𝑘𝑎𝑎𝑞𝑞,𝑘𝑘𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑞𝑞= 1, 2, 3 [11] Chain term The Helmholtz free energy contribution for the formation of molecules, 𝐴𝐴𝑐𝑐ℎ𝑎𝑎𝑀𝑀𝑚𝑚, is obtained with eq. [12], 𝐴𝐴𝑐𝑐ℎ𝑎𝑎𝑀𝑀𝑚𝑚 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇= −�𝑥𝑥 𝑀𝑀 ��𝜈𝜈 𝑘𝑘,𝑀𝑀 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑣𝑣 𝑘𝑘 ∗ 𝑆𝑆 𝑘𝑘 −1�· ln 𝑔𝑔 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀(𝜎𝜎� 𝑀𝑀𝑀𝑀 ;𝜁𝜁 𝑥𝑥 ) 𝑁𝑁𝐶𝐶 𝑀𝑀=1 [12] where 𝜁𝜁𝑥𝑥 is the packaging fraction of a hypothetical fluid, calculated with eq. [18], and 𝑔𝑔𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀(𝜎𝜎�𝑀𝑀𝑀𝑀;𝜁𝜁𝑥𝑥) the value of the RDF, at a distance 𝜎𝜎�𝑀𝑀𝑀𝑀, calculated in eq. [19]. The molecular fraction of a group k in a molecule 𝑖𝑖, 𝑧𝑧𝑘𝑘,𝑀𝑀 is obtained with eq. [13]. 𝑧𝑧𝑘𝑘,𝑀𝑀=𝑣𝑣𝑘𝑘,𝑀𝑀 𝑣𝑣𝑘𝑘∗ 𝑆𝑆𝑘𝑘 ∑𝑣𝑣𝑘𝑘,𝑀𝑀𝑣𝑣𝑘𝑘∗𝑆𝑆𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [13] The chain free energy contribution is obtained using average molecular parameters, 𝜎𝜎�𝑀𝑀𝑀𝑀, 𝑑𝑑𝑀𝑀𝑀𝑀, 𝜀𝜀𝑀𝑀𝑀𝑀 and 𝜆𝜆𝑀𝑀𝑀𝑀, calculated with the following equations. 𝜎𝜎�𝑀𝑀𝑀𝑀3= ��𝑧𝑧𝑘𝑘,𝑀𝑀 𝑧𝑧𝑘𝑘,𝑀𝑀 𝜎𝜎�𝑘𝑘𝑘𝑘 3 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [14] 𝑑𝑑𝑀𝑀𝑀𝑀 3= ��𝑧𝑧𝑘𝑘,𝑀𝑀 𝑧𝑧𝑘𝑘,𝑀𝑀 𝑑𝑑𝑘𝑘𝑘𝑘 3 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [15] 𝜀𝜀𝑀𝑀𝑀𝑀= ��𝑧𝑧𝑘𝑘,𝑀𝑀 𝑧𝑧𝑘𝑘,𝑀𝑀 𝜀𝜀𝑘𝑘𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [16] 𝜆𝜆𝑀𝑀𝑀𝑀= ��𝑧𝑧𝑘𝑘,𝑀𝑀 𝑧𝑧𝑘𝑘,𝑀𝑀 𝜆𝜆𝑘𝑘𝑘𝑘 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [17] The packing fraction of a hypothetical fluid of diameter 𝑑𝑑𝑥𝑥3=∑ ∑ 𝑥𝑥𝑎𝑎,𝑘𝑘𝑥𝑥𝑎𝑎,𝑘𝑘𝑑𝑑𝑘𝑘,𝑘𝑘 3 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 is obtained by: 𝜁𝜁𝑥𝑥=𝜋𝜋𝜌𝜌𝑎𝑎 6��𝑥𝑥𝑎𝑎,𝑘𝑘𝑥𝑥𝑎𝑎,𝑘𝑘𝑑𝑑𝑘𝑘𝑘𝑘 3 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [18] Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 16 The RDF used is a second-order expansion, 𝑔𝑔𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀𝑀𝑀(𝜎𝜎�𝑀𝑀𝑀𝑀;𝜁𝜁𝑥𝑥)=𝑔𝑔𝑑𝑑 𝐻𝐻𝑆𝑆(𝜎𝜎�𝑀𝑀𝑀𝑀;𝜁𝜁𝑥𝑥) exp �𝛽𝛽 𝜀𝜀𝑀𝑀𝑀𝑀 𝑔𝑔1(𝜎𝜎�𝑀𝑀𝑀𝑀) 𝑔𝑔𝑑𝑑 𝐻𝐻𝑆𝑆(𝜎𝜎�𝑀𝑀𝑀𝑀;𝜁𝜁𝑥𝑥)+(𝛽𝛽 𝜀𝜀𝑀𝑀𝑀𝑀)2 𝑔𝑔2(𝜎𝜎�𝑀𝑀𝑀𝑀) 𝑔𝑔𝑑𝑑 𝐻𝐻𝑆𝑆(𝜎𝜎�𝑀𝑀𝑀𝑀;𝜁𝜁𝑥𝑥)� [19] where 𝑔𝑔𝑑𝑑 𝐻𝐻𝑆𝑆(𝜎𝜎�𝑀𝑀𝑀𝑀;𝜁𝜁𝑥𝑥) is the RDF of a system of hard spheres of diameter 𝑑𝑑𝑀𝑀𝑀𝑀 evaluated at a distance 𝜎𝜎�𝑀𝑀𝑀𝑀 and a packing fraction of 𝜁𝜁𝑥𝑥. The RDF terms are approximated ([20]) by the value at the contact distance 𝑑𝑑𝑀𝑀𝑀𝑀, 𝑔𝑔𝑞𝑞(𝜎𝜎�𝑀𝑀𝑀𝑀) ≈ 𝑔𝑔𝑞𝑞�𝑑𝑑𝑀𝑀𝑀𝑀� 𝑞𝑞= 1, 2 [20] Association term The association interaction of molecules due to the definition of bonding sites is calculated with eq. [21], following the Wertheim TPT1 form27, summing over the number of compounds, 𝑁𝑁𝐶𝐶, the number of groups, 𝑁𝑁𝐺𝐺, and over the number of site types on each group, 𝑁𝑁𝑆𝑆𝑆𝑆,𝑘𝑘: 𝐴𝐴𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑐𝑐 𝑁𝑁𝑘𝑘𝐻𝐻𝑇𝑇=�𝑥𝑥𝑀𝑀 𝑁𝑁𝐶𝐶 𝑀𝑀=1 �𝑣𝑣𝑘𝑘,𝑀𝑀 𝑁𝑁𝐺𝐺 𝑘𝑘=1 �𝑛𝑛𝑘𝑘,𝑎𝑎 𝑁𝑁𝑆𝑆𝑆𝑆,𝑘𝑘 𝑎𝑎=1 �𝑙𝑙𝑛𝑛 𝑋𝑋𝑀𝑀,𝑘𝑘,𝑎𝑎+1−𝑋𝑋𝑀𝑀,𝑘𝑘,𝑎𝑎 2� [21] where 𝑛𝑛𝑘𝑘,𝑎𝑎 is the number of sites of type a on a group k and 𝑋𝑋𝑀𝑀,𝑘𝑘,𝑎𝑎 the fraction of molecules of component 𝑖𝑖 that are not bonded at a site a on group k, calculated with eq. [22]: 𝑋𝑋 𝑀𝑀,𝑘𝑘,𝑎𝑎 =�1 + 𝜌𝜌�𝑥𝑥 𝑗𝑗 𝑁𝑁𝐶𝐶 𝑗𝑗=1 �𝜈𝜈 𝑘𝑘,𝑗𝑗 𝑁𝑁_𝐺𝐺 𝑘𝑘=1 �𝑛𝑛 𝑘𝑘,𝑎𝑎 𝑋𝑋 𝑗𝑗,𝑘𝑘,𝑎𝑎 ∆ 𝑀𝑀𝑗𝑗,𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝑁𝑁𝑆𝑆𝑆𝑆,𝑘𝑘 𝑎𝑎=1 �−1 [22] where ∆𝑀𝑀𝑗𝑗,𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 means the overall strength of association between a site of type a on a group k of component 𝑖𝑖 and a site of type b on a group 𝑙𝑙 of component 𝑗𝑗, obtained with eq. [23] ∆𝑀𝑀𝑗𝑗,𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎=𝐹𝐹𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎𝐼𝐼𝑀𝑀𝑗𝑗,𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 [23] where 𝐹𝐹𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 is calculated with eq. [24], the temperature-density polynomial correlation of the association integral, 𝐼𝐼𝑀𝑀𝑗𝑗,𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎, is obtained from eq. [25] and 𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 is the bonding volume parameter. 𝐹𝐹𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎=exp �𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 𝑘𝑘𝐻𝐻𝑇𝑇�−1 [24] 𝐼𝐼𝑀𝑀𝑗𝑗,𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎=��𝑐𝑐𝑝𝑝𝑞𝑞(𝜌𝜌𝑎𝑎𝜎𝜎𝑥𝑥3)�𝑘𝑘𝐻𝐻𝑇𝑇 𝜀𝜀𝑀𝑀𝑗𝑗�𝑞𝑞 10−𝑝𝑝 𝑞𝑞=0 10 𝑝𝑝=0 [25] 𝑐𝑐𝑝𝑝𝑞𝑞 are coefficients obtained from bibliography28, and parameters 𝜎𝜎𝑥𝑥3, 𝜀𝜀𝑀𝑀𝑗𝑗 and 𝜎𝜎�𝑀𝑀𝑗𝑗 are obtained using the following equations. 𝜎𝜎𝑥𝑥3=��𝑥𝑥𝑎𝑎,𝑘𝑘𝑥𝑥𝑎𝑎,𝑘𝑘𝜎𝜎𝑘𝑘𝑘𝑘 3 𝑁𝑁𝐺𝐺 𝑘𝑘=1 𝑁𝑁𝐺𝐺 𝑘𝑘=1 [26] 𝜀𝜀𝑀𝑀𝑗𝑗=�𝜎𝜎�𝑀𝑀𝑀𝑀3𝜎𝜎�𝑗𝑗𝑗𝑗 3 𝜎𝜎�𝑀𝑀𝑗𝑗 �𝜀𝜀𝑀𝑀𝑀𝑀𝜀𝜀𝑗𝑗𝑗𝑗 [27] 2. Development of new functional groups 17 𝜎𝜎�𝑀𝑀𝑗𝑗=𝜎𝜎�𝑀𝑀𝑀𝑀+𝜎𝜎�𝑗𝑗𝑗𝑗 2 [28] 2.1.3. Combining rules As well as the self-interaction parameters, in order to use the described method, the unlike intermolecular interaction parameters are required. To obtain the unlike parameters, combing rules that described the parameters as a function of the self-interaction parameters of the two functional groups are used when possible. These relations are commonly used to facilitate the estimation of binary or mixtures behaviour, without having to optimise all the cross-interaction parameters3. These combining rules are described below, in eq. [10-15]. Each intermolecular parameter is calculated as a combination of the functional-group parameters. The unlike segment diameter, σkl, is calculated following eq. [29] by a simple arithmetic mean of the self-interaction segment diameters of groups k and l3. 𝜎𝜎𝑘𝑘𝑘𝑘=𝜎𝜎𝑘𝑘𝑘𝑘+𝜎𝜎𝑘𝑘𝑘𝑘 2 [29] An arithmetic mean is also used in eq. [30] to obtain the unlike Barker-Henderson hard-sphere diameter, dkl, instead of using the more-rigorous approach of the numerical integration of the segment diameter, reducing the computational cost1. 𝑑𝑑𝑘𝑘𝑘𝑘=𝑑𝑑𝑘𝑘𝑘𝑘+𝑑𝑑𝑘𝑘𝑘𝑘 2 [30] The unlike dispersive energy, εkl, is obtained by using an augmented geometric mean [31]1. 𝜀𝜀𝑘𝑘𝑘𝑘=�𝜎𝜎𝑘𝑘𝑘𝑘 3𝜎𝜎𝑘𝑘𝑘𝑘3 𝜎𝜎𝑘𝑘𝑘𝑘 3�𝜀𝜀𝑘𝑘𝑘𝑘𝜀𝜀𝑘𝑘𝑘𝑘 [31] The combining rule for both, the attractive and repulsive ranges of potential, 𝜆𝜆𝑘𝑘𝑘𝑘 𝑎𝑎 and 𝜆𝜆𝑘𝑘𝑘𝑘 𝑟𝑟, is described in eq. [32]1. 𝜆𝜆𝑘𝑘𝑘𝑘= 3 + �(𝜆𝜆𝑘𝑘𝑘𝑘−3)(𝜆𝜆𝑘𝑘𝑘𝑘−3) [32] In the case of the association parameters, eq. [33] collects the combining rule for the unlike association energy, 𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 , as a geometric mean, and eq. [34] the one for the unlike volume bonding ,𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎1. 𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 =�𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 𝜀𝜀𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 𝐻𝐻𝐻𝐻 [33] 𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎=��𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 3+�𝐾𝐾𝑘𝑘𝑘𝑘,𝑎𝑎𝑎𝑎 3 2�3 [34] 2.1.4. Functional groups The SAFT-γ Mie equation of state is a group-contribution approach in which the fluids are represented by their functionals groups and the interactions between them. For the main objective of the project, the functional groups of interest are collected in Table 1. In that table, group interaction parameters are represented in each cell, where the group on the left is the k group, and the group on the top is the l group. When k and l are different functional Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 18 groups, the cell represents the unlike interaction, whereas when both groups are the same, the self-interaction is represented. The ones that appear with the reference are those which has been previously estimated and the ones with the x refer to those that are estimated in the current work. CR refers to the use of combining rules to estimate the value of the parameters instead of optimising the parameters themselves by fitting them to experimental data. Table 1. Summary of the functional groups and the unlike interactions used in the current work. Each cell represents the group-group interaction parameters. CR: combining rule. *: to be published. x: estimated in the current work. CH3 CH3 3 CH2 CH2 3 3 CH2F CH2F x x x CHF2 CHF2 x x x x CHF CHF x x x x x CF3 CF3 x x x x x * CF2 CF2 x x x x CR * * As a reminder, each functional group of a molecule is defined by the following parameters, the number of segments, 𝜈𝜈𝑘𝑘∗, the shape factor, Sk, the segment diameter, σkk, the attractive and repulsive ranges of potential, 𝜆𝜆𝑘𝑘𝑘𝑘 a and 𝜆𝜆𝑘𝑘𝑘𝑘 r, and the like dispersive energy, εkk. Like parameters of the alkyl and perfluoroalkyl functional groups are collected in Table 2 . The attractive range of potential, 𝜆𝜆𝑘𝑘𝑘𝑘 a, is assigned to London-dispersion value of 6 for every functional group. Table 2. Like group parameters for use within the SAFT-γ Mie group-contribution approach. *: to be published k Group k vk * Sk 𝝀𝝀𝒌𝒌𝒌𝒌 𝐫𝐫 𝝀𝝀𝒌𝒌𝒌𝒌 𝐚𝐚 σkk /Å (εkk/kB) /K Ref. 1 CH3 1 0.57255 15.050 6.0000 4.0772 256.777 3 2 CH2 1 0.22932 19.871 6.0000 4.8801 473.39 3 3 CF3 1 0.54490 28.904 6.0000 4.9310 325.00 * 4 CF2 1 0.27520 33.963 6.0000 5.1980 459.92 * The unlike interaction between two functional groups is represented by the unlike segment diameter, σkl, the attractive and repulsive range of potential, 𝜆𝜆𝑘𝑘𝑘𝑘 r, the unlike dispersive energy, εkl. The unlike interaction group parameters are collected in Table 3, whereas the rest of parameters are obtained by the combining rules [equation]. Table 3. Unlike group interaction parameters for use within the SAFTγ Mie group-contribution approach. *: to be published k l Group k Group l (εkl/kB)/ K Ref. 1 2 CH3 CH2 350.77 3 3 4 CF3 CF2 390.00 * The remaining parameters are estimated in the current work. When all the parameters of Table 1 are estimated, the group contribution approach can be used for all the possible molecules involving the mentioned groups. 2. Development of new functional groups 19 2.1.5. Parameter estimation Within the SAFT-γ Mie equation of state, molecules are defined by their functional groups, characterised by their parameters and the interaction between them. Parameters for the alkyl and perfluoroalkyl have been obtained in previous works (Table 2). The remaining functional groups are characterised in the current work, following the next methodology2,3. A functional group is represented by the parameters cited in the section 2.1.1. Molecular model. The unlike interaction is obtained by the combining rules, summarised in section 2.1.3. Combining rules, involving the like parameters of the functional groups, except for the unlike dispersive energy, 𝜀𝜀𝑘𝑘𝑘𝑘. The number of segments, 𝜈𝜈𝑘𝑘∗, is set to 1 for all the groups in the current work. To sum up, the following parameters have been estimated with this methodology: the shape factor, 𝑆𝑆𝑘𝑘, the segment diameter, 𝜎𝜎𝑘𝑘𝑘𝑘, the attractive and repulsive ranges of potential, 𝜆𝜆𝑘𝑘𝑘𝑘 a and 𝜆𝜆𝑘𝑘𝑘𝑘 r, the like dispersive energy, 𝜀𝜀𝑘𝑘𝑘𝑘, and the unlike dispersive energy, 𝜀𝜀𝑘𝑘𝑘𝑘. These parameters of the functional groups of interest are estimated. To do so, experimental data are compared with the group contribution approach results, and the difference between both is minimised varying the parameters considered for the estimation. The experimental data of interest for parameter estimation have proved to be VLE (Vapour-liquid equilibrium) data3, and more precisely, the vapour pressure, pvap, and the saturated liquid density, 𝜌𝜌𝑘𝑘𝑀𝑀𝑞𝑞 𝑎𝑎𝑎𝑎𝑠𝑠. In the case of experimental data from binary mixtures, that are used in the case that no pure experimental data are available, bubble pressure, pBubble, and dew pressure, pDew, are used to obtain the parameters. The experimental data and the model data are compared using the equation [35], that sets the objective function to minimise in order to reduce the error between the SAFT model, with the parameters to optimise, and the experimental data gathered. This objective function is the result of the weighted addition of the square of the relative error in the vapour pressure and the square of the relative error in the saturation liquid density3. Variables ω1 and ω2 represent the weights of each property in the final function. In this particular case, both weights are assigned the value of 𝜔𝜔1=𝜔𝜔2= 1, so both properties have equal weighting in the objective function. min Ω 𝑖𝑖 𝑚𝑚𝑎𝑎𝑗𝑗 =𝜔𝜔 1 � �𝑝𝑝𝑣𝑣𝑎𝑎𝑝𝑝 𝑀𝑀𝑥𝑥𝑝𝑝(𝑇𝑇𝑚𝑚)−𝑝𝑝𝑣𝑣𝑎𝑎𝑝𝑝 𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐(𝑇𝑇𝑚𝑚,𝛺𝛺) 𝑃𝑃𝑣𝑣𝑎𝑎𝑝𝑝 𝑀𝑀𝑥𝑥𝑝𝑝(𝑇𝑇𝑚𝑚) �2 𝑁𝑁𝑃𝑃𝑣𝑣𝑎𝑎𝑣𝑣 𝑚𝑚 +𝜔𝜔2� � 𝜌𝜌𝐿𝐿,𝑎𝑎𝑎𝑎𝑠𝑠 𝑀𝑀𝑥𝑥𝑝𝑝 (𝑇𝑇𝑚𝑚)−𝜌𝜌𝐿𝐿,𝑎𝑎𝑎𝑎𝑠𝑠 𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐(𝑇𝑇𝑚𝑚,𝛺𝛺) 𝜌𝜌𝑘𝑘,𝑎𝑎𝑎𝑎𝑠𝑠 𝑀𝑀𝑥𝑥𝑝𝑝(𝑇𝑇𝑚𝑚)� 2 𝑁𝑁𝜌𝜌𝑘𝑘,𝑠𝑠𝑎𝑎𝑠𝑠 𝑚𝑚 [35] When the experimental data used to estimate the functional groups are binary VLE data equation [36] is used instead. The objective function to minimise in this case is the addition of the relative error between both, the experimental and calculated dew and bubble pressure. min Ω 𝑖𝑖 𝑚𝑚𝑎𝑎𝑗𝑗 =𝜔𝜔 1 � �𝑃𝑃𝐻𝐻𝐵𝐵𝑎𝑎𝑎𝑎𝑘𝑘𝑀𝑀 𝑀𝑀𝑥𝑥𝑝𝑝 (𝑇𝑇𝑚𝑚,𝑛𝑛)−𝑃𝑃𝐻𝐻𝐵𝐵𝑎𝑎𝑎𝑎𝑘𝑘𝑀𝑀 𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐 (𝑇𝑇𝑚𝑚,𝑛𝑛,𝛺𝛺) 𝑃𝑃𝐻𝐻𝐵𝐵𝑎𝑎𝑎𝑎𝑘𝑘𝑀𝑀 𝑀𝑀𝑥𝑥𝑝𝑝 (𝑇𝑇𝑚𝑚,𝑛𝑛) �2 𝑁𝑁𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑘𝑘𝑏𝑏 𝑚𝑚 +𝜔𝜔2� � 𝑃𝑃𝐷𝐷𝑀𝑀𝐷𝐷 𝑀𝑀𝑥𝑥𝑝𝑝(𝑇𝑇𝑚𝑚,𝑛𝑛)−𝑃𝑃𝐷𝐷𝑀𝑀𝐷𝐷 𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐(𝑇𝑇𝑚𝑚,𝑛𝑛,𝛺𝛺) 𝑃𝑃𝐷𝐷𝑀𝑀𝐷𝐷 𝑀𝑀𝑥𝑥𝑝𝑝(𝑇𝑇𝑚𝑚,𝑛𝑛)� 2 𝑁𝑁𝑃𝑃𝐷𝐷𝑏𝑏𝐷𝐷 𝑚𝑚 [36] The variable Ω (present in equations [35] and [36]) represents the vector of parameters that are going to be estimated by minimising the objective function. This set of parameters could belong Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 20 only to one functional group or to a wide range of groups. Furthermore, this set could include only self-interaction parameters, only unlike interaction parameters or both of them. Once the parameters to estimate are selected, upper and lower bounds are set, so the parameters obtained drop into the feasible range. The resulting problem is a NonLinear Problem, NLP, in which function [35], or [36] in case a binary system is used, is minimised modifying the set of parameters Ω, that is bounded by ΩL and ΩU. The NLP solver used belongs to the category of multiple-starting-points solvers, in which a Sobol sequence is used to specify the initial guess of the parameters estimated in each step. The software used in this step is python, and more precisely gSAFTmm29, a compiled version of SAFT-γ Mie. Experimental data The experimental data selection was carried out taking into account the presence of the functional groups to be estimated, and also the absence of other functional groups or interactions not available in the literature, reducing the parameters to estimate to the ones identified previously as the relevant ones for the current work. The compounds used belong to the category of refrigerants, and were selected from a wide range of molecules, using the experimental data availability, of the selected properties, and the quality of the data as the factors to decide which set of experimental data employ in the current work. In Table 4 the experimental data used in the parameter optimisation are summarised, including the compound, the temperature range, and number of points for both properties of interest. According to previous work3, in order to obtain the more accurate parameters to use this group contribution approach, the VLE experimental data used in the estimation process should cover the range from the triple point to 90% of the critical temperature of the fluid. This criterion has been used in the current work to obtain the desired parameters. The main reason of avoiding experimental data near the critical region is to prevent the possible loss of the physical significance of the parameters due to an improvement of the critical region description. It has been proved that, for an analytical equation of state, is not possible to capture at the same time both the critical and the subcritical regions accurately. For this reason, experimental data employed are limited to 90% of the critical temperature, ensuring an adequate model for the subcritical region. A summary of the compound used in this process, as well as the temperature range of the data available and the number of points, for both properties employed, the pure vapour pressure and the liquid saturation density is represented in Table 4. Experimental data from only one mixture, 1,1,1,2-tetrafluoroethane (R-134a) and 1,1,1,2,3,3,3heptafluoropropane (R-227ea)30, were used in the current parameter optimisation. 2. Development of new functional groups 21 Table 4. Experimental data of pure refrigerants. TTP: triple point temperature. Tcr: critical point temperature. T range: range of temperatures of the experimental data employed. n: number of experimental points. Ref.: bibliography reference of the experimental data. Name Molecular Formula TTP /K Tcr /K Vap. Pressure Density T range /K n Ref. T range /K n Ref. Pentafluoroethane C2HF5 172.52 339.26 223 - 303 29 31 248 - 306 16 1,1,2,2-tetrafluoroethane C2H2F4 120.00 391.85 150 - 350 21 32 150 - 350 21 32 1,1,1,2-tetrafluoroethane C2H2F4 169.85 374.07 204 - 333 34 33 253 - 333 18 34 1,1,2-trifluoroethane C2H3F3 189.20 425.00 220 - 380 15 32 314 - 378 13 35 1,1,1-trifluoroethane C2H3F3 161.00 346.25 251 - 311 31 36 230 - 310 13 37 1,1-difluoroethane C2H4F2 150.00 386.50 243 - 343 54 38 243 - 343 26 38 Fluoroethane C2H5F 130.00 375.00 180 - 330 16 32 180 - 340 17 32 1,1,1,2,3,3,3-heptafluoropropane C3HF7 146.35 375.04 233 - 338 54 39 243 - 337 38 40 1,1,1,2,3,3-hexafluoropropane C3H2F6 242.00 412.21 293 - 363 17 41 262 - 343 37 42 1,1,1,3,3,3-hexafluoropropane C3H2F6 179.52 398.07 271 - 359 68 39 262 - 353 38 42 1,1,2,2,3-pentafluoropropane C3H3F5 200.00 447.57 250 - 400 16 32 200 - 400 21 32 1,1,1,2,2-pentafluoropropane C3H3F5 120.00 381.65 280 - 343 14 43 296 - 344 6 43 1,1,1,3,3-pentafluoropropane C3H3F5 171.05 430.65 235 - 386 101 39 250 - 314 65 44 1-fluoropropane C5H11F 130.00 419.00 230 - 370 15 32 200 - 370 18 32 2-Fluoropropane C3H7F 130.00 410.40 230 - 370 15 32 200 - 370 18 32 1,1,1,2,2,3,3,4,4-nonafluorobutane C4HF9 130.00 413.00 270 - 370 11 32 298 - 373 6 43 1,1,1,2,2,3,3,4-octafluorobutane C4H2F8 130.00 433.65 260 - 390 14 32 200 - 390 20 32 1,1,1,3,3-pentafluorobutane C4H5F5 239.00 459.91 298 - 413 20 45 289 - 413 8 45 2-fluorobutane C4H9F 140.00 448.00 216 - 298 6 46 200 - 400 21 32 1-fluoropentane C3H7F 150.00 496.70 300 - 440 15 32 200 - 440 25 32 Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 28 According to the comparison between experimental and calculated values for the saturated density, the wide majority of the points fall into the ±5% deviation area. Furthermore, there is no evidence of higher average deviations at either high or low values of the density. Finally, both properties used for the parameter estimation are analysed calculating the deviation between experimental data and the data calculated with the model proposed at the same conditions for each compound. The Absolute Deviation, AD, represents the mean deviation from experimental data and calculated data, in absolute terms, whereas the Average Absolute Deviation, AAD, is the absolute deviation divided by the experimental value, expressed as a percentage. 𝐴𝐴𝐴𝐴=1 𝑁𝑁�|𝑅𝑅𝑀𝑀𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐−𝑅𝑅𝑀𝑀𝑀𝑀𝑥𝑥𝑝𝑝| 𝑁𝑁 𝑀𝑀 [37] 𝐴𝐴𝐴𝐴𝐴𝐴=1 𝑁𝑁�|𝑅𝑅𝑀𝑀𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐−𝑅𝑅𝑀𝑀𝑀𝑀𝑥𝑥𝑝𝑝| |𝑅𝑅𝑀𝑀𝑀𝑀𝑥𝑥𝑝𝑝|·100 𝑁𝑁 𝑀𝑀 [38] Both deviations for the vapour pressure and the saturated liquid density are collected in Table 8 for all the compounds with experimental data available, considering all the points included in the range between 2. Development of new functional groups 29 Table 8. Deviations from experimental data using the proposed parameters. AD: absolute deviation. AAD: average absolute deviation. Name Molecular Formula AD AAD % p vap /MPa ρ liq,sat /kg·m-3 p vap ρ liq,sat Pentafluoroethane C 2 HF 5 0.02 12.43 2.50 0.93 1,1,2,2-tetrafluoroethane C 2 H 2 F 4 0.01 17.32 1.96 1.25 1,1,1,2-tetrafluoroethane C 2 H 2 F 4 0.01 12.64 3.34 1.00 1,1,2-trifluoroethane C 2 H 3 F 3 0.06 6.28 5.33 0.57 1,1,1-trifluoroethane C 2 H 3 F 3 0.03 24.59 3.13 2.61 1,1-difluoroethane C 2 H 4 F 2 0.02 43.90 2.34 4.95 Fluoroethane C 2 H 5 F 0.10 8.98 4.73 1.23 1,1,1,2,3,3,3-heptafluoropropane C 3 HF 7 0.01 37.55 1.61 2.65 1,1,1,2,3,3-hexafluoropropane C 3 H 2 F 6 0.01 40.66 0.85 2.91 1,1,1,3,3,3-hexafluoropropane C 3 H 2 F 6 0.03 46.19 4.50 3.44 1,1,2,2,3-pentafluoropropane C 3 H 3 F 5 0.05 6.38 7.77 0.45 1,1,1,2,2-pentafluoropropane C 3 H 3 F 5 0.01 25.69 1.85 2.40 1,1,1,3,3-pentafluoropropane C 3 H 3 F 5 0.03 34.43 6.09 2.50 2-Fluoropropane C 3 H 7 F 0.00 21.35 0.34 3.03 1,1,1,2,2,3,3,4,4-nonafluorobutane C 4 HF 9 0.00 15.49 0.69 1.22 1,1,1,2,2,3,3,4-octafluorobutane C 4 H 2 F 8 0.02 15.61 5.90 1.01 1,1,1,3,3-pentafluorobutane C 4 H 5 F 5 0.01 4.00 2.98 0.40 1-fluoropropane C 5 H 11 F 0.01 6.31 2.63 0.87 1-fluoropentane C 3 H 7 F 0.01 6.19 2.52 0.84 2-fluorobutane C 4 H 9 F 0.00 8.43 1.67 1.16 Only four AAD calculated are higher than 5% for the vapour pressure, with an overall AAD of 3.14%, so it can be considered that the parameters proposed represent properly the vapour pressure of the refrigerants proposed. The same can be assumed for the saturated liquid density, with only three compounds with an AAD over 3%, and an overall AAD of 1.77%. According to the results shown, both the deviations and the figures, the models proposed for the groups involved are adequate to obtain an approach of the pure VLE, as the vapour pressure and the saturated liquid density, for the temperature range from the triple point to a temperature of 90%·Tcr. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 30 2.2.2. Properties prediction Once all the parameters needed are optimised to capture the experimental VLE, other properties of interest are analysed. Following the main objective of the current work, the optimisation of ORC, thermodynamic properties are estimated and compared with experimental data from bibliography. Regarding the thermodynamic properties, in the current work we focus the efforts on temperature—entropy (T-s diagrams) diagrams and pressure—enthalpy (p-h diagrams) ones. Other thermodynamic properties studied are the isobaric heat capacity, cp, the speed of sound, us, the isobaric coefficient of expansion, 𝛼𝛼𝑝𝑝, the isothermal compressibility, 𝛽𝛽𝑆𝑆, and the Joule-Thomson coefficient, 𝜇𝜇𝐽𝐽𝑆𝑆. In addition to these properties, the binary VLE is studied to know the extent to which the proposed parameters are useful and can be used. As a working fluid for an ORC, a blend of organic molecules can be used, leading in some cases to promising results. Thermodynamic properties Finally, entropy and enthalpy are analysed for six compounds. The estimated properties are compared with the experimental data from NIST32. Both thermodynamic properties are crucial to simulate cycles to obtain a power output, such as ORCs. Enthalpies are used to estimate the power and energy exchanged in the cycle, and entropies are useful for the isentropic efficiency of both the pump and the expander of the cycle. The specific entropy is analysed along with the temperature, by plotting the T-s diagram estimated with the group contribution approach and the experimental T-s diagram. To analyse the accuracy of the studied method, six fluorinated hydrocarbons are analysed, plotting the results in Fig. 10. Fig. 10. Temperature-entropy (T-s) diagrams. s: specific entropy, J·(mol·K)-1. T: temperature, K. Curves: calculated entropies with the proposed model. Points: experimental entropies. Reference states: s = 1 J·(g·K)-1 at 0°C. From the displayed diagrams, it can be concluded that the bigger deviations on the estimation of the entropy, for the studied functional groups, take place near the critical point, at the highest temperature of the VLE. This deviation can be explained as a cumulative error, as the proposed method fails to fit the experimental data, with the same parameters, in the critical region and the rest of VLE. However, the entropy estimation, represented in the diagrams with the continuous line, reproduce adequately the entropy in both the vapour and liquid phase. In addition to the plots, the deviations from experimental data are collected in Table 9. The absolute deviation [37] and the average absolute deviation [38] are split for the vapour and the liquid phase, and the overall deviations are calculated too. Both deviations are analysed for the entire range of temperatures, from 200K to pcr of each compound. 2. Development of new functional groups 31 Table 9. Entropy deviations between experimental and calculated data. AD: absolute deviation. AAD: average absolute deviation. Compound Vapour Liquid AD /J·(mol·K)-1 AAD % AD /J·(mol·K)-1 AAD % AD /J·(mol·K)-1 AAD % Pentafluoroethane 1.7044 0.99 1.3640 1.07 1.5342 1.03 1,1,1-trifluoroethane 2.8041 2.08 2.2075 2.54 2.5058 2.31 1,1,1,2,3,3,3-heptafluoropropane 1.1268 0.45 0.9825 0.46 1.0546 0.46 1,1,1,2-tetrafluoroethane 1.7445 1.02 0.9068 0.68 1.3257 0.85 1,1,1,2,3,3-hexafluoropropane 0.9075 0.36 0.4229 0.18 0.6652 0.27 1,1-difluoroethane 2.8182 2.17 1.5573 1.97 2.1878 2.07 As reflected in the deviations calculated and the T-s diagrams, the proposed parameters for the selected functional groups within the SAFT-γ Mie method can reproduce with a deviation under 3% the entropy of the six molecules selected. The maximum deviation calculated corresponds to the 1,1,1-trifluoroethane, the compound with a higher deviation in the vapour pressure, according to Table 8. The lowest deviation, corresponding to the 1,1,1,2,3,3-hexafluoropropane, reflects the low deviation in the vapour pressure for this molecule. The enthalpy is represented with the pressure in the p-h diagram (Fig. 11). Fig. 11. Pressure-enthalpy (p-h) diagrams. h: specific enthalpy, kJ·(mol)-1. p: pressure, MPa. Curves: calculated enthalpies with the proposed model. Points: experimental enthalpies. Reference states: h = 200 kJ·kg-1 at 0°C. As for the entropy, the proposed approach predicts the enthalpy properly from low temperatures near the triple point, until temperatures near the critical point. The higher deviations are found to occur in the critical region, due to the lack of accuracy prediction the VLE at this temperatures and pressures for the studied compounds. The calculated deviations ([37][38]) for the enthalpy of the compounds represented in the displayed diagrams are collected in Table 10. The AD and AAD are calculated for the saturated vapour phase enthalpy, the saturated liquid phase enthalpy and the overall for each compound. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 32 Table 10. Enthalpy deviations between experimental and calculated data. AD: absolute deviation. AAD: average absolute deviation. Compound Vapour Liquid AD /kJ·(mol)-1 AAD % AD /kJ·(mol)-1 AAD % AD /kJ·(mol)-1 AAD % Pentafluoroethane 0.5325 1.39 0.6810 2.09 0.6067 1.74 1,1,1-trifluoroethane 0.7299 2.34 1.3580 5.48 1.0440 3.91 1,1,1,2,3,3,3-heptafluoropropane 0.4288 0.71 0.7429 1.46 0.5859 1.09 1,1,1,2-tetrafluoroethane 2.0457 5.05 0.3098 1.55 1.1778 3.30 1,1,1,2,3,3-hexafluoropropane 0.4169 0.65 0.7100 1.23 0.5635 0.94 1,1-difluoroethane 1.0191 3.02 0.3084 1.58 0.6638 2.30 The tendency shown in the deviations for the enthalpy is the same as for the entropy, obtaining a lower deviation for those compounds with a better prediction of the VLE. It can be assumed that the proposed approach is a valid one to estimate the enthalpy, as reflected in the deviations table and the p-h diagrams. All the calculated deviations are below 4% average absolute deviation. Finally, both main thermodynamic properties, enthalpy and entropy, can be estimated adequately with the SAFT-γ Mie equation of state, using the parameter estimated by adjusting the vapour pressure and the saturated liquid density. The estimation of these properties allow the use of this accurate group contribution approach to model ORCs, as a further step of the current work. Other thermodynamic properties The calorific properties involved in this validation step are defined below. The isobaric heat capacity, 𝑐𝑐𝑝𝑝=�𝜕𝜕ℎ 𝜕𝜕𝑇𝑇�𝑝𝑝 [39] the Joule-Thomson coefficient, 𝜇𝜇 JT =�𝜕𝜕𝑇𝑇 𝜕𝜕𝑝𝑝�ℎ=−�𝜕𝜕ℎ 𝜕𝜕𝑝𝑝�𝑆𝑆 �𝜕𝜕ℎ 𝜕𝜕𝑇𝑇�𝑝𝑝 =−𝑉𝑉−𝑇𝑇�𝜕𝜕𝑉𝑉 𝜕𝜕𝑇𝑇�𝑝𝑝 𝑐𝑐𝑝𝑝 [40] the speed of sound, 𝑢𝑢𝑎𝑎2=�𝜕𝜕𝑝𝑝 𝜕𝜕𝜌𝜌�𝑎𝑎 [41] the isobaric coefficient of expansion, 𝛼𝛼𝑝𝑝=1 𝑉𝑉·�𝜕𝜕𝑉𝑉 𝜕𝜕𝑇𝑇�𝑝𝑝 [42] and the isothermal compressibility, 2. Development of new functional groups 33 𝛽𝛽𝑆𝑆= −1 𝑉𝑉·�𝜕𝜕𝑉𝑉 𝜕𝜕𝑝𝑝�𝑆𝑆 [43] These properties are estimated for three fluorinated hydrocarbons, pentafluoroethane, 1,1,1,2tetrafluoroethane and 1,1,1,2,3,3,3-heptafluoroethane, and compared with pseudo experimental data available from NIST/TRC32. These properties depend on the temperature and pressure at the same time, so the plots represent the model, cope with the experimental data, at fixed values of the pressure, reflecting the temperature effect at each temperature. First of all, pentafluoroethane (R-125) calorific performance is analysed and plotted in Fig. 12. Fig. 12. Calorific properties of pentafluoroethane. Upper row: Left: Isobaric heat capacity. Right: Joule-Thomson coefficient. Lower row: From left to right: Speed of sound, isobaric coefficient of expansion and isothermal compressibility. As reflected in the plots, the proposed method is able to estimate the calorific properties selected with a good accuracy, except for the Joule-Thomson coefficient at low temperatures, where a higher deviation is obtained. The AAD calculated for the isobaric heat capacity, 𝑐𝑐𝑝𝑝, is 2.75%, whereas the AAD for the Joule-Thomson coefficient, 𝜇𝜇JT, is 9.01%. The second compound analysed, the 1,1,1,2-tetrafluoroethane (R-134a), is represented in Fig. 13. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 34 Fig. 13. Calorific properties of 1,1,1,2-tetrafluoroethane. Upper row: Left: Isobaric heat capacity. Right: Joule-Thomson coefficient. Lower row: From left to right: Speed of sound, isobaric coefficient of expansion and isothermal compressibility. The performance of the proposed approach on the calorific properties of R-134a show a bigger deviation, reaching an AAD of 17.52% for the Joule-Thomson coefficient, and an AAD of 2.06% for the isobaric heat capacity. Furthermore, the plots of the remaining properties studied have bigger deviations, as shown for the speed of sound, compared with pentafluoroethane predictions. Calorific properties of the last compound analysed, the 1,1,1,2,3,3,3-heptafluoropropane (R227ea), are plotted in Fig. 14. Fig. 14. Calorific properties of 1,1,1,2,3,3,3-heptafluoropropane. Upper row: Left: Isobaric heat capacity. Right: JouleThomson coefficient. Lower row: From left to right: Speed of sound, isobaric coefficient of expansion and isothermal compressibility. The AAD calculated for the 𝑐𝑐𝑝𝑝 is 1.74%, whereas the AAD for 𝜇𝜇JT is 9.75%. The predicted properties have good agreement with experimental data, performing better at low temperatures, near the triple point, and at temperatures over 400K. Bigger deviations are found 2. Development of new functional groups 35 to take place near the changing phase region. This deviation can be explained as a cumulative error from the VLE estimation. The deviations between experimental and model data are calculated for both, the isobaric heat capacity and the Joule-Thomson coefficient. The AD and the AAD are collected in Table 11. Table 11. Isobaric heat capacity ,𝑐𝑐𝑝𝑝, and Joule-Thomson coefficient, 𝜇𝜇𝐽𝐽𝑆𝑆, deviations Compound 𝒄𝒄𝒑𝒑 𝝁𝝁𝐉𝐉𝐉𝐉 AD /J·(mol·K)-1 AAD% AD /K·(MPa)-1 AAD % Pentafluoroethane 4.3582 2.76 1.0044 9.01 1,1,1,2-tetrafluoroethane 3.2682 2.06 2.5025 17.52 1,1,1,2,3,3,3-heptafluoropropane 3.1612 1.74 1.2141 9.75 The description of the isobaric heat capacity and the Joule-Thomson Coefficient provided by the model studied in the current work provides an adequate description of the experimental data. Furthermore, the isobaric heat capacity, 𝑐𝑐𝑝𝑝, has a low AAD for the three compounds analysed at this step, in all the cases under 3%. This property of the working fluid is widely used for process modelling. In the case of the Joule-Thomson coefficient, 𝜇𝜇JT, bigger deviations are obtained when comparing the results using the proposed model with the experimental data collected. One of the main reasons of this deviation with respect to the experimental data is the idealheat-capacity contribution, estimated in the proposed method with the Joback-Reid correlation12. This correlation does not perform well for halogenated compounds. To reduce this error, that is found to be bigger for fluorinated compounds, a new correlation is being developed47. Binary VLE The interaction and behaviour of the mixtures of fluorinated hydrocarbons with themselves, alkanes and perfluoroalkanes is analysed with the proposed parameters for the functional groups within the SAFT-γ Mie by means of the binary vapour-liquid equilibrium (VLE). Binary VLE is represented in the current work in pressure—composition (pxy) diagrams, due to the availability of experimental data. In this type of diagram, equilibrium pressure is represented as a function of the mixture composition at constant temperature. The upper curve corresponds to the bubble curve, in which the bubble pressure is represented at each composition. The region over this curve corresponds to the liquid phase, and the region between the bubble and dew curves, the coexistence of a vapour and a liquid phase. The area below the dew curve corresponds to the vapour phase. In each figure, two binary VLE plots are represented. Each curve represents the isothermal VLE, at the temperature shown in each legend. The x axis represents the concentration, in mole fraction, of the second compound of the mixture represented in the plot. Fig. 15 represents, on the left side, the VLE of the mixture hexafluroethane / 1,1,1,2tetrafluoroethane, at four different temperatures. The right plot corresponds to the mixture of pentafluoroehtane / 1,1,1,2,3,3,3-heptafluoroethane. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 36 Fig. 15. Left: hexafluoroethane/1,1,1,2-tetrafluoroethane. Right: pentafluoroethane/1,1,1,2,3,3,3heptafluoropropane. Both phase diagrams are adequately predicted with the proposed parameters and equation of state. However, in both plots, the deviation increases at high values of the composition of the second molecule, near the pure regions (𝑥𝑥= 1). To explain the deviation near the pure compound VLE, it is found that the temperature considered is near the critical point of the second compound, making the deviation in the pure vapour pressure bigger. As explained before, the experimental data selected for the parameter optimisation was taken until 90% of Tcr because of analytical EoS are not able to capture the critical and subcritical regions with the same parameters, leading to higher deviation near the critical point for the pure compound. Fig. 16. Left: pentafluoroethane/1,1,1,3,3,3-hexafluoropropane. Right: pentafluoroethane/1,1,1,3,3pentafluoropropane. The right plot of Fig. 16 corresponds to the binary VLE, at two constant temperatures, of pentafluoroethane/1,1,1,3,3,3,3-hexafluoropropane. The pxy diagram reflects a good fitting between experimental and predicted data, resulting in an AAD of 3.94%. The right plot, corresponding to the blend of pentafluoroethane/1,1,1,3,3-pentafluoropropane, also reflects the agreement of experimental and model data, with an AAD of 8.52%. 2. Development of new functional groups 37 Fig. 17. Left: 1,1,1,2-tetrafluoroethane/1,1,1,2,3,3,3-heptafluoropropane. Right: 1,1,1,2-tetrafluoroethane/1,1,1,3,3pentafluoropropane. In Fig. 17, the mixture of 1,1,1,2-tetrafluroethane/1,1,1,2,3,3,3-heptafluoropropane (left plot) and with 1,1,1,3,3-pentafluoropropane, are collected. The first mixture was adequately predicted with the proposed method, with and AAD of 1.33%, whereas the plot on the right reflects a disagreement between experimental and predicted points, overpredicting the equilibria pressures. However, the model is able to predict the shape of the VLE, although the values of the pressure do not match, predicting the VLE in a qualitative way, but not in a quantitative way. Fig. 18. Left: 1,1,1-trifluoroethane/1,1-difluoroethane. Right: 1,1,1-trifluoroethane/1,1,1,3,3,3-hexafluoropropane. Both binary VLEs plotted in Fig. 18 reflect a good agreement of the model with the experimental points, with an AAD of 4.14% for the mixture on the and 3.74% for the right one. Fig. 19. Left: 1,1-difluoroethane/1,1,1,2,3,3,3-heptafluoropropane. Right: 1,1-difluoroethane/1,1,1,3,3pentafluoropropane. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 44 reduced boiling temperature, and 𝐶𝐶𝑀𝑀𝑘𝑘Bthe coefficient of the group 𝑖𝑖 for the thermal conductivity, that takes the values collected in Table 16. Table 16. Coefficients for liquid thermal conductivity51. 𝐶𝐶𝑀𝑀𝑘𝑘𝑘𝑘: liquid thermal conductivity coefficient for group 𝑖𝑖. Functional group 𝑪𝑪𝒊𝒊𝒌𝒌𝐋𝐋/𝐖𝐖·𝐦𝐦−𝟏𝟏·𝐊𝐊−𝟏𝟏 CH3 0.0545 CH2 -0.0008 CF3 0.0474 CF2 -0.0094 CH2F 0.0502 CHF -0.0090 CHF2 0.0420 Vapour thermal conductivity, kV The proposed method for the vapour thermal conductivity50,52 is not a group contribution, however, it depends on previously calculated properties, as the dynamic viscosity or the heat capacity. 𝑘𝑘V=3.75 ·𝜂𝜂V·𝑅𝑅 𝑀𝑀𝑚𝑚′·𝜓𝜓 [58] where kV is the vapour thermal conductivity in W·m-1·K-1, R is the ideal gas constant (𝑅𝑅= 8.314 J · mol−1· K−1), M’ is the molar mass in kg·mol-1 and ηV the vapour dynamic viscosity in Pa·s. Factor 𝜓𝜓 is calculated with eq. [59]. 𝜓𝜓= 1 −𝛼𝛼(0.215 + 0.28288 ·𝛼𝛼−1.061 ·𝛽𝛽+ 0.26665 ·𝑍𝑍) 0.6366 +𝛽𝛽·𝑍𝑍+ 1.061 ·𝛼𝛼·𝛽𝛽 [59] where factors α, β and Z are calculated with equations [60],[61] and [62] respectively. 𝛼𝛼=𝑐𝑐𝑉𝑉 𝑅𝑅−3 2 [60] 𝛽𝛽= 0.7862 −0.7109 ·𝜔𝜔+ 1.3168 ·𝜔𝜔2 [61] 𝑍𝑍= 2 + 10.5 · 𝑇𝑇r2 [62] where cV is the isochoric heat capacity in J·mol-1·K-1, ω the acentric factor of the molecule and 𝑇𝑇r=𝑇𝑇/𝑇𝑇cr the reduced temperature. The acentric factor mentioned above, ω, is a property of the fluid and it does not depend on the temperature or other factors. Its value is obtained with eq. [63], and it is defined at a reduced temperature of 0.7. 𝜔𝜔=−1−log10𝑝𝑝vap(𝑇𝑇r= 0.7) 𝑝𝑝cr [63] To sum up, in Table 17 all the methods involving the estimation of the transport properties of interest described previously are collected with the corresponding references to the bibliography. 3. Organic Rankine Cycle analysis 45 Table 17. Summary of the methods/correlations used for transport properties Property Methods Reference Normal boiling point Joback-Reid 12 Critical temperature SAFT-γ Mie / Joback-Reid 2,12 Critical pressure SAFT-γ Mie / Joback-Reid 2,12 Critical volume SAFT-γ Mie / Joback-Reid 2,12 Surface tension Sastri-Rao 48 Viscosity Liquid Sastri-Rao 49 Vapour Reichenberg 53 Thermal conductivity Liquid Sastri-Rao 51 Vapour Chung 50,52 3.1.2. Organic Rankine Cycle (ORC) model Rankine cycles are used to obtain electric power in a turbine from a heat source. In the case of ORCs, the objective is to convert the heat from a low-temperature heat source into useful electric power. In order to increase the efficiency and the net power output of the cycle, compared to steam Rankine cycles, organic compounds are used as working fluids as they show a better performance than water in these conditions of low temperature availability of the heat19,20. Among the possible configurations of ORCs, we distinguish between pure working fluid ORC (b), or just ORC, when the working fluids is a pure organic compound, and blend ORC (c) when the working fluid used is a mixture of two, or more, organic compounds. Additionally, depending on the condition of the fluid at the output of the evaporator, superheated cycles (b), if the output of the evaporator is a superheated vapour, and partially evaporated, if the output is in the vapour-liquid phase, can be implemented (Fig. 20). An ORC operates following the next schema: a saturated liquid at pressure p1 [1], is compressed in a pump to a pressure p2 [2], then the fluid is heated until a saturated liquid at pressure p2 is obtained [2’], the fluid is evaporated [3’], and superheated ΔTsh, until T3 [3]. The pressure drop in the condenser is neglected, leading to a constant pressure in the whole heat exchanger. The power is generated by expanding the working fluid from [3] to a lower pressure, p4 [4] in the turbine. Finally, the vapour is cooled until the saturation temperature [4’] and condensed, reaching the initial point [1]. As well as in the heater, the cooler is supposed to operate at constant pressure, neglecting any pressure drops. The common cycle is represented in the T-s diagram for a pure component in Fig. 20 (b). The heat source is considered to have a constant heat flow, 𝑚𝑚󰇗ℎ𝑐𝑐𝑝𝑝ℎ, and is characterised by three temperatures, the inlet temperature, Thi, the outlet temperature, Tho, and the pinch temperature, Thp, that corresponds to the point of the heat exchanger in which the temperature difference between the heat source and the working fluid is the lowest. This point occurs when the heat source is at Thp, and the working fluid at [2’]. The cooling source is also represented by a constant heat flow, 𝑚𝑚𝑐𝑐 󰇗𝑐𝑐𝑝𝑝𝑐𝑐, and three temperatures, the inlet temperature, Tci, the outlet temperature, Tco, and the pinch point of the cooler, Tcp. In this case the pinch is the temperature difference between Tcp and the saturated vapour [4’]. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 46 Cycles for blends have the same points as the pure ORC, but the main difference is the fact that evaporation and condensation temperatures are not constant, as shown in Fig. 20. In these cycles, another variable appears, which is the composition of each compound in the working fluid. Nevertheless, both models follow the same equations, indeed, one can consider the pure ORC as a blend cycle with a fixed composition of unity. The use of blends as the working fluid can improve the performance of the ORC, by improving the power output and the thermal efficiency 22. For the partially evaporated cycles for pure working fluids, the superheat temperature, ΔTsh, is equal to 0. Thermodynamic properties, mainly specific entropy and enthalpy, of [3] are calculated as a weighted property function of the vapour fraction and the liquid and vapour saturated properties. Points [4] and [4s] can also be in the liquid-vapour phase, depending on the conditions of [3]. In that case, their thermodynamic properties are calculated in a similar way. The compression step (1  2) is characterised by the isentropic efficiency of the pump, ηpump, that is considered to have a fix value. For the expansion step (3  4), in the same way, a fixed value for the isentropic efficiency, ηturbine, for the expander is assumed. 2 14 3 T hi T ci T ho T co T s 1 4 3 2 2s 2' 3' 4' 4s T ci T ci T cp T ho T hi T hp a) b) T s 1 4 3 2 2s 2' 3' 4' 4s T ci T ci T cp T ho T hi T hp c) d) T s 14 3 2 2s 2' 3' 4' 4s T cp = T ci T ci T ho T hi T hp Fig. 20. a) ORC simplified flow diagram, with the main four stages. 1  2: compression. 2  3: evaporation. 3  4: expansion. 4  1: condensation. b) T-s diagram of an ORC with a pure working fluid, including the heat and cooling source. c) T-s diagram of an ORC with a blend as the working fluid. d) T-s diagram of a partially-evaporated ORC with a pure working fluid. Green: working fluid. Blue: cooling source. 1-4: ORC main states. s: isentropic process. ‘: at saturated conditions. 3. Organic Rankine Cycle analysis 47 The model employed in the current work to obtain an optimal ORC, and then the optimal operation conditions and working fluid, is described below, including the equations. This model is a pure completely evaporated ORC (Fig. 20 b)). The variable z is a normalised variable used to characterised the outlet of the evaporator and it is characterised by �𝑥𝑥vap=𝑧𝑧; 0 ≤𝑧𝑧< 1 ∆𝑇𝑇sh=(𝑧𝑧−1)· (𝑇𝑇ℎ𝑀𝑀−𝑇𝑇3′); 1 ≤𝑧𝑧≤ 2 [64] where 𝑥𝑥vap is the vapour fraction and ∆𝑇𝑇sh the superheat temperature. Compression stage 1  2 From saturated liquid at T1 (1) the working fluid is compressed, with an isentropic efficiency [67], to a pressure p2. The suffix ‘s’ corresponds to the isentropic process, so the point 2s is the point at p2 with the same entropy as the point 1. The suffix ‘in’ corresponds to the heat/power input to the system, whereas the suffix ‘out’ to the output of the system. This stage is characterised by the following equations. 𝑇𝑇1=𝑇𝑇𝑎𝑎𝑎𝑎𝑠𝑠(𝑝𝑝1) [65] 𝑠𝑠L(𝑇𝑇2𝑎𝑎,𝑝𝑝2) = 𝑠𝑠L(𝑇𝑇1,𝑝𝑝1) [66] 𝜂𝜂𝑝𝑝𝐵𝐵𝑚𝑚𝑝𝑝=ℎL(𝑇𝑇2𝑎𝑎,𝑝𝑝2) −ℎL(𝑇𝑇1,𝑝𝑝1) ℎL(𝑇𝑇2,𝑝𝑝2)−ℎL(𝑇𝑇1,𝑝𝑝1) [67] 𝑝𝑝2r=𝑝𝑝2 𝑝𝑝cr [68] 𝑤𝑤𝑀𝑀𝑚𝑚=ℎL(𝑇𝑇2,𝑝𝑝2)−ℎL(𝑇𝑇1,𝑝𝑝1) [69] The reduced pressure in the evaporator, p2r, is obtained with eq. [68], where pcr represents the critical pressure of the working fluid calculated with the Joback-Reid12 correlation [51]. The specific power of the pump, win, is calculated with eq. [69]. Evaporation stage 2  3 The working fluid is heated up until a saturated liquid is obtained (2’), then is evaporated (3’), and finally superheated to a temperature T3. The following equations characterised the evaporation stage. 𝑇𝑇2′=𝑇𝑇𝑎𝑎𝑎𝑎𝑠𝑠(𝑝𝑝2) [70] 𝑇𝑇3′=𝑇𝑇2′ [71] 𝑇𝑇3=𝑇𝑇3′+∆𝑇𝑇sh [72] ∆𝑇𝑇sh=(𝑧𝑧−1)· (𝑇𝑇ℎ𝑀𝑀−𝑇𝑇3′) [73] 𝑞𝑞in=ℎV(𝑇𝑇3,𝑝𝑝2)−ℎL(𝑇𝑇2,𝑝𝑝2) [74] Variable z is a normalised variable to represent the extent to which the vapour is superheated [64]. The specific heat input, qin, is obtained with eq. [74]. Expansion stage 3  4 From superheated vapour (3), the working fluid is expanded (4), obtaining a specific power, wout. The expansion is characterised by an isentropic efficiency, 𝜂𝜂turbine [76]. 𝑠𝑠V(𝑇𝑇4𝑎𝑎,𝑝𝑝1)=𝑠𝑠V(𝑇𝑇3,𝑝𝑝2) [75] Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 48 𝜂𝜂turbine=ℎV(𝑇𝑇4,𝑝𝑝1) −ℎV(𝑇𝑇3,𝑝𝑝2) ℎV(𝑇𝑇4𝑎𝑎,𝑝𝑝1)−ℎV(𝑇𝑇3,𝑝𝑝2) [76] 𝑤𝑤𝑚𝑚𝐵𝐵𝑠𝑠=ℎV(𝑇𝑇3,𝑝𝑝3)−ℎV(𝑇𝑇4,𝑝𝑝4) [77] Condensation stage 4  1 Finally, the fluid is cooled down to the saturated vapour (4’) and condensed back to the initial point (1). 𝑇𝑇4′=𝑇𝑇𝑎𝑎𝑎𝑎𝑠𝑠(𝑝𝑝1) [78] 𝑞𝑞𝑚𝑚𝐵𝐵𝑠𝑠=ℎV(𝑇𝑇4,𝑝𝑝1)−ℎL(𝑇𝑇1,𝑝𝑝1) [79] An “if” condition is implemented to ensure both points, (4) and (4s), are out of the two-phase region and the working fluid is then superheated vapour. Heat source The heat source is defined by the heat flow, 𝑚𝑚󰇗ℎ·𝑐𝑐𝑝𝑝,ℎ, and three temperatures, the inlet temperature, 𝑇𝑇ℎ𝑀𝑀, the outlet temperature, 𝑇𝑇ℎ𝑚𝑚 and the temperature at the pinch point, 𝑇𝑇ℎ𝑝𝑝. All the heat released by the heat source is considered to be absorbed by the working fluid. 𝑚𝑚󰇗𝐷𝐷𝑤𝑤=𝑚𝑚󰇗ℎ·𝑐𝑐𝑝𝑝,ℎ·(𝑇𝑇ℎ𝑀𝑀−𝑇𝑇ℎ𝑚𝑚) 𝑞𝑞𝑀𝑀𝑚𝑚 [80] 𝑚𝑚󰇗ℎ·𝑐𝑐𝑝𝑝,ℎ·�𝑇𝑇ℎ𝑀𝑀−𝑇𝑇ℎ𝑝𝑝�=𝑚𝑚󰇗𝐷𝐷𝑤𝑤·�ℎ𝑉𝑉(𝑇𝑇3,𝑝𝑝2)−ℎ𝐿𝐿(𝑇𝑇2′,𝑝𝑝2)� [81] 𝑃𝑃𝑃𝑃ℎ=𝑇𝑇ℎ𝑝𝑝−𝑇𝑇2′ [82] The pinch point, [82], is the point at which the temperature difference between the heat source and the working fluid is the lower one. In the described model it occurs in the saturated liquid point (2’) for the working fluid. Cooling source The cooling source is considered to be cooling water, with an inlet temperature of 𝑇𝑇𝑐𝑐𝑀𝑀. 𝑚𝑚󰇗𝐷𝐷𝑤𝑤·𝑞𝑞𝑚𝑚𝐵𝐵𝑠𝑠=𝑚𝑚󰇗𝑐𝑐·𝑐𝑐𝑝𝑝,𝑐𝑐·(𝑇𝑇co−𝑇𝑇ci) [83] 𝑚𝑚󰇗ℎ·𝑐𝑐𝑝𝑝,ℎ·�𝑇𝑇cp−𝑇𝑇ci�=𝑚𝑚󰇗𝐷𝐷𝑤𝑤·�ℎV(𝑇𝑇4′,𝑝𝑝1)−ℎL(𝑇𝑇1,𝑝𝑝1)� [84] 𝑃𝑃𝑃𝑃𝑐𝑐=𝑇𝑇4′−𝑇𝑇𝑐𝑐𝑝𝑝 [85] The cooling pinch point, [85], takes place at the saturated vapour point (4’). Finally, the performance of the cycle is measured with two variables, the thermal efficiency, 𝜂𝜂𝑠𝑠ℎ=𝑤𝑤𝑚𝑚𝐵𝐵𝑠𝑠−𝑤𝑤𝑀𝑀𝑚𝑚 𝑞𝑞𝑀𝑀𝑚𝑚 [86] and the net power output, 𝑊𝑊󰇗=𝑚𝑚󰇗𝐷𝐷𝑤𝑤· (𝑤𝑤𝑚𝑚𝐵𝐵𝑠𝑠−𝑤𝑤𝑀𝑀𝑚𝑚) [87] Once the model equations and variables are collected, the resulting model is analysed to make it feasible. The structural analysis of the cycle involves solving the set of equations described above, incorporating the fixed values (see Table 18) that characterise the model. The total number of equations describing the model is 23, involving a total of 34 variables, some of which represent parameters that must be selected in order for the model to be completely specified. 3. Organic Rankine Cycle analysis 49 The number of fixed variables to consider is seven. These values are summarised in the Table 18 and they were collected from literature to make the results comparable to other works of ORC optimisation19,20,24. Table 18. Constant values of the ORC model proposed. ηpump ηturbine 𝒎𝒎󰇗𝒉𝒉·𝒄𝒄𝒑𝒑,𝒉𝒉 /𝐤𝐤𝐖𝐖·𝐊𝐊−𝟏𝟏 𝒎𝒎󰇗𝒄𝒄 /𝐤𝐤𝐤𝐤·𝐬𝐬−𝟏𝟏 𝒄𝒄𝒑𝒑,𝒄𝒄 /𝐤𝐤𝐉𝐉·(𝐊𝐊𝐤𝐤𝐤𝐤)−𝟏𝟏 𝑻𝑻𝐜𝐜𝐜𝐜 /𝐊𝐊 𝑻𝑻𝐡𝐡𝐜𝐜/𝐊𝐊 70% 80% 4.2 5 4.2 288 [423, 523, 623] In addition to the constants of the model, some variables of the model are constrained. Table 19 collects the bounds, if considered, of the model variables. Table 19. Bounds of the ORC model proposed 𝑷𝑷𝑷𝑷𝒄𝒄𝐋𝐋 /𝐊𝐊 𝒑𝒑𝟏𝟏 𝐋𝐋 /𝐛𝐛𝐚𝐚𝐫𝐫 𝒎𝒎󰇗𝒘𝒘𝒘𝒘 𝐋𝐋 /𝐤𝐤𝐤𝐤·𝐬𝐬−𝟏𝟏 5 0.25 0 The resulting parameter from the structural analysis are collected in Table 20. Besides these variables, the working fluid selection is considered. In the group contribution approach employed in the current work, a molecule is represented by the functional groups and their multiplicity. Table 20. Parameters of the ORC model proposed Parameters Lower bound Upper bond Units 𝑇𝑇1 288 383 K 𝑝𝑝2r 0.001 0.85 - 𝑧𝑧 1 2 - 𝑃𝑃𝑃𝑃h 10 200 K As a final step of the modelling of the ORC, the described model is implemented in gPROMS29 and validated reproducing the same cycles simulated in some works of the bibliography18–21, so the results are the same. By doing this last step, the model developed can be compared with the rest cycles of the bibliography, making the results comparable. 3.1.3. ORC optimisation The performance of a Rankine cycle can be measured in several ways, nevertheless, two properties are widely used for this objective, the thermal efficiency, ηth, and the net power output, 𝑊𝑊󰇗𝑠𝑠. First of all, the thermal efficiency, described in eq. [86], has been used as the reference of the cycle performance in several studies18,54, and it has proved to be a good characteristic to optimise for low-temperature cycles, such as solar applications55. On the other hand, the net power output, described in eq. [87], is a crucial tool to measure the performance of the cycle and it has been used among several publications18–20,56,57. As shown in Fig. 30, the maximum thermal efficiency is obtained at the highest evaporator pressure, leading sometimes to negative power outputs, whereas the maximum power output is achieved at intermediate pressures, so both variables cannot be optimised with the same parameters. In this study, the net power is selected as the optimisation variable to be maximised, as it is considered to have a higher impact on the cycle performance. As listed in the previous chapter (ref Chapter 2.3), the proposed ORC model has four degrees of freedom: the temperature in the condenser, T1; the reduced pressure in the evaporator, p2r; the Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 50 conditions of the output of the evaporator, z; and the working fluid, defined by the makeup of the molecule, in terms of the nature and number of its constituent groups. Therefore, the optimisation problem is a mixed integer non-linear programming (MINLP), as it concerns continuous variables, as well as integer variables for the working fluid description. This optimisation problem can be described with equations from [88] to [93]. max 𝑥𝑥,𝑦𝑦𝑊𝑊󰇗𝑠𝑠(𝑥𝑥,𝑦𝑦) [88] Subject to: ℎ(𝑥𝑥,𝑦𝑦)= 0 [89] 𝑔𝑔(𝑥𝑥,𝑦𝑦)≤0 [90] 𝐶𝐶𝑛𝑛≤𝑑𝑑 [91] 𝑥𝑥∈𝑋𝑋 [92] 𝑦𝑦∈𝑌𝑌 [93] The net power output is set in equation [88] as the process variable to be maximised subject to the conditions set out in equations [89] to [93]. In equation [89], the set h represents the equality constraints of the problem, such as mass and energy balances or thermodynamic relations. On the other hand, the set of equations g(x,y) [90] represents the inequality constraints of the problem that sets the bounds of the process variables. Equation [91] represents the bounds and restrictions for chemical feasibility of the working fluid considered. x indicates the set of continuous variables of the model [92], like the temperatures, pressures, etc., whereas y represents the discrete variables [93], in other words, the multiplicity of each contributing group of the working fluid. To solve the resulting Mixed Integer Nonlinear Programming (MINLP) problem, an Outerapproximation (OA) framework is employed to solve the optimisation problem. This OA framework splits the optimisation problem in the primal problem and the master problem. The primal problem, is the Nonlinear subproblem (NLP). It is the first step of each iteration, in which the nonlinear variables are set to constant values, turning the problem into an NLP problem. The master problem, the Mixed Integer Problem (MILP), is defined by linearising the solution and the variables around the primal problem solution. The output of the master problem is employed in the next primal problem as the input variables. The OA framework employed in the current work is a multiple start solver. Once the problem is initialised, the primal problem (NLP) is solved by fixing the integer variables, 𝑦𝑦, to constant values. If no feasible solution is found at this step, the primal problem is restarted. Otherwise, the variables and the objective function are linearised and the master problem (MILP) is attempted. This process is repeated until the termination criteria are reached. As a multiple start algorithm, the MINLP problem is solved until the maximum number of iterations is reached, being represented by the outer loop in Fig. 21. The termination criteria mentioned above refer to those operating conditions at which the objective function calculated, the net power output, is a local optima. 3. Organic Rankine Cycle analysis 51 Primal problem NLP Master problem MILP Yes No Termination criteria Optimal solution Initialisation Optimal solution No Yes Initial guess Number of iteration +1 No Yes Max. iterations Fig. 21. Outer-approximation (OA) algorithm employed Implementation Both the ORC model and the optimisation problem are implemented in gPROMS29. Regarding the optimisation process described previously, the solution options configured in the software used are explained below. The outer method selected to solve the MINLP is the NLPMSO solver of gPROMS, a multiplestart solver. Is an hybrid stochastic/deterministic approach for global optimisation by solving the optimisation problem in many initial guesses. The inner solver selected is an outerapproximation (OA) algorithm to solve the MINLP optimisation problems by splitting the problem in two subproblems, the primal problem (NLP) and the master problem (MILP); this solver is designated by the acronym OAERAP. The NLP solver selected to solve the non-linear problem employs a sequential quadratic programming (SQP) method; this solver is designated as NLPSQP (NonLinear Programming Sequential Quadratic Programming). The solver selected for the MILP subproblem, the master problem, uses a Branch-and-bound algorithm to cope with the integer variables of the system described; this method acronym is LPSOLVE, Linear Programming Solver. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 52 3.2. Results and discussion 3.2.1. Thermophysical properties for ORC modelling In this section, the transport properties of interest are analysed, comparing the experimental data from NIST/TRC Web Thermo Tables (WTT)32 with the results from the correlations and methods explained in 3.1.1. Thermophysical properties for ORC modelling. To analyse each property, the average absolute deviation is calculated for each property, 𝐴𝐴𝐴𝐴𝐴𝐴% = 1 𝑁𝑁𝑅𝑅�|𝑅𝑅𝑀𝑀𝑀𝑀𝑥𝑥𝑝𝑝−𝑅𝑅𝑀𝑀𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐| 𝑅𝑅𝑀𝑀𝑀𝑀𝑥𝑥𝑝𝑝 𝑁𝑁𝑅𝑅 𝑀𝑀=1 ·100 [94] where NR is the number of data points of the property R, 𝑅𝑅𝑀𝑀𝑀𝑀𝑥𝑥𝑝𝑝 the experimental value of the property and 𝑅𝑅𝑀𝑀𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐 the calculated one, at the 𝑖𝑖𝑠𝑠ℎ conditions. These properties are analysed for the compounds selected in Table 4, so trustable experimental data are available. Normal boiling point To estimate the normal boiling point, TB, two methods are proposed, the Joback-Reid correlation12 and the SAFT-γ Mie approach3. Both methods are analysed in order to select the more accurate one. The results are plotted in Fig. 22, here the results for the same compounds calculated are plotted against the experimental data. The dashed lines represent a 10 K deviation between calculated and experimental data. Normal boiling point calculated with SAFT, the yellow diamonds, for all the compounds studied fall within the ±10 K deviation area. The Joback-Reid correlation, represented with the green circles, underestimates in some cases the normal boiling point, as the points are beneath the zero-deviation line. Fig. 22. Parity plot of calculated and experimental normal boiling point temperatures, TB, highlighting TB calculated using SAFT-γ Mie (orange open diamonds) and the Joback-Reid correlation (green open circles). Central line: parity line, calculated point equals experimental one. Dashed lines: ±10K deviation The absolute deviation, AD, calculated for the SAFT-γ Mie is 0.97 K and 0.37% the average absolute deviation, AAD. For the Joback-Reid method, the AD is 14.37 K and the AAD obtained is 6.36%. Based on these results, the SAFTγ Mie method is selected as the most suitable one to estimate the normal boiling point for the molecules studied in the current work. 3. Organic Rankine Cycle analysis 53 Critical point For the critical point temperature, pressure and volume, as well as in the boiling point, the SAFT method and the Joback-Reid correlation are proposed. Both methods are analysed for each property. Critical temperature, Tcr Critical temperatures calculated with both methods, 𝑇𝑇𝑐𝑐𝑟𝑟 𝑐𝑐𝑎𝑎𝑘𝑘𝑐𝑐, are plotted against the experimental critical temperatures, 𝑇𝑇𝑐𝑐𝑟𝑟 𝑀𝑀𝑥𝑥𝑝𝑝 in Fig. 23. The dashed lines represent the ±10 K deviation area. The Joback-Reid method underpredicts the critical temperature in most of the compounds studied. For its part, results obtained with the SAFT method proposed tend to overpredict the critical temperature with a lower deviation than the other method. Fig. 23. Parity plot of calculated and experimental critical temperatures, Tcr, highlighting Tcr calculated using SAFT-γ Mie (orange open diamonds) and the Joback-Reid correlation (green open circles). Central line: parity line, calculated point equals experimental one. Dashed lines: ±10K deviation The AD for the SAFT-γ Mie is 9.15 K and 2.30% the average absolute deviation, AAD. For the Joback-Reid method, the AD is 20.99K and the AAD obtained is 5.08%. In accordance with these results, the SAFT method is selected to estimate the critical temperature for the compounds. Critical pressure, pcr The experimental and calculated critical pressures for the compounds of interest are plotted in Fig. 24, here the Joback-Reid correlation proves a better performance. The dashed lines define the ±0.5MPa deviation area. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 60 Fig. 30. Left: Net power output as a function of the reduce pressure in the evaporator, p2r. Right: thermal efficiency as a function of the reduced pressure in the evaporator, p2r. Lines: current work model. Dashed lines: reference model As reflected in the figure, both models, the continuous and the dashed lines have the same trend and they are virtually indistinguishable from one another, concluding that the model implemented perform in a similar way, as the operating conditions and the thermodynamic approach employed are the same. In the second paper, White et al. (2018)19 considered either, partially and completely evaporated cycles. The variable z [64] represents the vapour fraction when it takes values from 0 to 1, and the extent of the superheat when takes values from 1 to 2. The power output is represented, for various values of the pressure in the evaporator, as a function of the variable z. Fig. 31. Power output as a function of variable z for n-pentane (upper left), n-hexane (upper right) and isopentane (lower). Lines: current work model. Dashed lines: reference model The dashed curves represent the cycle modelled in the reference, whereas the continues curves are from the current work. It can be concluded that the output of both models is essentially the same. There are no bigger deviations at either higher or lower values of the pressure in the evaporator. 3. Organic Rankine Cycle analysis 61 Finally, it can be concluded that the ORC model implemented in the current work is able to represent other previously implemented cycle, obtaining the same values for the same operating conditions. ORC optimiser validation Once the ORC model implemented is validated, the NLP solver selection is validated, to ensure a correct behaviour of the solvers seleceted. To accomplish this step, results from two papers are selected. Schilling et al. (2017)23 presented the results of a CAMD optimisation of the ORC, optimising simultaneously the working fluid and the process variables. The equation of state employed in the reference is the PC-SAFT approach. The outperforming working fluids obtained were selected to be optimised with the code used in the current work. In Table 22 the net power output obtained in the reference for each molecule is collected together with the optimised net power output obtained as a result of an NLP optimisation of the operation conditions. Table 22. Net power output comparison between the Schilling et al. (2017) paper and the current work Compound Schilling et al.(2017)23 Current work AAD W opt /MW W opt /MW propane 1.59 1.66206 4.53% propene 1.57 1.68429 7.28% isobutane 1.56 1.59364 2.16% isobutene 1.55 1.54633 0.24% n-butane 1.55 1.56677 1.08% 1-butene 1.54 1.56001 1.30% 2-butene 1.53 1.53726 0.47% neopentane 1.55 1.58206 2.07% dimethyl ether 1.53 1.58448 3.56% ethyl methyl ether 1.53 1.53824 0.54% According to the results displayed in the table, the NLP solvers of both works, give a similar value of the optimal net power output for the given conditions. The differences between both works can be the fact that a different EoS is being used in, so the VLE and thermodynamic properties may differ, whereby small differences in AAD are inevitable. In the previous work it was also reported the operation conditions of the optimal net power output of each working fluid. In Fig. 32 three of the operating conditions are compared for the optimal point. These conditions are the condenser pressure, p1, (upper left figure), the evaporator pressure, p2, (upper right) and the mass flow of working fluid, 𝑚𝑚󰇗𝐷𝐷𝑤𝑤(lower). Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 62 Fig. 32. Comparison of the optimised process variables for each compound between Schilling et al. (2017) reference and the current work. Upper left: condenser pressure. Upper right: evaporator pressure. Lower: working fluid mass flow. As reflected in the three plots, not only the power output value is of the same order, but also both operating pressures, condenser and evaporator pressures, are of the same order. The mass flow rates of working fluid obtained for the optimal conditions take very similar values. The second paper used is Bowskill et al. (2020)18. The operating conditions reflected in the papers were used to estimate the optimal net power output and the parameters to obtain that optimal point. The EoS used is SAFT-γ Mie, the same approach used in the current work. Both optimal points are collected, for two different conditions, in Table 23. 3. Organic Rankine Cycle analysis 63 Table 23. Net power output comparison between the Bowskill et al. (2020) paper and the current work Compound Case I Case II Bowskill et al. (2020)18 Current work AAD % Bowskill et al. (2020)18 Current work p2r AAD% W opt /MW W opt /MW W opt /MW W opt /MW propane 1.662 1.66101 0.06 6.288 7.43469 0.8 18.24 propene 1.661 1.66272 0.10 5.779 6.76287 0.8 17.02 n-butane 1.567 1.56646 0.03 8.347 8.35181 0.8 0.06 1-butene 1.56 1.55979 0.01 8.185 8.50693 0.8 3.93 2-butene 1.537 1.53712 0.01 7.53 8.71895 0.8 15.79 butadiene 1.536 1.53584 0.01 6.496 8.83064 0.8 35.94 1-pentene 1.191 1.19092 0.01 5.828 5.82522 0.33 0.05 1,4-pentadiene 1.201 1.20039 0.05 5.771 5.76878 0.31 0.04 ethyl methyl ether 1.538 1.53807 0.00 7.45 7.23113 0.8 2.94 diethyl ether 1.164 1.1633 0.06 5.752 5.7497 0.32 0.04 For the case I, the results only differ a little, as reflected in the small AADs calculated. In the case of the case II, bigger deviations are obtained for some compounds. These deviations are found when the upper bound set for the reduce pressure in the evaporator is reached. The reduced pressure, 𝑝𝑝2𝑟𝑟=𝑝𝑝2/𝑝𝑝𝑐𝑐𝑟𝑟, depends on the critical pressure. In the current work, the critical pressure is estimated with the Joback-Reid correlation, as explained in the chapter 3.2.1. Thermophysical properties for ORC modelling. In the reference, the critical pressure is estimated with the SAFT-γ Mie approach. A big difference when estimating the critical pressure has been reported previously in the current work. For the ORC model and optimisation, the important parameter is the pressure in the evaporator, the reduced pressure is used only to bound the pressures of the model, to operate in the subcritical region. Despite this deviation in some molecules in case II, the NLP solver used in the current work is valid to obtain the optimal operating conditions of ORCs. At this point, the framework for the computer-aided molecular and process design has been analysed and implemented in gPROMS29. The optimal working fluid and operating conditions of ORCs determination by running the selected algorithm remains as future work. Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 64 4. Conclusions Development of new functional groups SAFT-γ Mie approach is used, in the current work, to estimate the pure and binary VLE, as some thermodynamic properties like enthalpies and entropies, heat capacities, coefficients of expansion and so on. These properties are useful for the thermodynamic modelling of cycles, in particular organic Rankine cycles (ORCs). SAFT-γ Mie belongs to the SAFT-type equations of state family and it is also a group-contribution method, making it suitable to evaluate the thermodynamic properties of the working fluids that can be formed from not only the available groups, but also the ones developed in the current work. The first objective is the extend the range of application of SAFT-γ Mie approach. Previous works developed the framework to use this approach for alkanes, alkenes, branched alkanes, ethers, perfluoroalkanes and so on. Hydrofluorocarbons (HFCs) are alkanes in which some of the hydrogen atoms have been replaced by fluorine atoms. Then, the functional groups present in these compounds are alkyl, perfluoroalkyl and hydrofluoroallkyl groups. Hydrofluoroalkyl groups had not been developed within the SAFT-γ Mie in previous works, whereas the other groups are available. HFCs are fluids with several applications, among which their use as working fluid in cycles is outlined. These compounds have replaced Chlorofluorocarbons (CFCs) in refrigeration cycles and they show a good performance as working fluid for power cycles, like ORCs. To enable the modelling of cycles that use HFCs as the working fluid, new groups are developed within the SAFT-γ Mie framework, allowing a complete thermodynamic description of this family of compounds. To obtain the parameters needed to employ new groups within the SAFT-γ Mie approach, some selected properties, calculated with the model, are compared to experimental data, and then the parameters of the groups are optimised to obtain the best fitting. In this step, vapour pressure and saturated liquid density are commonly used to train the parameters. After the optimisation stage, the results obtained are validated by predicting other properties and comparing this prediction with experimental data available. The parameters obtained in this work, corresponding to the hydrofluoroalkyl groups, are optimised to capture the VLE behaviour of some HFCs. This is done by fitting the calculated VLE, in particular vapour pressures and saturated liquid densities, to the experimental data from bibliography. The fitting is analysed, obtaining a good accordance between the VLE calculated with the model and the experimental VLE. To ensure an adequate functioning of the SAFT-γ Mie approach with the optimised parameters, some properties that are not used in the training stage are predicted and compared with the experimental data. These properties are the binary VLE and thermodynamic properties, such as specific enthalpies and entropies, isobaric heat capacities and other derivative thermodynamic properties. The model is validated for the thermodynamic properties, obtaining a good agreement with experimental data. In the case of binary VLE, the results are adequate for blends of similar HFCs, but when predicting the VLE of unsimilar HFCs it should be noted the failure of the proposed parameter to capture the VLE. To solve this issue, secondary-order group parameters should be used. To conclude, the SAFT-γ Mie approach is a useful tool to model the thermodynamic behaviour of fluids. To use this approach to model the desired fluids, group parameters are needed. These parameters allow the use of the SAFT-γ Mie approach to reproduce the VLE behaviour of HFCs, 4. Conclusions 65 as well as predict fundamental properties, like calorific and thermodynamic ones. Regarding the binary VLE prediction, it should be noted that the developed parameters are able to predict the binary interaction in some cases, in those situations comprising unlike molecules, the proposed parameters are unable to predict the VLE. The parameters developed, along with the SAFT-γ Mie equation of state, are useful for thermodynamic modelling of cycles, like ORCs. ORC analysis Organic Rankine cycles (ORCs) are power cycles, as steam Rankine cycles or Kalina cycles, employed to obtain a power output using a heat source. In particular, ORCs employ organic compounds as working fluid to improve the thermodynamic performance of the cycle at certain conditions. They increase the power output, compared with steam Rankine cycles, when the heat source is available at a low temperature, making them more suitable to recover waste heat. ORCs are composed of four stages: compression, evaporation, expansion and condensation. The repetition of these four stages leads to obtaining a net power output. It can be distinguished between partially evaporated cycles, in which the output of the evaporator is a mixture of liquid and vapour, or completely evaporated cycles, where the output is a superheated vapour. In relation to the critical point, it can be distinguished between subcritical ORCs and supercritical ORCs, when the pressure in the evaporator takes values over the critical pressure. Regarding the working fluid, mixtures can be used, meaning a non-isothermal evaporation and condensation stages. In the current work the simplest cycle is considered, with a pure working fluid, subcritical and a completely evaporated cycle. However, the thermodynamic efficiency, together with the cost estimation, of other cycle configurations remains as a future objective. The selected configuration of the ORC is modelled to analyse the performance of the HFCs as working fluid, as well as the operating conditions of the cycle. The SAFT-γ Mie approach is employed to model the thermodynamic properties of the working fluid of the cycle, through the four stages described above. The model is validated by comparing the output of the model with results from previous works, ensuring the same operating conditions lead to the same power output and process variables. In addition to the thermodynamic modelling, other thermophysical properties are needed, to completely defined the ORC model. These properties are the critical point, the normal boiling point, the surface tension and transport properties. They are used to bound the cycle, ensuring and subcritical ORC and to implement the sizing and cost correlations. The final objective would be to optimise the cycle, obtaining the optimal working fluid and the optimal conditions. To reach this objective, a computer-aided molecular and process design (CAMPD) framework is established, in which the net power output is set to be maximised modifying the operating conditions of the cycle and the working fluid employed. An outerapproximation (OA) algorithm is proposed to solve the resulting mixed integer nonlinear programming (MINLP) optimisation problem. The solution of this established optimisation problem remains as future work to be carried out. The simultaneous optimisation of the working fluid and cycle parameters is a crucial step to obtain an economically viable ORC. To conclude, the model of ORC can be used to optimise the thermodynamic performance of the cycle, reaching the optimal conditions of the cycle. To provide a more detailed description of the cycle, including the equipment sizing and the cost estimation, transport properties models are analysed. These models are group contribution based, so the same working fluid description for the thermodynamic model (SAFT-γ Mie) and the transport properties model. The use of group contribution methods allows a higher number of molecules to be screened during the working Development of a SAFT-γ Mie GC approach to modelling HFCs and its application in the CAMPD of ORCs 66 fluid selection. Finally, an optimisation of the ORC is proposed as future work. This optimisation tries to find the optimal working fluid for the ORC, formed with the functional groups developed in the first section and with previously estimated functional groups, and the optimal operating conditions for that working fluid, stablishing the computer-aided molecular and process design (CAMPD) framework. Bibliography 67 Bibliography 1. 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