From molecules to dollars: integrating molecular design into thermoeconomic process design using consistent thermodynamic modeling†
Received
7th April 2017
, Accepted 9th June 2017
First published on 9th June 2017
The right molecules are often the key to overall process performance and economics of many energy and chemical conversion processes, such as, e.g., solvents for CO_{2} capture or working fluids for organic Rankine cycles. However, the process settings also impact the choices at the molecular level. Thus, ultimately, the process and the molecules have to be optimized simultaneously to obtain a thermoeconomically optimal process. For a detailed design of the process and also the equipment, a thermodynamic model is required for both equilibrium and transport properties. We present an approach for the integrated thermoeconomic design of the process, equipment and molecule on the basis of a comprehensive, thermodynamically consistent model of the molecule. For this purpose, we developed models for transport properties based on entropyscaling of the perturbedchain statistical associating fluid theory (PCSAFT) equation of state. Thereby, a single model predicts both equilibrium and transport properties in our optimizationbased approach for the integrated design of the process, equipment and molecule, the socalled 1stage CoMT–CAMD approach. The predicted transport properties allow for the design and sizing of unit operations as degrees of freedom during the optimization. Computeraided molecular design allows the design of novel molecules tailored to the specific process while considering safety and environmental issues. The presented approach is exemplified for the design of an organic Rankine cycle showing the merits of detailed sizing of heat exchangers with different heat transfer types and the rotating equipment as part of the optimization. Singleobjective optimization is used to obtain a ranking of potential working fluids. The detailed tradeoff between the total capital investment and the net power output of the ORC is studied using multiobjective optimization. Thus, the 1stage CoMT–CAMD approach allows for efficient and holistic designs linking the molecular scale to economics.
Design, System, Application
The design of molecules is usually not a goal by itself but should enable optimal performance at the process level. Here, we aim at the design of economically optimal molecules. Thus, we have to capture the impact of the molecular structure on process economics. To capture this impact, we integrate molecular design directly into thermoeconomic design optimization. Thermoeconomic design considers both the process and the employed equipment. The proposed integrated design allows us to quantify the trade offs between molecular design, process settings and equipment sizing. We argue that the key to an efficient integrated design method is a thermodynamically consistent model for both equilibrium and transport properties. Here, we employ the physically based PCSAFT equation of state. PCSAFT has been extended beyond equilibrium properties to predict also viscosities and heat conductivities of fluids. We demonstrate the integrated thermoeconomic design for an organic Rankine cycle to convert lowtemperature heat into mechanical work. The design considers detailed correlations for sizing of the heat exchangers and the rotating equipment. The presented approach should be applicable to general problems for the integrated thermoeconomic design of fluid molecules in energy and chemical engineering ranging, e.g., from refrigerants in heat pumps to solvents for CO_{2} absorption.

1. Introduction
The key to energy and chemical conversion technologies is often the right choice of the molecules used as processing materials,^{1}e.g., solvents^{2–6} and adsorption materials^{7,8} for CO_{2} capture or CO_{2}/CH_{4} separation^{9} as well as methane storage,^{10} refrigerants,^{11,12} working fluids,^{13–17} conductor materials^{18} or polymers.^{19,20} Only the right molecule enables optimal and sustainable processes. However, choosing the right molecule is inherently complex as the choice of molecules depends directly on the design of the process itself. Thus, the process and molecule have to be designed simultaneously to obtain the optimal combination. In addition, the design has to reflect environmental restrictions and safety issues. However, the practically unlimited number of potential molecules renders the direct solution of the integrated design impossible in practice.^{2}
Systematic design approaches have therefore been proposed to select or even design the right molecule. These design approaches crucially depend on the employed selection criterion which should reflect the actual process performance. A general classification of the design approaches is schematically illustrated in Fig. 1 based on the employed selection criterion: heuristic, thermodynamic or thermoeconomic criteria. The performance prediction is the basis to identify the optimal molecule. Thus, the quality of the performance prediction decides how reliable an approach is in finding the optimal molecules, i.e., hitting the target in Fig. 1.

 Fig. 1 Schematic illustration of process and molecule design using heuristic, thermodynamic or thermoeconomic prediction methods as selection criteria (from left to right). The molecules can be chosen from a database or individually designed.  
Heuristic performance indicators allow separating the molecular design from the process design, since no information about the process is needed. The heuristics employed for molecule design merely rely on the physical properties of the molecule, e.g., boiling point or selectivity. The heuristic assessment of process performance requires experience which might be unreliable. An important heuristic method is the screening of existing molecule databases. A highthroughput screening approach is presented by Simon et al.^{10} Here, Monte Carlo simulations are used to identify adsorption materials for methane storage based on heuristic process indicators for storage capacity. Schwöbel et al.^{21} present a highthroughput screening approach for working fluid selection of ORCs based on a physical domain reduction and subsequent process simulations using COSMORS. Joos et al.^{7} present a multiobjective screening approach for selection of absorption materials for CO_{2} capture. Here, a postPareto search algorithm is used to select absorption materials considering heuristic performance indicators for selectivity and uptake. However, screening approaches are naturally limited to the considered databases. To design novel and promising molecules, systematic computeraided molecular design (CAMD) approaches have been developed. Therein, group contribution (GC) approaches have been combined with heuristic performance indicators for a variety of problems, e.g., for refrigerants,^{12,22} solvents for metal degreasing and crystallization^{12} or for extraction–distillation processes.^{23} To circumvent the use of GC approaches that require measurement data for parameterization and group additivity, Scheffczyk et al.^{24} recently proposed an optimizationbased CAMD approach for solvents of extraction processes using quantum mechanicsbased predictions by COSMORS and heuristic process indicators. While heuristics are efficient, the process module information is missing leading to suboptimal results. For the case of adsorption materials, First et al.^{9} therefore only preselect a set of potential adsorption materials for CO_{2}/CH_{4} separation using a heuristic screening approach. The preselected adsorption materials are then assessed in a subsequent thermoeconomic process optimization to verify their performance. However, Kossack et al.^{25} show that different objective functions for preselection and process optimization can still lead to overall suboptimal solutions. These authors preselect an entrainer for extractive distillation based on the separation selectivity. However, the selected entrainer is undesirable in terms of economics.
To identify suitable selection criteria, targeting approaches have been developed aiming at the identification of favorable molecular properties, socalled targets. Targeting approaches commonly solve the design problem in two stages: in a first stage, favorable target properties are identified. Based on these targets, real molecules are designed in a second stage. Recent reviews recommend using process models for the identification of favorable target properties of the molecules to obtain an overall optimal solution.^{2,13,14,26} For this purpose, the molecular design has to be integrated into process design. The integration requires a thermodynamic model to capture all interactions between molecules and processes (see Fig. 1). If an equilibrium model is employed, a thermodynamic objective function can be used to assess the process performance (e.g., the thermal efficiency). However, by integrating CAMD into process design, discrete degrees of freedom are added to the process optimization. Therefore, the integrated design results in largescale, challenging mixed integer nonlinear programming (MINLP) problems.^{27} A thermodynamic approach for the integrated design of Organic Rankine Cycles (ORC) and working fluid mixtures based on CAMD has been proposed by Papadopoulos et al.^{28} A tailored solution algorithm relaxes the feasibility constraints of one mixture component to design an optimal first component. Then, the identified optimal component is fixed to identify the corresponding optimal second component. The resulting optimal mixtures are shown to outperform proposed mixtures for ORCs from the literature.^{29} Targeting approaches have also been developed, e.g., for product design for metal degreasing based on property clustering techniques,^{30} solvent design based on groupcontribution prediction^{31} or refrigerant design based on a cubic equation of state (EoS).^{11}
In our previous work, we presented a targeting approach for integrated design of the process and solvent, the socalled continuousmolecular targeting (CoMT).^{3} Within this approach, the solvent is modelled using the perturbedchain statistical associating fluid theory (PCSAFT) equation of state.^{32} In a first stage, molecular targets are identified by relaxation of the pure component parameters representing the solvent in PCSAFT. This relaxation transforms the integrated design problem into a nonlinear programming (NLP) problem. Due to this relaxation, the optimization results in a hypothetical optimal solvent, the target. In a second stage, the socalled structure mapping, real solvents are identified from a database using a Taylor approximation of the objective function to estimate the objective function values. The CoMT framework for database search has been applied to the integrated design of solvents for CO_{2} capture and storage^{3,5} and working fluids for ORCs.^{15} Lampe et al.^{33} extended the structure mapping of the CoMT framework by a CAMD formulation using a group contribution approach of PCSAFT.^{34} Thereby, a mixedinteger quadratic programming (MIQP) problem is solved in the structuremapping stage with the Taylor approximation as the objective function and the molecular structure of the working fluid as the degree of freedom. The resulting CoMT–CAMD approach allows designing novel molecules. Recently, we directly linked the CoMT framework^{3} to a CAMD formulation.^{35} This optimizationbased approach, socalled 1stage CoMT–CAMD, solves the resulting MINLP efficiently in one stage. The MINLP is solved using an outerapproximation algorithm extended by a relaxation strategy. Since an equilibrium thermodynamic model is employed, only thermodynamic objective functions, such as the power output of an ORC, can be considered. The 1stage CoMT–CAMD approach has been exemplified by the integrated design of the process and pure working fluid of ORCs. Screening approaches can also be applied with a thermodynamic objective function if the computational effort can be reduced. Scheffczyk et al.^{36} present a highthroughput screening approach for more than 4600 solvents for extraction–distillation using a thermodynamic objective function and COSMORS. Pinchbased shortcut models of the process^{37,38} are used to reduce the high computational effort of such a screening approach.
However, the thermodynamically optimal molecule can differ from the thermoeconomically optimal molecule.^{39} Thus, systematic approaches for thermoeconomic optimization have been developed to obtain an overall optimal solution. A thermoeconomic objective function requires the integration of equipment design into the process and molecular design to quantify the investment cost of the equipment. To size equipment, a model for the transport properties of the molecule is needed to capture all transportrelated tradeoffs (see Fig. 1).
The group around Adjiman, Galindo and Jackson pioneered in this area by presenting a thermoeconomic approach for the integrated process and solvent design for physical CO_{2} absorption.^{40,41} Here, the search space is limited to the design of linear alkanes, which are modelled by the physically based statistical associating fluid theory for potentials of variable attractive range (SAFTVR).^{42} The economics are predicted from the equipment sizes, which are estimated based on heuristic correlations for equipment sizing depending on operation conditions. Viscosity is predicted based on an empirical correlation for nalkanes. They extended their approach in the work of Burger et al.^{4} using a hierarchical approach for the integrated thermoeconomic design of the process and solvent for physical CO_{2} absorption. Within this approach, the equilibrium properties of the solvent are modelled using the SAFTγ Mie equation of state.^{43,44} The design space of the solvent is extended beyond linear alkanes by groups for linear alkyl ethers. Recently, the same group proposed an optimizationbased approach for the integrated design of absorption processes extending an outerapproximation formulation with a physically driven domain reduction.^{45} In this approach, infeasible process and molecular design regions are removed to ensure numerical robustness of the MINLP optimization. Zhou et al.^{46} present an integrated thermoeconomic approach for solvent design of a Diels–Alder reaction using solvent descriptors determined from quantum chemical density functional theory calculations. Here, a reaction kinetics model is combined with a GC approach. Since no model for transport properties is considered, the equipment sizing for cost estimation is performed based on heuristic guidelines. Recently, the same authors^{47} proposed a hybrid stochastic–deterministic approach for the integrated thermoeconomic design of the process and solvent exemplified for absorption–desorption. The economics are estimated using heuristic sizing models for absorption–desorption equipment.
Thus, the lack of a consistent model for transport properties has enforced equipment sizing based on empirical correlations and heuristics. Thereby, the detailed tradeoffs between processes, equipment and molecules are neglected. The prediction of transport properties is less mature than that of equilibrium properties.^{48} However, recent progress enables predicting transport properties based on the PCSAFT equation of state and entropyscaling.^{49–51} Thereby, PCSAFT yields a consistent model for both equilibrium and transport properties with a small set of physically based parameters.
In this work, the prediction of transport properties based on PCSAFT and entropyscaling is directly integrated into the 1stage CoMT–CAMD design approach. Thereby, we enable the detailed sizing of the equipment within the design of the process and molecule yielding an overall optimal thermoeconomic process. Herein, the PCSAFT equation of state provides a thermodynamically consistent model for both equilibrium and transport properties. Thus, the design of the molecule, process and equipment is linked in a thermodynamically consistent way capturing all crucial interactions. A CAMD formulation allows designing molecules as degrees of freedom in the optimization. To ensure the design of a safe and environmentally benign process, additional safety and environmental limitations are considered. The 1stage CoMT–CAMD approach enables identifying the thermoeconomically optimal molecule and the corresponding optimal process in one single MINLP optimization.
The paper is structured as follows: in section 2, the framework of the 1stage CoMT–CAMD approach for integrated thermoeconomic design is presented. In section 3, the model of the organic Rankine cycle is presented, which is considered as a case study in this work. Designing an ORC process allows us to show detailed modelling of the process equipment during the integrated design of the process and molecule. Here, an axial turbine and heat exchangers for singlephase, evaporation and condensation heat transfer are modelled and sized during the optimization. The results are presented and validated in section 4. Conclusions are drawn in section 5.
2. Framework for thermoeconomic optimization in molecular design
The proposed method aims at modelling the economic consequences of changing the molecular structure for energy and chemical conversion technologies. The model for thermoeconomic optimization in molecular design can be decomposed into 6 levels (see Fig. 2): 1 main level, 3 design levels and 2 connector levels. The main level is the economic level (level 1). Here, the economics of the process are represented serving as assessment criteria. To estimate the total capital investment, the equipment has to be designed using detailed models for equipment sizing (level 2 – design). The detailed sizing of the equipment requires a model for the transport properties of the molecule (level 3 – connector). The equipment is sized according to the specifications from process design which requires a detailed model of the process (level 4 – design). To determine all states of the process, a model for the equilibrium properties of the molecule is required (level 5 – connector). In this work, both equilibrium and transport properties are modelled using the physically based PCSAFT equation of state yielding a consistent model for both connector levels. To integrate the molecular design as a degree of freedom into the process design, a computeraided molecular design formulation is used allowing the design of novel, promising molecules (level 6 – design). To capture all interactions between the economics, process, equipment, and molecule, simultaneous design of all levels is performed.

 Fig. 2 Schematic illustration (left) and the corresponding MINLP problem formulation (right) of the presented 1stage CoMT–CAMD approach for integrated thermoeconomic design of the process, equipment and molecule.  
Gani^{27} proposed a generic MINLP problem formulation for the integrated design of the process and molecule. The MINLP is originally formulated as singleobjective optimization. In this work, we consider the generalized multiobjective optimization problem, which is given in problem (1) (Fig. 2). The model optimizes a set of thermoeconomic objective functions (f_{1}, f_{2},…, f_{k})^{T}. These objectives depend on process and equipment variables x (e.g., pressure levels), equilibrium properties Θ (e.g., enthalpies) and transport properties κ (e.g., viscosities). The process and equipment models encompass equality constraints g_{1}, p_{1} as well as inequality constraints g_{2}, p_{2}, respectively (see section 2.1). The equilibrium properties Θ and transport properties κ of the molecule are both calculated using the PCSAFT equation of state.^{32} In PCSAFT, a molecule is represented by a set of pure component parameters z (see section 2.2). A CAMD formulation is used to integrate the molecular design into the process design. Here, each molecule is characterized by the functional groups constituting its molecular structure. The integer vector y^{s} contains the number of occurrences of a certain group of the molecule. To link PCSAFT and the CAMD formulation, a group contribution (GC) approach is employed for calculating the PCSAFT pure component parameters z from the molecular structure y^{s} of the molecule. Structural feasibility of the molecular structure is ensured by additional equality F_{1}·y^{s} = 0 and inequality F_{2}·y^{s} ≤ 0 constraints.^{52,53} These constraints ensure a proper connectivity of the designed molecular structure, e.g., by the octet rule or additional bonding rules. Additionally, the molecular design is constrained by limitations on nonconventional properties nc (e.g., environmental or safety constraints), as presented in section 2.3.
The degrees of freedom of the full MINLP problem are the process and equipment variables x as well as the molecular structure of the molecule y^{s}. The integrated thermoeconomic design of the process and molecule is achieved by solving the MINLP given in problem (1). The optimization results in the optimal molecular structure and the corresponding optimal process and equipment. Integercuts^{54} can be used to obtain a ranking of optimal molecules allowing the consideration of further aspects, which are not covered by the optimization model. For a detailed description of the optimization strategy, see section 2.4.
2.1. Process and equipment models
The process and equipment models depend on three kinds of variables: process and equipment variables x (e.g., mass flow rates or pressure levels), equilibrium properties Θ (e.g., enthalpy or entropy) and transport properties κ (e.g., viscosity or thermal conductivity). Knowledge of the process allows determining the operating cost, while knowledge of the equipment determines the investment cost. Thus, any objective function can be considered for a detailed economic assessment of the process (e.g., specific investment cost or net present value). The general problem formulation also allows using a thermodynamic objective function (e.g., net power output for ORCs or energy demands for extraction–distillation). In this case, no detailed equipment sizing and, thus, no model for transport properties are required.
The equality constraints are composed of process constraints p_{1}(x, Θ) = 0 (e.g., mass balances or energy balances) and equipment constraints g_{1}(x, Θ, κ) = 0 (e.g., heat transfer correlations). The inequality constraints consist of process constraints p_{2}(x, Θ) ≤ 0 (e.g., pressure limits) and equipment constraints g_{2}(x, Θ, κ) ≤ 0 (e.g., velocity limits in the heat exchangers). In general, structural degrees of freedom can additionally be regarded in the optimization considering different flowsheets of the process. In this work, only continuous process variables are considered.
2.2. Equilibrium and transport properties based on PCSAFT and CAMD
To calculate the equilibrium and transport properties of the molecule, a thermodynamically consistent model is used: the perturbedchain statistical associating fluid theory (PCSAFT)^{32} including contributions for polar components.^{55,56} In PCSAFT, each molecule is described by perturbedchains of spherical segments. A set of typically 3 to 7 pure component parameters is required to represent a molecule. In this work, associative and quadrupolar molecules are excluded from the design space so that 4 pure component parameters are sufficient to characterize a molecule; two of them describe the geometry of the chains: the segment number m and the segment diameter σ. The van der Waals attraction is described by the segment dispersion energy ε/k and the dipole interaction by the dipole moment μ.
To link PCSAFT to the CAMD formulation, the homosegmented GC approach of PCSAFT^{34} is used. With this GC approach, the pure component parameters can be calculated from the molecular structure y^{s} of the molecule. The group contributions have been adjusted to measured vapor–liquid equilibria and liquid densities from a database.^{34} To calculate the pure component parameters from the molecular structure, the mixing rules of Vijande et al.^{57} are applied:

 (2) 
Here, n_{i} denotes the number of functional groups of type i of the molecular structure represented by the vector y^{s} = (n_{1}, n_{2},…, n_{l})^{T}. The contribution of group i is contained in the vector ẑ_{i} = (m_{i}, σ_{i}, (ε/k)_{i}, μ_{i})^{T}. Since a combination of the pure component parameters is used in the GC approach, this parameter combination is also used to define the vector z = (m, mσ^{3}, mε/k, μ)^{T} in problem (1).
PCSAFT calculates both equilibrium and transport properties in every state of the process based on a consistent thermodynamic picture. The model is based on the residual Helmholtz energy. Thus, a reference property is needed to calculate absolute equilibrium properties Θ. Here, we use the heat capacity of the ideal gas c^{ig}_{p} (T), which is calculated from the molecular structure using Joback's firstorder GC approach.^{58} For a given pressure p and temperature T, absolute caloric properties such as the absolute enthalpy h can be calculated as the sum of a residual contribution (res) and an ideal contribution (id), which is based on an arbitrary reference temperature T_{0}:

 (3) 
Besides the heat capacity of the ideal gas c^{ig}_{p}, the molar mass M is also calculated from the molecular structure.
The calculation of transport properties κ from PCSAFT is based on Rosenfeld's entropyscaling.^{59,60} Therein, the transport properties of a molecule are found to be a monovariable function of the residual entropy s^{res}. However, the function varies for different molecules. A GC approach for transport properties based on entropyscaling and PCSAFT is presented for viscosities η by LötgeringLin and Gross^{49} and for thermal conductivities λ by Hopp and Gross.^{50,51} In these approaches, the transport properties κ = (η, λ)^{T} are defined as a product of a reduced transport property κ* = (η*, λ*)^{T} and a reference transport property κ_{ref} = (η_{ref}, λ_{ref})^{T}:

 (4) 
The reduced transport property κ* is described through a thirdorder polynomial depending on the residual entropy s^{res}(T, p, z) calculated from PCSAFT:

 (5) 
with reduced residual entropy

 (6) 
where
k_{B} denotes the Boltzmann constant. The scaling factors of the polynomial (
A_{κ} to
D_{κ}) depend on the molecule and vary for the different transport properties. Therefore, the scaling factors are calculated from the molecular structure of the molecule using a firstorder GC approach:

 (7) 
The vector _{i} = (A_{κ,i}, B_{κ,i}, C_{κ,i})^{T} contains the group contributions of group i. The scaling factor is a model constant and set to for viscosity and for thermal conductivity.
LötgeringLin and Gross^{49} use the Chapman–Enskog viscosity η_{CE} as reference viscosity η_{ref}, which is related to the pure component parameters of PCSAFT:

 (8) 
N
_{A} denotes the Avogadro constant and Ω^{(2,2)}* indicates the reduced collision integral, which is calculated using an empirical approximation.^{61}
As reference thermal conductivity λ_{ref}, the Chapman–Enskog thermal conductivity λ_{CE} is insufficient, since it neglects vibrational degrees of freedom in regions of low densities. Thus, an additional contribution λ^{vib} is proposed by Hopp and Gross^{50} to cover the behavior of the gas phase. Thereby, the reference thermal conductivity is calculated as:

 (9) 
with

 (10) 
and

 (11) 
The critical point (T_{critical}, p_{critical}) is also calculated using PCSAFT. With these GC approaches, PCSAFT predicts the viscosity and thermal conductivity accurately from the molecular structure of the molecule for a given state. Compared to measurement data for nhexane, the average errors are 5.5% for the viscosity and 5.2% for the thermal conductivity, showing the good accuracy of these approaches (see Fig. 3). In this work, a preliminary GC approach for the thermal conductivity is used.^{50} The extension of the full entropyscaling approach for thermal conductivities^{51} to a GC approach is currently in progress.

 Fig. 3 Predicted viscosities^{49} (top) and thermal conductivities^{50} (bottom) of nhexane depending on temperature using GC PCSAFT in comparison to measurement data.  
Hopp and Gross also work on an approach to predict the selfdiffusion coefficients based on entropyscaling and PCSAFT.^{51} Diffusion is important to design processes in process engineering and would enhance the applicability of the thermoeconomic 1stage CoMT–CAMD approach to further types of processes.
In this work, GC approaches are used to predict the heat capacity of the ideal gas c^{ig}_{p}, molar mass M, pure component parameters of PCSAFT z, and the scaling factors for transport properties A_{κ} to D_{κ}. The groups considered within the presented approach are limited by the measurement data available to adjust the group contributions. To prevent extrapolation from the databases, a limitation of the molecular design space is regarded in the inequality constraints F_{2}·y^{s} ≤ 0 in problem (1). Thereby, a high accuracy is ensured.^{35} While large databases of experimental data exist for pure component parameters and viscosity, there is only scarce data for thermal conductivity. Thus, the employed groups are limited to the current state of the group contribution method for the thermal conductivity: –CH_{3}, –CH_{2}–, >CH– and >C< for branched alkanes, CH_{2} and CH– for 1alkenes, C^{arom} and CH^{arom} for aromatics with alkyl side groups and –CHO for aldehydes. Additionally, methane CH_{4} and ethane C_{2}H_{6} are defined as separated groups to increase the accuracy of these small molecules. Here, the heat capacity of the ideal gas is calculated using a moleculespecific correlation.^{62} Additional groups can easily be integrated into the approach, as soon as the GC approach for thermal conductivity is further developed.
2.3. Nonconventional properties
To ensure the design of safe and environmentally benign processes, the molecular design space is constrained by limitations of nonconventional properties, i.e., flammability, autoignition temperature (AIT), toxicity, and the environmental properties global warming potential (GWP) as well as ozone depletion potential (ODP) (Table 1). The nonconventional properties are calculated from the molecular structure of the molecule using firstorder GC approaches. A detailed description can be found in the ESI† (section S1).
Table 1 Nonconventional properties limited in the integrated thermoeconomic design approach, the used GC approaches and the considered limitation
Property 
GC approach 
Limitation 
Flammability 
FNumber Γ 
Kondo et al.^{63} 
Γ < 0.8 (ref. 63) 
AIT 
Albahri and George^{64} 
T
_{max,process} + 30 K ≤ AIT 
Toxicity 
96h LC50 
Martin and Young^{65} 
96h LC50 ≥ 10 mg l^{−1} (ref. 66) 
Environmental impact 
GWP 
Hukkerikar et al.^{67} 
GWP ≤ 150 (ref. 68) 
ODP 
Hukkerikar et al.^{67} 
ODP = 0 
The limitations on GWP and OPD are implemented to demonstrate the general applicability of the consideration of environmental properties in the presented design approach. Since the considered groups have no significant environmental impact, both limitations have no influence on the optimization result of the current study. The limitations become relevant as soon as halogenated groups are added to the approach.
2.4. Optimization strategy
The 1stage CoMT–CAMD approach enables single and multiobjective optimization. The result of a singleobjective optimization is one single optimal molecule and the corresponding optimal process. Integercuts^{54} are used to obtain a ranking of optimal molecules. Here, the MINLP is solved repeatedly, wherein integercuts constrain the feasible design space to exclude previous solutions. The result of a multiobjective optimization is a Pareto front, which can be calculated using multiobjective optimization strategies (e.g., normalized constraint method^{69} or epsilonconstraint method^{70}).
The optimization problem is solved using the software GAMS (version 24.6.1 (ref. 71)). However, the process model and PCSAFT contain demanding iterative calculations for, e.g., boiling points. Thus, the process calculation and thermodynamic calculations of PCSAFT are performed in external functions (Fig. 4). Here, GAMS forwards all molecule property parameters calculated from the molecular structure y^{s} and all process variables x to the external functions. The molecule property parameters are the pure component parameters of PCSAFT z, the parameters needed to calculate the heat capacity of the ideal gas c^{ig}_{p} and the molar mass M as well as the coefficients to calculate the reduced transport properties (A_{κ} to D_{κ}). From the external functions, the objective function values f, the inequality constraints of the process p_{2} as well as equipment g_{2} and the firstorder derivatives with respect to all parameters, which are forwarded to the external functions, are returned to the optimization problem in GAMS. This provides a blackbox model to GAMS, which enables a very stable computation of the iterative calculations.

 Fig. 4 Schematic illustration of the interaction between GAMS and the external functions.  
However, the external functions prevent the use of global MINLP solvers, since these still need explicit models today.^{72} Thus, the MINLP is solved using the local, deterministic MINLP solver DICOPT,^{73} which combines outerapproximation^{74} with a relaxation strategy. In DICOPT, series of nonlinear program (NLP) subproblems and mixedinteger linear program (MILP) master problems are solved. Initially, relaxation of problem (1) is solved to obtain a hypothetical optimal molecule, the socalled target. The relaxation problem is identical to the CoMT problem in the CoMT–CAMD approach.^{3} Afterwards, an optimal integer solution is identified using outerapproximation. As subsolvers, the NLP solver SNOPT^{75} and the MILP solver CPLEX^{76} are used.
3. Design of an organic Rankine cycle
The integrated thermoeconomic design of the equipment, process and molecule is a crucial challenge in the design of Organic Rankine Cycles (ORC) as recently reviewed by Linke et al.^{26} ORCs can be used to transform lowtemperature heat stemming from renewable heat sources into electrical power.^{77} Renewable heat sources can be, e.g., solar heat,^{78} geothermal heat^{79} or biomass.^{80} Additionally, lowtemperature waste heat from industry^{81} and automotive applications^{82} can be exploited. Due to the low exergy content of these heat sources, ORCs have to be tailored to the specific application to ensure an economic process.^{13} The operating principle of an ORC follows four steps (Fig. 5): firstly, the molecule, the socalled working fluid, is pumped to the upper pressure level p_{evap} (1 → 2). Then, the working fluid is preheated, evaporated and optionally superheated exploiting the heat source (2 → 3). In a third step, the working fluid is expanded to the lower pressure level p_{cond} in a turbine (3 → 4). A generator transforms the work of the turbine into electrical power. The working fluid is desuperheated and condensed to boiling liquid in the last step (4 → 1). In summary, an ORC encompasses three types of heat transfer (single phase, evaporation and condensation) and the design of several types of rotating equipment (turbine, pump, generator and gearbox). To exploit the renewable heat source in the most economical way, the process, equipment and working fluid have to be optimized simultaneously.

 Fig. 5 ORC process in (a) flowsheet and (b) temperature–entropy diagram.  
In this work, the presented 1stage CoMT–CAMD approach for integrated thermoeconomic design of the process and molecule is applied to the design of an ORC for waste heat recovery. The general specifications of the case study are given in section 3.1. In section 3.2, the considered objective functions are described. The relevant assumptions for designing the heat exchangers and the rotating equipment are given in sections 3.3 and 3.4, respectively.
3.1. General specifications of the case study
In the case study, a subcritical, nonregenerated ORC is considered (Fig. 5). Industrial waste water is used as a heat source with a mass flow rate of ṁ_{HS} = 10 kg s^{−1} and an inlet temperature of T^{in}_{HS} = 150 °C (Table 2).^{83} The ORC is cooled using cooling water with a mass flow rate of ṁ_{CW} = 175 kg s^{−1} and an inlet temperature of T^{in}_{CW} = 15 °C. The heating is performed in two heat exchangers in series: a preheater for singlephase heat transfer to boiling liquid and an evaporator for evaporation and optional superheating. A detailed description of the thermodynamic model of the ORC process is given in the ESI† (section S2).
Table 2 Specifications of the ORC case study
Parameter 
Symbol 
Value 
Parameter 
Symbol 
Value 
Flow rate (heat source) 
ṁ
_{HS}

10 kg s^{−1} 
Isentropic turbine efficiency 
η
_{T,is}

0.8 
Temperature (heat source) 
T
^{in}_{HS}

150 °C 
Isentropic pump efficiency 
η
_{P,is}

0.75 
Heat capacity (heat source) 
c
_{p,HS}

4.2 kJ kg^{−1} K^{−1} 
Generator efficiency 
η
_{G}

0.98 
Flow rate (cooling water) 
ṁ
_{CW}

175 kg s^{−1} 
Min. absolute pressure 
p
_{min}

1 bar 
Temperature (cooling water) 
T
^{in}_{CW}

15 °C 
Min. reduced pressure 
p
^{red}_{min}

10^{−2} 
Heat capacity (cooling water) 
c
_{p,CW}

4.2 kJ kg^{−1} K^{−1} 
Max. absolute pressure 
p
_{max}

50 bar 
Min. steam quality (turbine outlet) 
φ
_{min}

0.95 
Max. reduced pressure 
p
^{red}_{max}

0.8 
Max. segment number 
n
_{max}

25 



The process degrees of freedom are the mass flow rate of the working fluid ṁ_{wf}, the reduced operating pressure levels of the condenser p^{red}_{cond} and the evaporator p^{red}_{evap} and the degree of superheating at the turbine inlet ΔT_{sh}. Reduced pressures are defined as:

 (12) 
where
p denotes the absolute operating pressure level and
p_{critical} denotes the absolute pressure at the critical point as calculated from PCSAFT. Using reduced pressure levels as process variables allows simple bound constraints instead of implicit constraints to ensure a subcritical process (here
p^{red} ≤ 0.8). Furthermore, computations are more stable by using reduced pressure levels since a subcritical process operation is ensured as an initial guess in all subproblems of the MINLP optimization. Pressure drops in the preheater, evaporator and condenser are neglected. Constant isentropic efficiencies are assumed for the pump
η_{P,is} and the turbine
η_{T,is}. The steam quality at the turbine outlet is limited to be at least
φ_{min} = 0.95 to prevent droplet erosion on turbine blades. Feasible heat transfer is ensured by additional process constraints on the minimal approach temperature in the heat exchangers. Additionally, the operation pressures are constrained to minimal and maximal values for both the absolute pressures (
p_{min},
p_{max}) and the reduced pressures (
p^{red}_{min},
p^{red}_{max}). The maximal number of molecular groups is limited to
corresponding to 2648 structurally feasible molecular structures, which fulfill the CAMD constraints.
3.2. Objective function
For thermoeconomic evaluation of a process, several objective functions can be used, e.g., specific investment cost or net present value. The specific investment costs SIC are defined as: 
 (13) 
where TCI denotes the total capital investment and P_{net} indicates the net power output of the process. The advantage of the specific investment costs is that no assumptions have to be made regarding payback periods etc.
The total capital investment TCI is calculated as the sum of the purchasedequipment cost PEC_{i} multiplied by factors w for additional direct and indirect costs as:^{84}

 (14) 
where PEC
_{RE,k} denotes the purchasedequipment cost of the rotating equipment k and PEC
_{HE,j} indicates the purchasedequipment cost of the heat exchanger j. The factors
w_{1} = 3.7 and
w_{2} = 3.1 consider the direct cost for,
e.g., installation, piping, electrical equipment and service facilities as well as the indirect cost for,
e.g., engineering, supervision and construction.
^{84,85} In comparison to
w_{1}, the factor
w_{2} neglects the additional direct costs for installation, since these costs are already regarded in the purchasedequipment cost of the heat exchangers PEC
_{HE,j} (see section 3.3).
The net power output P_{net} is calculated as:

 (15) 
where
η_{G} denotes the efficiency of the generator given in
Table 2 and
h_{i} denotes the enthalpy at state i.
3.3. Purchasedequipment costs of the heat exchangers
The heat exchangers are modelled as shell and tube heat exchangers in counterflow control without shell baffles. Hall et al.^{86} present a cost correlation for the purchasedequipment costs of shell and tube heat exchangers depending on the heat exchanger area A_{HE}. The cost correlation is available for several material combinations of shell and tubes and already includes the installation costs. Since the organic working fluid of the process is not known during the design, the cost correlation for stainless steel for the shell and tubes is used to prevent corrosion. The cost correlation is defined as: 
 (16) 
Originally, the cost correlation originates from 1982. The Chemical Engineering Plant Cost Index (CEPCI) is used to account for inflation and development of raw material prices. The CEPCI is relatively specific to the United States. However, the presented approach is independent of the cost models and conversion factors, which can be individually selected by the user for the designed application and its location. Using the CEPCI, the current PEC_{HE} is calculated as:

 (17) 
with CEPCI
_{2016} = 556.8 and CEPCI
_{1982} = 314.
^{87}
The heat exchanger area can be calculated as:

 (18) 
where
denotes the transferred heat flow, Δ
ϑ_{ln} indicates the logarithmic temperature difference in the heat exchanger
^{88} and
k_{HE} represents the heat transmission coefficient. The heat transmission coefficient
k_{HE} is defined as:

 (19) 
where
d_{o} and
d_{i} denote the outer and inner diameters of the tubes, respectively,
α_{o} and
α_{i} denote the outer and inner heat transfer coefficients, respectively, and
λ_{Tu} = 16 W m
^{−1} K
^{−1} indicates the thermal conductivity of the tubes. The outer and inner diameters of the tubes are fixed to
d_{o} = 20 mm and
d_{i} = 16 mm. To fully specify the design of the shell and tube heat exchanger (
Fig. 6), the number of tubes
n_{tubes} is found from the maximal allowed velocity in the tubes. The outside diameter of the shell
D_{shell} in turn is found from the maximal allowed velocity in the shell. The maximal velocities are
c_{max,l} = 1.5 m s
^{−1} for liquids and
c_{max,v} = 20 m s
^{−1} for vapors.
^{89}

 Fig. 6 Degrees of freedom of the considered shell and tube heat exchangers in counterflow control without shell baffles.  
The heat transmission coefficient k_{HE} in eqn (19) depends on the inner and outer specific heat transfer coefficients α_{o} and α_{i}. The heat transfer coefficients are calculated for single phase, evaporation and condensation using specific heat transfer correlations (Table 3). Within these correlations, the heat transfer is described by dimensionless parameters such as Reynolds and Nusselt numbers, which strongly depend on the transport properties viscosity η and thermal conductivity λ. It is assumed that the working fluid is on the shell side and the water for heating and cooling is on the tube side. This configuration enables better cleaning of the heat exchangers. Since the heating and cooling medium is fixed to water, specific correlations for the properties of water are used.^{90} The heat transfer correlations for flow boiling in the evaporator and filmwise condensation in the condenser depend on the steam quality φ and, thus, represent the local heat transfer. Therefore, discretization of the steam quality φ is performed to calculate the heat exchanger areas for the evaporator and the condenser. The heat transfer for flow boiling additionally depends on the heat exchanger area itself enforcing an iterative calculation of the heat exchanger area. A detailed description of the correlations is given in the ESI† (section S3).
Table 3 Correlations for the different heat transfer types in the shell and tube heat exchangers of the ORC and additional features of the calculation of the heat exchanger area
Heat exchanger 
Side 
Type 
Correlation 
Features 
Preheater 
Shell 
Single phase, forced convection 
Gnielinski^{91} 

Tube 
Single phase, forced convection 
Gnielinski^{91} 

Evaporator 
Shell 
Flow boiling 
Gungor and Winterton^{92} 
Discrete, iterative 
Tube 
Single phase, forced convection 
Gnielinski^{91} 

Condenser 
Shell 
Filmwise condensation 
Numrich and Müller^{93} 
Discrete 
Tube 
Single phase, forced convection 
Gnielinski^{91} 

3.4. Purchasedequipment costs of the rotating equipment
The purchasedequipment costs PEC_{RC} of rotating equipment, i.e., pump, generator, gearbox and turbine, are calculated using the recent cost correlations from Astolfi et al.^{94} The proposed cost correlations are based on experience gained by the authors in the cooperation with manufacturers of ORC plants and turbines. The correlations are used to capture a detailed tradeoff between the working fluid and purchasedequipment cost. However, inaccuracies can occur, since the correlations are used in an extended range of equipment sizes in comparison to the original reference. Since the cost correlations are provided in Euros, an exchange rate of k_{€→$} = 1.114 US$ per € (4th August 2016)^{95} is considered. The purchasedequipment cost of the pump PEC_{RC,P} is calculated based on the pump power input P_{P} as: 
 (20) 
with C_{P,0} = 14000 € and P_{P,0} = 200 kW. The purchasedequipment cost of the generator PEC_{RC,G} is defined based on the net power output P_{net} by: 
 (21) 
with C_{G,0} = 200000 € and P_{G,0} = 5000 kW. The purchasedequipment costs of the gearbox PEC_{RC,GB} are regarded as 40% of the generator cost.^{94} The purchasedequipment cost of the turbine PEC_{RC,T} cannot be deduced from its power output P_{T}, since the costs vary significantly for the same power output but different enthalpy drops, volume ratios and volume flow rates.^{94} Thus, Astolfi et al.^{94} proposed a cost correlation for axial turbines depending on the number of turbine stages n_{st} and the last stage size parameter SP: 
 (22) 
with C_{T,0} = 1230000 €, n_{st,0} = 2 and SP_{0} = 0.18 m. The size parameter SP is defined by: 
 (23) 
where denotes the volume flow of the last stage and Δh_{is,st} indicates the isentropic enthalpy drop of one stage. The size parameter SP is proportional to the stage diameter, which serves as a measure for the purchasedequipment costs. By using this detailed cost correlation, the number of turbine stages n_{st} has to be considered as an additional integer degree of freedom within the optimization. As recommended by Astolfi et al.,^{94} two additional turbine design constraints are taken into account during the optimization: to avoid high Mach numbers and large blade heights, the maximal isentropic volume ratio of one stage V^{ratio}_{is,st} is limited to: 
 (24) 
where V^{ratio}_{is,T} denotes the overall isentropic volume ratio of the turbine. To avoid high mechanical stresses, the maximal isentropic enthalpy drop of one stage Δh_{is,st} is limited to: 
 (25) 
where Δh_{is,T} denotes the overall isentropic enthalpy drop of the turbine.
4. Results and discussion
The integrated thermoeconomic design of the organic Rankine cycle is defined by 4 continuous process degrees of freedom , 1 integer equipment degree of freedom x_{T} = n_{st} and 55 binary degrees of freedom describing the molecule in the CAMD formulation. A binary notation of the number of functional groups is used instead of an integer notation for an easy implementation of the integercuts.^{54} The 1stage CoMT–CAMD approach is firstly applied to a singleobjective optimization of the ORC using the specific investment cost SIC as the objective function in section 4.1. In section 4.2, the result of a multiobjective optimization is presented to demonstrate the tradeoff between the total capital investment TCI and the net power output P_{net} of the ORC. Finally, the results are validated in section 4.3.
4.1. Results of singleobjective optimization
The 1stage CoMT–CAMD approach is applied to optimize the specific investment cost. Initially, the relaxed problem is solved in the socalled CoMT step leading to a value of SIC = 3058 US$ per kW. This target value could be achieved for a hypothetical working fluid. It serves as a lower bound on the objective function of all molecular structures captured by the 1stage CoMT–CAMD approach. 1stage CoMT–CAMD with integercuts is used to calculate a ranking of the top 10 real working fluids shown in Table 4. For this case study, the approach identifies mainly shortchained alkanes and alkenes. Benzenes and aldehydes are not identified.
Table 4 The target and the top 10 molecular structures identified by a thermoeconomic optimization using the 1stage CoMT–CAMD approach with integercuts, the specific investment cost SIC, the net power output P_{net}, the total capital investment TCI and the rank (real) resulting from an individual process optimization (see section 4.3.1)
Rank 
Name 
SIC (US$ per kW) 
P
_{net} (kW) 
TCI (10^{6} US$) 
Rank (real) 
— 
Target 
3058 
434 
1.33 
— 
1 
Propene 
3318 
422 
1.40 
1 
2 
Propane 
3476 
393 
1.37 
2 
3 
But1ene 
4646 
328 
1.52 
3 
4 
Isobutane 
4722 
326 
1.54 
4 
5 
nButane 
5040 
324 
1.63 
5 
6 
Neopentane 
6397 
296 
1.89 
6 
7 
3Methylbut1ene 
6741 
276 
1.86 
7 
8 
Pent1ene 
7241 
262 
1.90 
8 
9 
Pentane 
7781 
251 
1.95 
10 
10 
Isopentane 
7242 
265 
1.92 
9 
Propene is identified as the best working fluid with specific investment costs of SIC = 3318 US$ per kW. The objective function value is 8.5% higher than the target value. Propane shows a similar objective function value to that of propene (SIC = 3476 US$ per kW). For the 3rd rank, the predicted specific investment cost already increases significantly. The following working fluids are not competitive for this case study.
For propene, the turbine costs constitute the major part of the total purchasedequipment cost with a share of 46% (Fig. 7). About a third of the total purchasedequipment cost accounts for the heat exchangers. Here, the evaporator creates the smallest purchasedequipment cost with a share of 8%, although the transferred heat is similar to that of the preheater. This fact results from the benefited flow boiling heat transfer in the evaporator in comparison to the singlephase heat transfer in the preheater. As a result of the optimization, the optimal approach temperature is found to be ΔT = 8.4 K in the preheater and evaporator and ΔT = 9.1 K in the condenser. This approach temperature is a compromise between the low purchasedequipment cost of the heat exchanger, which decreases for high approach temperatures, and a high net power output.

 Fig. 7 Cost distribution of the thermoeconomic optimal purchasedequipment cost for propene.  
By applying the 1stage CoMT–CAMD approach without constraining the nonconventional properties (section 2.3), acetaldehyde is additionally identified in the top 10. However, acetaldehyde is excluded from the design space because of its low autoignition temperature of AIT = 140 °C.^{96} In this case study, the other nonconventional properties do not affect the result. However, all identified alkenes are classified as strongly flammable, which has to be considered in a subsequent assessment of the ranking.
In comparison to a thermoeconomic design, Table 5 shows the top 5 working fluids identified by a thermodynamic design of the considered ORC using the net power output P_{net} as an objective function. In this case, no tradeoff between the objective function and the minimal approach temperature in the heat exchangers is reflected in the model, so the lower bound of the minimal approach temperature is set to ΔT^{lo}_{min} = 2 K.
Table 5 The target and the top 5 molecular structures identified by a thermodynamic optimization using the 1stage CoMT–CAMD approach with integercuts, the net power output P_{net}, the specific investment costs SIC and the total capital investment TCI
Rank 
Name 
P
_{net}/kW 
SIC/US$ per kW 
TCI/10^{6} US$ 
— 
Target 
634 
9227 
5.85 
1 
Propane 
589 
6097 
3.59 
2 
Propene 
563 
5934 
3.34 
3 
But1ene 
479 
6659 
3.19 
4 
Isobutane 
489 
6620 
3.24 
5 
Neopentane 
472 
9436 
4.46 
The optimal net power output of the target is P_{net} = 634 kW. The optimal real working fluid identified by 1stage CoMT–CAMD is propane with a net power output of P_{net} = 589 kW. The optimal thermoeconomic process conditions for one working fluid differ from the optimal thermodynamic process conditions, since the tradeoff between maximal net power output P_{net} and minimal total capital investment TCI is considered within a thermoeconomic optimization.
4.2. Results of multiobjective optimization
For better visualization of the tradeoff between the net power output P_{net} and the total capital investment TCI, a multiobjective optimization is performed using the normal constraint method.^{69} Since a local MINLP solver is used, the normal constraint method is repeated wherein the direction of movement of the normal constraint is changed. The resulting Pareto front is filtered according to Pareto dominance to increase the accuracy of the solution. To identify design tradeoffs, nonaggregated objective functions should be chosen in multiobjective optimization, which do not comprise a tradeoff itself.^{97} Thus, the net power output P_{net} of the ORC is considered as one objective function and the total capital investment TCI as the other objective function. As for the singleobjective optimization of the net power output, the lower bound of the minimal approach temperature is set to ΔT^{lo}_{min} = 2 K. Additionally, the lower bound of the net power output is set to P^{lo}_{net} = 200 kW to ensure the validity of the equipment design correlations for the given conditions. As seen in the singleobjective optimization (Table 5), propane maximizes the net power output (Fig. 8). However, the total capital investment is high for a high net power output, since the low approach temperatures in the heat exchangers result in large heat exchanger areas. The strong increase of the total capital investment for high net power output can lead to misleading results if a thermodynamic objective function is solely considered in the integrated design of the process and molecule. Minimal total capital investment is achieved for minimal net power output using propene as a working fluid. The optimal working fluid changes at P_{net} = 540 kW. In the range of net power outputs below P_{net} ≤ 526 kW, the turbine can be operated with one turbine stage using propene as the working fluid. For higher net power output P_{net} ≥ 526 kW, two turbine stages are necessary to satisfy the turbine constraints in eqn (24) and (25) resulting in a discontinuity in the Pareto front.

 Fig. 8 Pareto front resulting from multiobjective optimization of the net power output P_{net}versus the total capital investment TCI. The optimal working fluids along the Pareto front are propane (marker +) and propene (marker x). Additionally, the optimal specific investment cost SIC, net power output P_{net} and total capital investment TCI are marked.  
In Fig. 8, the optimal specific investment cost is also marked. The optimal operating conditions and the optimal working fluid vary between the objective functions net power output P_{net}, total capital investment TCI and specific investment cost SIC. Thus, the objective function has to be chosen deliberately in advance.
4.3. Validation
The thermoeconomic evaluation of the ORC process is the key to a reliable selection of the working fluid. Accordingly, the results are validated with respect to three main aspects: firstly, the quality of the optimal solution and the computational effort are analyzed from a numerical point of view (section 4.3.1). Secondly, uncertainties of the thermodynamic model are investigated in section 4.3.2. Finally, the results of 1stage CoMT–CAMD are compared to data from scientific publications and manufacturers to validate the employed cost models (section 4.3.3).
4.3.1. Numerical validation.
For a numerical validation of the results of 1stage CoMT–CAMD, a bruteforce, individual process optimization of all possible molecular structures was performed. Theoretically, the combinatorics would allow for more than 600 million combinations of the functional groups. However, the CAMD constraints for molecular feasibility reduce the actual set of structurally feasible molecules to 2648. We optimized the process individually for all of these structurally feasible molecules.
The resulting real rank is used to validate the optimization result of 1stage CoMT–CAMD which employed a nonglobal optimization algorithm. Still, the top 10 global optimal solutions are identified by the 1stage CoMT–CAMD approach. The order is nearly correct (see the last column of Table 4). Thus, the local MINLP solver DICOPT provides a very good solution for the presented case study, but integercuts are necessary to find all good solutions.
The computational effort for the 1stage CoMT–CAMD approach can be compared to the bruteforce computations by the number of function evaluations of the process model: while an individual process optimization of all possible molecular structures requires 803209 function evaluations, only 9786 function evaluations are needed to calculate a ranking of 10 working fluids with 1stage CoMT–CAMD, which corresponds to a saving of 98.8%. Thus, the 1stage CoMT–CAMD approach is an accurate and efficient method for the integrated design of the molecule, process and equipment.
4.3.2. Validation of the thermodynamic model.
The PCSAFT equation of state used in 1stage CoMT–CAMD provides a thermodynamically consistent model for both equilibrium and transport properties. However, according to G.E.P. Box,^{98} “all models are wrong” and PCSAFT is no exception: deviations from reality can occur from the parameterization, measurement data and assumptions of the PCSAFT model itself. To quantify the impact of the resulting uncertainties in our design approach, a Monte Carlo simulation study is performed for the process optimization of propene. In Monte Carlo simulations, statistical uncertainties in input parameters are propagated to the results of the model by repeated random samplings. Here, we impose uncertainties on the output properties computed by PCSAFT, which serve as input parameters for the process model. We assume a normal distribution and a coefficient of variation of CV = 10% for the transport properties (η, λ) based on LötgeringLin and Gross^{49} and Hopp and Gross^{50} and CV = 3% for the heat capacity of the ideal gas c^{ig}_{p} based on Poling et al.^{62} (see Table 6). For the saturation temperature T^{sat}, all residual properties (h^{res}, s^{res}, c^{res}_{p}) as well as the molar volume v, the authors assume a coefficient of variation of CV = 5%. Based on the scarce data available for the uncertainties of the output of PCSAFT, we believe that our assumptions for the coefficient of variation are conservative.
Table 6 Assumed coefficient of variation CV of the output parameters of the thermodynamic model and the resulting maximal deviation Δ_{max} as well as minimal deviation Δ_{min} of a test set of 1000 random samplings

η

λ

c
^{ig}_{p}

T
^{sat}

h
^{res}

s
^{res}

c
^{res}_{p}

v

CV 
10% 
10% 
3% 
5% 
5% 
5% 
5% 
5% 
Δ
_{max}

27.9% 
31.1% 
9.2% 
17.8% 
13.2% 
14.2% 
15.3% 
15.6% 
Δ
_{min}

−40.9% 
−29.3% 
−9.1% 
−17.9% 
−15.0% 
−15.2% 
−16.3% 
−15.5% 
A test set of 1000 random samplings is used for the Monte Carlo simulations, of which finally 810 samplings converged to an optimal solution and are considered for the assessment. Overall, a mean value of SIC = 3854 US$ per kW with a standard deviation of σ_{SIC} = 648 US$ per kW (CV = 16.8%) is calculated for propene corresponding to an average deviation of 16.2% to the expected value of SIC = 3318 US$ per kW calculated in 1stage CoMT–CAMD (see Fig. 9). The major source for the deviations in the objective function is uncertainties in the saturation temperature since they change the optimal pressure levels. For propene, pressure constraints are active in the optimum. Thus, any change in the saturation temperature forces the solution away from the optimal pressure levels. Thereby, the distribution is shifted to higher specific investment cost compared to the expected value calculated in 1stage CoMT–CAMD. Thus, the authors expect a similar trend for working fluids with the same active set of constraints at the optimum identified by 1stage CoMT–CAMD to that for propene. In contrast, a smaller shift is expected for working fluids without active constraints in the optimum. In the top 10 identified working fluids, only propane has the same active set of constraints as propene.

 Fig. 9 Histogram of the specific investment cost of propene from a Monte Carlo simulation considering uncertainties in the output parameters of the thermodynamic model. The mean value of the Monte Carlo simulations for propene is illustrated as well as the top 10 working fluids identified with 1stage CoMT–CAMD.  
Within the standard deviation of σ_{SIC} = 648 US$ per kW, the two top working fluids, propane and propene, cannot be distinguished by 1stage CoMT–CAMD. Propane and propene have very similar chemical structures and thus very similar performance in the process. In contrast, the candidates from rank 3 onwards give performance beyond the range of the standard deviations and thus can be clearly distinguished as inferior working fluids by our method. Thus, even based on the conservative uncertainties assumed by the authors, the presented 1stage CoMT–CAMD approach can provide significant information on the real ranking of molecules.
4.3.3. Validation of the cost models.
The validation of the cost models is performed with respect to two main aspects: firstly, the predicted specific purchasedequipment costs of the top 10 identified working fluids are compared to specific purchasedequipment costs from scientific publications and ORC manufacturers. Secondly, the cost distribution of the purchasedequipment cost of propene is compared to the cost distribution of a real ORC application for waste heat recovery. To ensure a fair comparison, we choose the specific purchasedequipment cost as a validation criterion to prevent deviations caused by the conversion factors w_{1} and w_{2}. These constant conversion factors vary in the literature and result in a systematic deviation without influence on the presented rankings.
Quoilin et al.^{14} compiled specific purchasedequipment costs of ORCs for waste heat recovery collected from ORC manufacturers and scientific publications, which are used as reference data. The specific purchasedequipment costs predicted for the top 10 working fluids show good agreement with the reference data (Fig. 10). In comparison to the reference data, we identify optimal working fluids in a range of low specific purchasedequipment costs showing the potential of the presented approach for thermoeconomic optimization. Please note that the results are converted into Euros to allow for a better comparison with the reference data.

 Fig. 10 Comparison of the predicted specific purchasedequipment costs of the top 10 working fluids identified with the 1stage CoMT–CAMD approach (squares) with specific purchasedequipment costs collected from ORC manufacturers and scientific publications^{14} (circles).  
Lemmens^{99} notes that the distribution of purchasedequipment costs commonly differs largely between estimated and real costs of the equipment. The purchasedequipment costs predicted by the 1stage CoMT–CAMD approach (exemplified for propene in Fig. 11a) show good agreement with the cost distribution of the real ORC for waste heat recovery presented by Lemmens^{99} in Fig. 11b. However, the design of the real ORC differs from our design: a radial turbine, plate heat exchangers and an air cooling system are used. Considering lower purchasedequipment costs for plate heat exchangers and higher costs for an air cooling system, a sufficient accuracy of the cost distribution predicted by the 1stage CoMT–CAMD approach is still expected. Since the aim of the presented approach is not a highly accurate prediction of the total capital investment of a process, but the thermoeconomic comparison of different working fluids during the design of the process, the models considered in this work show good agreement with real ORC applications.

 Fig. 11 Cost distribution of the purchasedequipment cost of (a) the predicted ORC for propene and (b) a real ORC.^{99}  
5. Conclusion
From molecules to dollars: in this work, we integrate molecular design into thermoeconomic design of a process. For this purpose, a Computeraided Molecular Design (CAMD) formulation is directly linked to models for process design and equipment sizing. In the resulting 1stage CoMT–CAMD approach, the link is enabled by the physically based perturbedchain associating fluid theory (PCSAFT). PCSAFT allows us to predict equilibrium as well as transport properties in a thermodynamically consistent way. The calculation of transport properties enables the detailed sizing of the equipment during the optimization. Heat exchangers are designed using detailed heat transfer models for single phase, evaporation and condensation. Thereby, the economic consequences of changing the molecule in the process are captured. To ensure safety and environmental protection, nonconventional properties of the molecule, e.g., flammability, toxicity or environmental properties, are considered. Thus, the approach enables a holistic design of the thermoeconomically optimal molecule and the corresponding optimal process and equipment in one single MINLP optimization. Because of the efficient computation, the 1stage CoMT–CAMD approach also allows multiobjective optimization to capture all processrelated tradeoffs.
The presented 1stage CoMT–CAMD approach is successfully applied to the integrated design of a subcritical organic Rankine cycle. This case study demonstrates the merits of detailed equipment models: sizing of heat exchangers considering the heat transfer for single phase, evaporation and condensation as well as the sizing of rotating equipment such as the turbine. The 1stage CoMT–CAMD approach identifies the most promising working fluids, which minimize the specific investment costs of the ORC. It is shown that the integrated thermodynamic design of the process and molecule results in a combination of the optimal process and working fluid with high total capital investment showing the significance of an integrated thermoeconomic design approach. A multiobjective optimization is used to more closely visualize the thermoeconomic tradeoff between the net power output and the total capital investment. We show that the predicted specific purchasedequipment cost and the cost sharing of the purchasedequipment cost show good accordance with real ORC applications. The 1stage CoMT–CAMD approach can directly be applied to any process design which is sufficiently well described by the equation of state, e.g., to refrigerant design, solvent design or working fluid design.
Nomenclature
Latin symbols
A
 Area 
A − D  Scaling factors of the Taylor polynomial 
c
 Velocity 
c
_{p}
 Heat capacity 
CV  Coefficient of variation 
d
 Diameter 
D
 Diameter of the shell 
f
 Objective function 
F
_{1}/F_{2}  CAMD constraints 
g
_{1}/g_{2}  Equipment constraints 
h
 Enthalpy 
k
 Heat transmission coefficient 
k
_{B}
 Boltzmann constant 
k
_{€→$}
 Exchange rate 
m
 Segment number 
ṁ
 Mass flow rate 
M
 Molar mass 
n
_{i}
 Number of functional group i 
nc
 Nonconventional fluid property constraints 
N
_{A}
 Avogadro's constant 
p
 Pressure 
p
_{1}/p_{2}  Process constraints 
P
 Power 
 Heat flow 
s
 Entropy 
SP  Size parameter 
T
 Temperature 
 Volume flow 
V
^{ratio}
 Volume ratio 
v
 Molar volume 
w
_{1}/w_{2}  Correction factors for direct/indirect cost 
x
 Process degrees of freedom 
y
^{S}
 Molecular structure 
z
 Pure component parameters 
Greek symbols
α
 Heat transfer coefficient 
Γ
 Fnumber 
Δ
 Difference 
ε/k  Segment dispersion energy 
η
 Efficiency 
η
 Viscosity 
Θ
 Equilibrium properties 
ϑ
 Temperature 
κ
 Transport properties 
λ
 Thermal conductivity 
μ
 Dipole moment 
σ
 Segment diameter 
σ
 Standard deviation 
φ
 Steam quality 
Ω^{(2,2)}*  Reduced collision integral 
Abbreviations
AIT  Autoignition Temperature 
CAMD  Computeraided Molecular Design 
CEPCI  Chemical Engineering Plant Cost Index 
CoMT  ContinuousMolecular Targeting 
DICOPT  Discrete and Continuous Optimizer 
EoS  Equation of State 
GC  Group Contribution 
GWP  Global Warming Potential 
MILP  MixedInteger Linear Program 
MINLP  MixedInteger Nonlinear Program 
MIQP  MixedInteger Quadratic Program 
NLP  Nonlinear Program 
ODP  Ozone Depletion Potential 
ORC  Organic Rankine Cycle 
PCSAFT  PerturbedChain Statistical Associating Fluid Theory 
PEC  PurchasedEquipment Cost 
SIC  Specific Investment Cost 
SNOPT  Sparse Nonlinear Optimizer 
TCI  Total Capital Investment 
Subscript
CE  Chapman–Enskog 
cond  Condensation 
critical  Critical state 
CW  Cooling water 
evap  Evaporation 
G  Generator 
HE  Heat exchangers 
HS  Heat source 
i  Inner 
is  Isentropic 
l  Liquid 
lb  Lower bound 
ln  Logarithmic 
max  Maximal 
min  Minimal 
o  Outer 
P  Pump 
pr  Process 
pre  Preheating 
RE  Rotating equipment 
ref  Reference 
sh  Super heating 
st  Stage 
T  Turbine 
Tu  Tubes 
ub  Upper bound 
wf  Working fluid 
v  Vapor 
0  Reference 
Superscript
id  Ideal 
ig  Ideal gas 
in  Inlet 
out  Outlet 
red  Reduced 
res  Residual 
sat  Saturation 
vib  Vibrational 
*  Reduced 
Acknowledgements
We thank the Deutsche Forschungsgemeinschaft (DFG) for funding this work (BA 2884/41 and GR 2948/21).
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Footnote 
† Electronic supplementary information (ESI) available: Detailed description of the nonconventional properties of the molecules, the thermodynamic model of the considered case study and the calculation of the heat transfer coefficients. See DOI: 10.1039/c7me00026j 

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