Thermo-Economic Multiobjective Optimization of a LOW Temperature

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ThermoEconomic Multiobjective Optimization of a Low Temperature Organic Rankine Cycle for Energy Recovery Ruben Omar Bernal-Lara, and Antonio Flores-Tlacuahuac Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01198 • Publication Date (Web): 20 Sep 2017 Downloaded from http://pubs.acs.org on September 26, 2017

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Industrial & Engineering Chemistry Research

ThermoEconomic Multiobjective Optimization of a Low Temperature Organic Rankine Cycle for Energy Recovery Ruben Omar Bernal-Lara† and Antonio Flores-Tlacuahuac∗,‡ †Departamento de Ingenieria y Ciencias Quimicas, Universidad Iberoamericana, Ciudad de Mexico, Mexico ‡Escuela de Ingenieria y Ciencias, Tecnologico de Monterrey, Campus Monterrey, N.L., 64849, Mexico E-mail: [email protected] Phone: +52(1) 55 4347 2804

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Abstract In last decades the scarcity of fossil fuels around the world has promoted the quest of new processes to produce power from alternative energy sources. The conventional Rankine cycle with water as working fluid is a good choice to convert heat into electricity, although it is inefficient for temperatures below 350 o C. Moreover, using organic compounds mixtures as working fluid the Rankine cycle (Called Organic Rankine Cycle) improves the cycle efficiency and the economy of the process for energy recovery from low temperature sources. In this work we propose a set of organic compounds to simultaneously obtain both optimum values of compositions of the mixture (used as working fluid) and optimal operating conditions of the cycle for a given low temperature source. As design goals we set the minimization of total annual cost and the maximization of thermal efficiency. Because these two goals are in conflict the design problem was posed as a non-linear multi-objective optimization problem with thermodynamic efficiency and total annual cost of the cycle as trade-off objective functions. In most cases binary mixtures are chosen for low cost and power ranges. On the other hand, as cycle efficient improves, leading to expensive cycle, pure working fluids are preferred.

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Introduction The high energy demand in the world in last years has become a focus of attention for researchers due to the constant quest of alternative processes to produce power in a economic way and low environmental impact. The recovery of energy from waste heat streams is an example of an alternative energy process seeking to meet energy demand and taking care of sustainability issues, 1 . 2 In fact, waste heat sources are commonly found in solar and geothermal sources, as well as in industrial process streams. Thermodynamic cycles have been widely studied to use waste heat sources and produce power. 3–5 Most of those cycles use water as working fluid for low-temperature energy recovery. The most common thermodynamic cycle is the water Rankine cycle. However, the thermodynamic efficiency of the Rankine cycle is low at temperatures below 370o C. 6 To improve the performance of the Rankine cycle for low-temperature energy recovery, the Organic Rankine Cycle (ORC), featuring organic compounds as working fluids, has been proposed. 1 Organic compounds are good candidates as working fluids because of their low boiling point, low medium vapor pressure and high vaporization enthalpy at low temperature ranges. Some authors have studied the effects of the selection criteria to find what fluid must be employed in ORC. 2,7,8 Wang et al. 9 studied factors like Ozone Depletion Potential (ODP), Global Warming Potential (GWP) and atmospheric life time considering them as decision variables in the selection of working fluids in ORC. Moreover, improved thermal efficiency results have been obtained when mixtures rather than pure organic components have been deployed. 10–13 A problem when considering organic fluid mixtures has to do with the determination of the optimal proportion of each mixture component. This issue adds complexity to the optimal design of organic mixtures for low-temperature energy recovery purposes. In this work we exploit the well known fact that equipment process design is strongly linked to the determination of process operating conditions. 14,15 By exploiting this relationship we expect to compute improved optimal solutions. Moreover, these two prob3



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lems will be embedded in an optimization framework where the optimal design of the organic mixture will also be addressed. In addition, in some works researchers seek maximizing thermal efficiency or power produced in ORC neglecting the operational and equipment costs. The alternative configurations, constructions and flow sheets proposed to increase the ORC efficiency (or power) can end up in high operational costs or expensive process equipments. Because of this an equilibrium between thermal efficiency and capital/operational cost of ORC is desirable. A solution taking account these two parameters is known as Thermo-economic optimization and this is the approach used in the present manuscript which seeks to maximize thermal efficiency at minimum cost. In summary, in this work we approach the optimal design of ORC systems by addressing the simultaneous optimization of equipment design, operating conditions and composition of the organic mixture such that thermal efficiency is maximized while at the same time process cost is minimized. Recognizing that commonly thermal efficiency and process economics are conflicting optimization goals, a multiobjective optimization approach has been deployed for the Rankine cycle optimal design.

Literature review In this section we provide a short literature review on the energy recovery issue of lowtemperature streams. Since the amount of related literature is large, we only cite some of the most relevant works related to our research problem. Recognizing that the type of energy recovery fluid by itself, is not the only driver that has an effect on thermal efficiency, Chen et al. 16 proposed a supercritical ORC to improve the cycle efficiency around 10-30% against a normal condition cycle and avoiding the two-phase region in the heating process. However, a supercritical cycle requires pressures close to critical point conditions increasing the operational risk and cost of the ORC. Another way to increase thermal efficiency is by heat integration in ORC to reduce

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Heat/Sink Sources. Tuo 17 studied the CO2 Transcritical Rankine Cycle (TRC) with and without reheat comparing their performance at the same operating conditions. However, a TRC requires high pressure conditions. Li and Dai 18 compared a conventional structure ORC and a regenerative ORC. In this last kind of ORC, the heat exchanged between the process streams is decreased by amounts related to heat source (HS) and cold sink (CS) used in the exchangers. Rusev 19 presented four different kinds of ORC configurations (regenerative, cascaded with economizer and cascaded with two and 3 heat sources) and their performance to produce power with satisfactory results. Moreover, the regenerative ORC forces to append an extra heat exchanger, process streams installation, extra maintenance, etc. Another alternative is a multi-stage ORC. Molina et al. 20 tested different configurations (Series and cascade) of HS and CS to exploit heat recovery. In this way efficiency is enhanced because of the heat loss reduction, but the capital cost of ORC increases. There are some studies about ORC thermo-economic optimization in the literature to produce power (or get the maximum thermal efficiency) at low cost. Lecompte et al. 21 consider the Specific Investment Cost (SIC) as objective function to obtain the ORC cost per work unit produced. Furthermore, a quasi-steady one year simulation was presented for a better evaluation of process economy. Dimitrova et al. 22 developed a multi-objective model for minimizing both fuel consumption and investment cost of technologies in ORC. The optimization methodology proposed by the authors was decomposed into four parts: a master multi-objective optimization, a thermo-economic simulation, a slave optimization and a techno-economic evaluation. However, the solution of the model is too complex and require decisions involving mixed programming. Wang et al. 23 also realized a multi-objective optimization with exergy efficiency and equipment purchase cost as objective functions, and also used a Pareto curve to determine the best scenario of ORC performance. Other authors 24,25 employed the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) to solve multiob-

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jective optimization in ORC, maximizing thermal efficiency and minimizing SIC or Heat Transfer Area Per Power Output Unit. The NSGA-II is a good choice to solve multiobjective problems but the time spent in the resolution is bigger than other simples methods with the similar results. In addition, Quoilin et al. 26 developed a methodology to maximize both the thermal efficiency and economic profitability of ORC for pure fluids. Schuster et al. 27 compiled the most ORC innovative systems using experimental data and their applications in industry. Further, Yu et al. 28 realized the simulation of an ORC recovering waste heat in exhaust gas and in jacket water of a diesel engine using R245fa as pure working fluid.

ORC System Description The ORC system can be deployed for low-grade temperature or waste heat recovery process for power conversion. The ORC process requires the same processing equipment than a conventional Rankine cycle, but enables operation at temperatures below 200◦ C. Besides, in this work water is replaced by a mixture of organic compounds as working fluid. The Conventional ORC flowsheet is shown in Figure 1 and can be described as a four stages cycle. (S1) Working fluid compression to PHigh (S2) Vapor phase heating at constant pressure. (S3) Gas expansion to PLow . (S4) Total condensation at constant pressure. Starting at the point S1 the working fluid is fed into a pump to compress it to PHigh . Then, the compressed fluid goes to a vaporizer (HEX) to boil the working fluid at constant pressure. A heat source (HS) is required for heat exchange in the vaporizer. The vapor 6


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Figure 1: Conventional ORC Flowsheet leaving the vaporizer enters to the turbine for power generation, and at the same time reducing the pressure of the working fluid to PLow . The released heat during the expansion of the working fluid is turned into power. Finally, the mixture leaves the turbine and goes to a condenser (HEX) to return to its initial liquid state and the cycle is repeated over and over. Same as with the vaporizer, a thermal source to achieve heat exchange is needed, in this case, cold sink (CS). Because a working fluid phase change occurs inside the heat exchangers it is important to know how the phase change affects the heat retired from the ORC. When a pure component is used as working fluid the phase change occurs in the same point (TBubble = TDew ) avoiding phase equilibrium conditions. Otherwise, if the working fluid is a mixture, (TBubble 6= TDew ) phase equilibrium takes place and affects the heat required and the net power produced by the ORC. 10 Imran et al. 11 found that zeotropic mixtures have a better performance in ORC and require less heat transfer area than pure working fluids. Considering this issue a mixture is selected instead a pure component as working fluid. Furthermore, for mixtures to present better performance than pure working fluids demands suitable operation parameters and mole fractions of mixtures. 29 Mixtures as 7



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ORC working fluids with non-isothermal phase change leads to efficiency increase. Pure fluids are suitable, if the pinch point shifts. Using a mixture as working fluid the shift can be adjusted to higher temperatures levels. Then, employing mixtures permit a larger set of solutions concerning the choice of working fluid for ORC power plants. 30,31 Step 1 takes place in the pump and step 3 occurs in the turbine (both pressure change equipment). The second and fourth steps indicate that exist phase change in the Rankine cycle. Thus, phase change and phase equilibrium occur inside the heat exchangers (vaporizer and condenser). For simplification of the process mathematical model some assumptions commonly used in this kind of problems were taken, 32 these are: • Steady State. • Thermodynamic equilibrium at inlet an outlet sections of each equipment. • Negligible kinetic and gravitational terms in the energy balances. • Negligible heat losses toward the environment in heat exchangers, pump and turbine (expander). • One-dimensional Flow. In this work, a set of ten organic compounds were selected for the mixture working fluid considering the good performance of those compounds evaluated in previous works. 10,18,21 Table 1 shows the ten organic compounds considered for this case of study.

Problem Definition Considering both the ORC system and assumptions described in previous section the problem statement is defined as follows. Given: • A geometry in the heat exchangers 8


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Table 1: Set of Organic Compounds considered for mixture working fluid ID R-22 R-134a R-152a R-245ca HC-270 R-C318 FC-4-1-12 R-600 R-600a R-601

Component Name Chlorodifluoromethane 1,1,1,2-Tetrafluoroethane 1,1-Difluoroethane 1,1,2,2,3-Pentafluoropropane Cyclopropane Octafluorocyclobutane Dodecafluoropentane n-Butane isoButane n-Pentane

• A set of organic compounds • A low-grade temperature heat/sink sources then simultaneously determine the operation conditions, composition of the mixture working fluid of an Organic Rankine Cycle such that the total annual cost (T AC) is minimized, while ORC efficiency is maximized.

Thermo-economic Optimization Model The problem formulated in section was solved by developing an optimization model to find systematically the optimal values of T AC, operation conditions and mixture compositions of ORC. The optimization model consist of 3 parts: Objective functions, Thermodynamic modeling and Economic modeling. Because two conflicting objective functions will be used, instead of merging both objective functions in a single goal, we approach the optimization problem as a multiobjective optimization problem such that the best trade-off between the conflicting goals can be reached.

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Objective Functions The aim of this work is to obtain the composition of a working fluid mixture at optimal operation conditions which maximizes the net work produced by the ORC and minimizing the T AC. The thermal efficiency dictated by the first law of thermodynamics takes into account implicitly the net work produced in the cycle. It was decided to use the thermal efficiency as objective function instead of the net work because is an indicator of the useful output/input energy ratio. 33 In this way the power generated is maximized and a better exploitation of the HS is achieved. The thermal efficiency is given by (all terms are defined in the Nomenclature section):

max Ω1 = η =

Wnet qin

(1)

Also the minimization of T AC is required, hence the second objective function is:

min Ω2 = T AC = CapCost + UtiCost

(2)

where UtiCost and CapCost are the utility and capital costs, respectively. Only this two cost parameters were considered because they are the dominant costs in magnitude and impact for any process. 14

Thermodynamic Modeling The physical and thermodynamic properties of organic components of the mixture selected as working fluid consider the following thermodynamic constraints previously developed and evaluated. 10 All the thermodynamic parameters of the pure components and critical/normal parameters were taken from PURE32 database of Aspen Plus v8.8 simulator. 34

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Molar Fractions Constraint The molar fraction of the mixture must obey the following equations (for both single and two-phase): NC X

zi = 1

(3)

i

NC X

xif pe = 1,

i

∀f, p, e

(4)

where NC is the number of components in the mixture, i is the subscript for mixture component, zi is the composition mixture, p denotes the phase L/V (Liquid/Vapor), e is the pressure level (Low/High) and f is the vaporization level (B, 1, 2, .., NΨ , D). Pressure Level Constraint The ORC operates at two different pressures for power generation purposes. The pump increases pressure and the turbine decreases it. To guarantee that high-pressure level is above low-pressure level the next equation is enforced:

PHigh ≥ PLow + ∆Pmin

(5)

Temperature Levels Constraints To achieve heat transfer in the exchangers there must exist a minimum temperature difference between the thermal sources and the working fluid. For a two-phase region the

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next constraints were applied:

S Tf,High + ∆Tmin ≤ TfHS ,

∀f

(6a)

S Tf,Low − ∆Tmin ≥ TfCS ,

∀f

(6b)

S being TfHS and TfCS the fluid temperatures of HS and CS, respectively. Moreover, ∆Tmin is

the minimum temperature approach. For the zones where there is no two-phase region, the following equations were used:

S Tˇsub,High + ∆Tmin ≤ T HS,OUT

(7a)

S Tˇsub,Low − ∆Tmin ≥ T CS,IN

(7b)

S Tˇover,High + ∆Tmin ≤ T HS,IN

(7c)

S Tˇover,Low − ∆Tmin ≥ T CS,OUT

(7d) (7e)

where ≥ T S,IN is the inlet temperature and ≥ T S,OUT is the outlet temperature both of the thermal source. For model simplification aims, the rigorous calculations of the mixture true critical points was not considered. Hence, operating close to the critical point temperature should be avoided. So, the next constraint was employed:

Tf e < Tc,i ,

∀i, f, e

(8)

To calculate the transport properties of the working fluid inside the heat exchangers the variable T me was employed. T me is defined as the mean arithmetic temperature of the working fluid temperatures (IN and OUT) in heat exchangers. For simplicity, the transport properties are calculated as one-phase fluid and to force this the next constraints 12


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were applied:

T mHigh ≤ TB,High

(9)

T mLow ≥ TD,Low

(10)

Equilibrium Calculations at bubble and dew point The Rachford-Rice equation was used to calculate the equilibrium conditions of the mixture: 10,35 NC X i

zi (1 − Kif e ) = 0, 1 + ψf (Kif e − 1)

(11)

∀f, e

where Ψf is the vapor fraction being f =0 at bubble point and f =1 at dew point. Kif e is the phase equilibrium ratio, and for non-ideal mixtures it was calculated by the Gamma−P hi formulation: 10,35

γif Le Pifsate φsat if e exp

ZPe

sat Pif e

Kif e =

vL,if e (P ) dP Rg T f e ,

φîf e Pe

∀i, f, e

(12)

deploying equation 12 the compositions at the equilibrium point are computed as follows:

xif Le =

zi , 1 + ψf (Kif e − 1)

∀i, f, e

(13)

xif V e =

zi Kif e , 1 + ψf (Kif e − 1)

∀i, f, e

(14)

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Activity Coefficients The UNIFAC model is employed to calculate the activity coefficients of the mixture working fluid. 36 The following group of equations were applied:

C R ln γif pe = ln γif pe + ln γif pe ,

(15a)

∀i, f, p, e

C The term γif pe is the combinatorial term and is calculated as follows:

C ln γif pe

= ln N C P

ri rj xjf pe

10 + qi ln 2

qi

N C P

rj xjf pe

j

ri

j

N C P

+ li −

qj xjf pe

N C P

rj xjf pe

j

j

ri =

NG X

(i)

ν k Rk

k

qi =

NG X

(i)

ν k Qk ,

k

li =

ri

NC X

xjf pe lj ,

j

∀i, f, p, e (15b)

∀i

(15c)

∀i

(15d)

10 (ri − qi ) − (ri − 1) , 2

(15e)

∀i

where k, m and n are the UNIFAC functional groups for pure components.The terms Rk , R Qk , ri , qi and li are dimensionless parameters. The residual term γif pe is given by:

R ln γif pe

=

NG X k

(i) ln Γkf e

ln Γkf pe



(i)

νk

(i) ln Γkf pe − ln Γkf e ,

NG  X  (i) = Qk 1 − ln θm Ψmkf e  m



!

NG  X  = Qk 1 − ln θmf pe Ψmkf e  m

(15f)

∀i, f, p, e 

NG (i) X θm Ψkmf e   − , N G P (i)  m θn Ψnmf e

∀i, k, f, e

(15g)

n

!



NG X θmf pe Ψkmf e   − , N G P  m θnf pe Ψnmf e n

14


∀k, f, p, e

(15h)

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(i)

(i) θm

Qm X m =N , G P (i) Qn Xn

∀i, m;

θmf pe =

n

Ψmnf e

(i) Xm

,

∀m, f, p, e

Qn Xnf pe

(15i)

n

amn = exp − Tf e

,

(i)

νm =N , G P (i) νn

Qm Xmf pe N G P

∀i, m;

Xmf e =

n

N C P i

(15j)

∀m, n, f, p, e N C P

(i)

xif pe νm

, i N G P (i) xif pe νn

∀m, f, p, e

(15k)

n

Fugacity Coefficients The equation of state (EOS) Predictive-Soave-Redlich-Kwong (PSRK) is used to calculate the fugacity coefficients due to good prediction of the behavior of organic mixtures. 37 The PSRK equation reads as follows:

ln φîf e =

βif e ˆ Zˆf e + βˆf e Zf e − 1 − ln Zˆf e − βf e + αif e ln , Zˆf e βˆf e

∀i, f, e

(16)

The molar partial αif e is given by:

αif e

1 = A1

βˆf e βif e ln γif V e + ln + −1 βif e βˆf e

!

+ αif e ,

∀i, f, e

(17)

The dimensionless parameters αif e and βif e are calculated by the Soave-Redlich-Kwong (SRK) conventional method for pure components:

βif e =

αif e =

0.08664 Pe , Tr,if e Pc,i

(18)

∀i, f, e

2 0.42748 1 0.5 , 1 + cω,i 1 − Tr,if e 0.08664 Tr,if e

∀i, f, e

(19)

where Pc,i is the critical pressure of i. The parameter cω,i is dependent of acentric factorωi as:

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cω,i = 0.48 + 1.574ωi − 0.176ωi2 The reduced temperature Tr,if e is calculated by:

Tr,if e =

Tf e , Tc,i

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(20)

(21)

∀i, f, e

with Tc,i and Tf e as critical and working fluid temperature. For the mixture βˆf e is used:

βˆf e =

NC X

xif V e βif e ,

i

(22)

∀f, e

For the PSRK-EOS a excess gibbs model like UNIFAC is employed: NC X GE f pe = xif pe ln γif pe , Rg T f e i

(23)

∀f, p, e

The PSRK model calculates the excess Gibbs energy in the vapor phase at an hypothetic liquid state, hence α ˆ f e is given by:

α ˆf e

1 =− 0.64663

NC X GE βˆf e fV e xif V e ln + Rg T f e βif e i

!

+

NC X i

xif V e αif e , ∀f, e

(24)

The compressibility factor for a pure component and a mixture reads as follows 38 respectively:

Zie = 1 + βiBe − αiBe βiBe

1 Zie − βiBe , ∀i, e Zie (Zie + βiBe )

1 Zˆf e − βˆf e , Zˆf e = 1 + βˆf e − α ˆ f e βˆf e Zˆf e Zˆf e + βˆf e

∀f, e

Vapor Pressure Calculations The vapor pressure is obtained by the Riedel corresponding states method: 39 16


(25)

(26)

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ln Pifsate = A+ R,i −

+ BR,i + + 6 Tr,if ln Tr,if e + DR,i + CR,i e, Tr,if e

∀i, f, e

(27)

the equation parameters are given by:

+ = −QR,i , DR,i

+ = 42QR,i + αRc,i CR,i

+ = −36QR,i , BR,i

A+ R,i = −35QR,i ,

(28)

QR,i = 0.0838 (3.758 − αRc,i )

αRc,i =

ψbR,i

0.3149204ψbR,i + ln (Pc,i /1.0135) 0.0838ψbR,i − ln Tb,i /Tc,i

Tb,i Tc,i + 42 ln − = −35 + 36 Tb,i Tc,i

Tb,i Tc,i

(29)

6

(30)

where Tb,i is the normal boiling temperature. Compressed Volume Calculation The Chang-Zao equation 40 as function of the pressure is used to obtain the compressed liquid volume as follows:

vL,if e (P ) =

sat vL,if e

(Dcz −Tr,if e ) Acz,if e Pc,i + Ccz

Bcz,i

P − Pifsate sat

Acz,if e Pc,i + Ccz P − Pif e

where Acz,if e and Bcz,i are parameters calculated by:

3 6 Acz,if e = a0 + a1 Tr,if e + a2 Tr,if e + a3 Tr,if e +

a4 Tr,if e

Bcz,i = bcz,0 + ωi bcz,1

,

∀i, f, e

(31)

(32) (33)

The symbols Ccz , Dcz , a0 through a4 , bcz,0 and bcz,1 are constants. The saturated liquid volume is calculated by the Rackett equation: 41

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(1−Tr,if e ) sat vL,if e = vc,i Zc,i

Page 18 of 64

2/7

,

(34)

∀i, f, e

where vc,i is the critical volume of i and Zc,i is the compressibility factor of i. Enthalpy Calculations Considering the mixture as an incompressible fluid the saturated liquid enthalpy is given by:

ˆ e = h0 + h

NC X i

T  ZBe ZPe zi  CpL,i (T ) dT + vL,iBe (P ) dP  + HeE , T0

∀e

(35)

P0

where h0 is the enthalpy at a reference state, zi is the composition, CpL i is the pure component liquid heat capacity as a function of temperature, and HeE is the excess enthalpy for mixture at TBe in liquid state. The liquid heat capacity by the corresponding states method is obtained as follows:

" # 1/3 0.4355 6.3 (1 − T /T ) 0.49 c,i IG +Rg ωi 4.2775 + + (T )+1.586Rg +Rg CpL,i (T ) = Cp,i 1 − T /Tc,i T /Tc,i 1 − T /Tc,i (36) IG where Cp,i is the heat capacity of ideal gas for pure species i and it is calculated as tem-

perature function by next equation: 42

IG Cp,i

(T ) = C1,i + C2,i

C3,i /T sinh (C3,i /T )

2

+ C4,i

C5,i /T cosh (C5,i /T )

2

(37)

where C1,i through C5,i are equation parameters for each i component. The enthalpy of the mixture at the saturated vapor condition is given by:

ê − HE + ê = h H e

NC X i

 ZTDe R IG ˆ eR , zi ∆hLV Cp,i (T ) dT  + H ie − Hie + 

TBe

18


∀e

(38)

Page 19 of 64

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R where ∆hLV ie is the change of vaporization enthalpy for each pure component i. Hie is the

ˆ eR is the enthalpy departure residual enthalpy from ideal gas of pure component i and H in vapor phase at TBe and Pe . The vaporization enthalpy ∆hLV ie is calculated by Watson equation 43 as follows:

∆hLV ie

=

∆hLV b,i

1 − Tr,iBe 1 − Tb,i /Tc,i

0.375

,

(39)

∀i, e

44 where ∆hLV b,i is the vaporization enthalpy at normal boiling point and is obtained by:

∆hLV b,i = 1.093Rg Tc,i Tb,i

ln Pc,i − 1.013 0.93 − Tb,i /Tc,i

(40)

The departure enthalpy HieR of pure component i reads as follows: HieR = Rg TBe

a′ie αiBe − Rg bi

Zie + Zie − 1, Zie + βiBe

ln

∀i, e

(41)

where a′ie and bi are parameters of SRK-EOS and are given by: 35

a′ie = −0.42748

−0.5 Rg2 Tc,i 0.5 Tr,iBe , cω,i + c2ω,i 1 − Tr,iBe Pc,i bi = 0.08664

∀i, e

Rg Tc,i Pc,i

(42) (43)

ˆ R reads as follows: The departure enthalpy for the mixture H e ˆ eR H = Rg TDe

a ˆ′ α ˆ De − e Rgˆb

!

ln

ZˆDe ZˆDe + βê

+ ZˆDe − 1,

∀e

(44)

where the parameters a ˆ′e and ˆb are obtained by PSRK model as follows: # NC NC X X ˆ β z R 1 ′ De g i ′ zi ln , GE + aie + a ˆ′e = ˆb A1 e T b A1 β i iDe i i "

ˆb =

NC X

z i bi

i

19


∀e

(45)

(46)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

where GE e follows:

′

T

Page 20 of 64

is the Gibbs excess energy temperature derivative from eq 23 and reads as

GE e

′

T

= Rg TDe

NC X

zi (ln γiDV e )′T +

i

GE DV e , TDe

(47)

∀e

at last, the liquid excess enthalpy HeE is given by: 38 N

C X HeE zi (ln γiBLe )′T , = − 2 Rg TBe i

(48)

∀e

Entropy calculations The entropy calculations consider the mixture as incompressible liquid and the same reference state used in enthalpy calculations. So, the saturated liquid entropy reads as follows:

sê = s0 +

NC X i

T  ZBs CpL,i (T )  zi  dT + SeE , T

(49)

∀e

T0

where s0 is the reference entropy and SeE is the excess entropy and is given by:

SeE =

HeE − GE BLe TBe

(50)

The saturated vapor entropy Sê is calculated as follows:

Sê = sê − SeE +

NC X i

zi



LV  ∆hie

TBe

R − Sie +

ZTDe

TBe

IG Cp,i



(T ) dT  + SêR , T

∀e

(51)

The departure entropy for a pure component and for the mixture are obtained as follows, respectively: R Sie a′ Zie = ln (Zie − βiBe ) − ie ln , Rg Rg bi Zie + βiBe

20


∀i, e

(52)

Page 21 of 64

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a ˆ′ SêR ZˆDe = ln ZˆDe − βê − e ln , Rg Rgˆb ZˆDe + βê

∀e

(53)

Isentropic and Non Isentropic Operation To achieve isentropic operation of both pump and turbine the next restrictions are required:

sˇsubS,High = sˇsubS,Low

(54)

SˇoverS,High = SˇoverS,Low

(55)

For the non-isentropic region, the next restrictions were used: ˇ overS,High − H ˇ overS,Low ˇ over,High − H ˇ over,Low = ηST H H ˇ subS,High − h ˇ subS,Low ˇ sub,High − h ˇ sub,Low = 1 h h ηSP

(56) (57)

where suffix subS, e and overS, e are sub-cooling isentropic point and overheating isentropic point respectively, at pressure level e. Besides, ηST and ηSP are the isentropic efficiencies for turbine and pump, respectively.

Energy Balances Applying an energy balance between the working fluid and thermal sources in heat exchangers, the heat added and removed by the ORC can be calculated deploying the next equations: qIN

ˇ sub,High ˇ over,High − h = H

qOUT

ˇ sub,Low ˇ over,Low − h = H

To obtain the required moles of each thermal source and carry out the balance:

21


(58)


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Page 22 of 64

nHS ∆HHS = qIN nCS ∆HCS

(59)

= qOUT

where nHS and nCS are the molar ratios of heat source and sink of the working fluid. Moreover, qIN and qOUT are heat flow in molar basis entering and removing, respectively, of the ORC. Whereas ∆HHS and ∆HCS are their respective molar enthalpy changes. In zones where phase equilibrium of the working fluid takes place, the energy balances are given by: ˆ High , ∀f ˆ High + h ˇ sub,High = ψf H ˆ High − h nHS ∆HHS,f + h ˆ Low , ∀f ˆ Low + h ˇ sub,Low = ψf H ˆ Low − h nCS ∆HCS,f + h

(60)

where ∆HHS,f and ∆HCS,f are the two-phase point enthalpy change for the heat source/sink and are calculated as follows:

∆HHS,f =

Z

TfHS T HS,OUT

CpL,S (T )dT ;

∆HCS,f =

Z

TfCS T CS,IN

CpL,CS (T )dT

∀f

(61)

Transport properties of Working Fluid The sizing of heat exchangers are needed to calculate the transport properties of the mixture working fluid at temperature T me . These properties are obtained as one-phase fluid (liquid phase for vaporizer and vapor phase for condenser) for ease of calculation. To achieve this, the constraints given in section were used (Eqs. 9 and 10). (1) Gas viscosity The mixture viscosity in the condenser in vapor phase is calculated by Wilke’s correlation: 35

µ ˆLow

Nc X yi µi,Low = N C P i yj φij j

22


(62)

Page 23 of 64

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where µLow is the mixture viscosity in low-level energy (condenser), yi is the molar fraction of component i in vapor phase and φij is the interaction factor between the components and is given by:

φij =

1+

µi,Low µj,Low

0.5

h 8 1+

Mi Mj

Mj Mi

0.25 2

i0.5

(63)

where Mi is the molecular weight of specie i, µiV is the pure gas viscosity and is calculated by method of Chung et al.: 35

µi,Low =

40.785 F ci (Mi T mLow )0.5 2/3

vc,i Ων,i

(64)

where F ci is a correction factor for polar substances and Ων,i is the collision integral term and is obtained by the Neufeld et al. 35 method as follows respectively:

4 F ci = 1 − 0.2576 ωi + 0.059035 δr,i + κi

Ων,i = AN eu (Ti∗ )−BN eu + CN eu [exp(−DN eu Ti∗ )] + EN eu [exp(−FN eu Ti∗ )]

(65) (66)

4 with constants AN eu through FN eu (see Table 9), δr,i is dimensionless dipolar mo-

ment, κ is a correction factor for high polar components and Ti∗ is a reduced temperature function given as follows:

Ti∗ = 1.2593 Trm,iLow where Trm,ipe is the reduced temperature based on T mipe . (2) Liquid Viscosity

23


(67)


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Page 24 of 64

The working fluid mixture viscosity in the condenser is treated as liquid fluid. The UNIFAC-Visco method 35 is used:

ln µ ˆHigh =

NC X

xi ln(µi,High

i

V mi ) − ln Vˆm +

GE L,High Rg T mHigh

(68)

where xi is the molar fraction of each component, V mi is the molar volume of a pure component and Vˆm is the molar volume of the mixture both at T miLHigh . The term µiLHigh stands for pure liquid viscosity and is calculated by the method proposed by Przezdziecki and Sridhar method: 35

µi,High =

1 Vps,i Eps,i (V mi − Vps,i )

(69)

with parameters Eps,i and Vps,i given by:

vc,i

Eps,i = − 1.12 + Vps,i

12.94 + 0.10Mi − 0.23Pc,i + 0.0424Tif p − 11.58(Tif p /Tc,i ) (70) fp V mi (71) = 0.0085 ωi Tc,i − 2.02 + fp 0.342(Ti /Tc,i ) + 0.894

where Tif p is the freezing point temperature and V mfi p is the molar volume for pure component at freezing point. (3) Gas Thermal Conductivity The mixture gas thermal conductivity reads as follows:

ˆ Low = λ

NC X yi λi,Low N C P i yj Λij j

where Λij is a parameter similar to φij and given by: 24


(72)

Page 25 of 64

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λi,Low λj,Low

0.5

ε 1+ Λij = h 8 1+

Mi Mj

Mj Mi

0.25 2

i0.5

(73)

with ε as constant and equal to 1. The pure gas thermal conductivity λi,Low is obtained employing the Roy and Thodos method: 35

λi,Low =

λr i,Low Γi

(74)

where λr i,Low is the dimensionless reduced thermal conductivity and is calculated by two contributions, the translational energy and the internal energy (like rotational, vibrational, etc.) as follows:

λr i,Low = (λΓ)tr, i + (λΓ)int, i

(75)

The functions (λΓ)tr, i and (λΓ)int, i read as follows:

(λΓ)tr,i = 8.757 [exp(0.0464 Trm i,Low ) − exp(−0.2412 Trm i,Low )]

(76)

(λΓ)int,i = CRT f (Trm )

(77)

with the parameter CRT and the reduced temperature-dependent function f (Trm ). The missing parameter Γi is given by:

Γi = 210

Tc,i Mi3 4 Pc,i

1/6

(78)

(4) Liquid Thermal Conductivity The mixture thermal conductivity in the vaporizer is calculated in liquid phase applying the Power-Law correlation: 35 25



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

N C P

−0.5

wi    i  ˆ λHigh =  2   λi,High 

Page 26 of 64

(79)

where wi is the mass fraction of i component and λi,High is the pure thermal conductivity in liquid phase obtained by Latini et al.: 35

λi,High =

ALat,i 1 − Trm i,High 1/6

Trm,iLHigh

0.38

(80)

where the parameter ALat,i given by:

ALat,i =

αLat A∗ Tb,i γLat MiβLat Tc,i

(81)

where A∗ , αLat, βLat and γLat are parameters of the equation. (5) Gas/Liquid Heat Capacity Finally, the heat capacities of the working fluid are calculated in the same way as before with the equations 36 and 37 but at Trm,ipe as appropriate. Transport Properties of Waste-Heat Sources The transport properties of thermal sources are needed too. In this work the heat/sink sources are considered as waste-heat water. The correlations of International Association of Properties of Water and Steam (IAPWS) 45–47 are used to estimate the properties of water because of their good prediction capability concerning the behavior of water in liquid/vapor phase. It is important to note that the temperature to estimate the transport properties of thermal sources is a mean arithmetic temperature as was done with the working fluid.

26


Page 27 of 64

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(1) Density The density as function of pressure and temperature is obtained from specific volume which is given by: 45

ρSe =

1 Rg T mSe ∗ = π γπ ∗ νsp PeS

(82)

where ρe is the density, νsp is the specific volume, T mSe is the mean arithmetic temperature of the waste-heat source (concerning to HS when e=High and CS if e=Low). P S is the pressure of the waste-heat source, π ∗ is a dimensionless pressure parameter and γπ∗ is the derivative of Gibbs free energy with respect to π ∗ and reads as follows:

γπ ∗ =

34 X i=1

−nπi Ii (7.1 − π ∗ )(Ii −1) (τ ∗ − 1.222)Ji

(83)

where τ ∗ is dimensionless temperature parameter like π ∗ . To calculate these two parameters the values of both references pressure and temperature are 165.3 bar and 1386 K, respectively. The coefficients Ii , Ji and nπi are shown in Table 10 (2) Viscosity The viscosity of thermal sources as function of temperature is given by: 46

µSe = µ0,e · µ1,e · µ2,e

(84)

where e stands for the pressure level as mentioned in sections and . The first factor of the product µ0,e represents the viscosity in the dilute-gas limit and is given by:

µ0,e

p 100 T¯e = 3 P HiW ¯e i i=0 T

27


(85)


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Page 28 of 64

where T¯e is the reduced dimensionless temperature and calculated by:

T mSe T¯e = Tcw

(86)

where Tcw is the critical temperature, Hi are coefficients shown in Table 11. The second factor µ1,e represents the contribution of finite density and is given by: "

# i X 5 6 X 1 = exp ρ¯e HijW (ρ¯e − 1)j ¯e − 1 T i=0 j=0

µ1,e

(87)

where ρ¯ is a dimensionless parameter as function of a reference density 46 and calculated in similar form to equation 86. The coefficients HijW are shown in table 12. It is pertinent to mention that the coefficients HijW omitted in the Table are equal to zero. The third factor µ2,e represents the critical enhancement of the viscosity and its effect was neglected because the operating conditions are not close to the critical points, so µ2,e is equal to 1. (3) Thermal conductivity For computing thermal conductivity the next equation is employed: 47

λSe = λ0,e · λ1,e · λ2,e

(88)

where λ0,e and λ1,e represent the same contributions that µ0,e and µ1,e , and are given by:

λ0,e

λ1,e

"

p T¯e = 4 P LW k k ¯ ( T e) k=0

!# i X 4 5 X 1 = exp ρ¯e LW ρe − 1)j − 1 ij (¯ ¯ Te i=0 j=0 28


(89)

(90)

Page 29 of 64

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W with coefficients LW k and Lij (see Tables 13 and 14). Similar to viscosity, the factor

λ2,e is equal to 1. (4) Heat Capacities The heat capacities of the heat/sink sources are obtained by the same equations deployed in section but taking into account the T mSe term.

Economic Modeling For addressing the economic modeling of the ORC system we need first to carry out the sizing of the heat exchangers and then evaluate the cycle economy by cost correlations. Heat Transfer Areas The heat exchangers were chosen as counter-current flow shell and tubes (one-pass shell and two-pass in tubes) equipment and basic geometry by heuristics. 48 The working fluid flows in shell side, whereas the waste-heat sources flows in tube side. The selected geometry is shown in Table 2. A Kern-based method 49 was employed for exchangers sizing and is presented in the next sections. Table 2: Selected Geometry of Heat exchangers dout (in) 0.75

din (in) Pt (in) 0.62 0.9875

Ltube (ft) Tube Pattern 20 Triangular

Baffles Cut (%) 25

Tube-Side Calculations From the energy balance in a heat exchanger (HEX):

AHE,e =

QHE,e FT, e Uc,e M LDTe

(91)

where AHE,e is the heat transfer area, QHE,e is the heat flow, FT, e is the temperature factor correction, Uc,e is the global heat transfer coefficient and M LDTe is the mean logarithmic 29



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Page 30 of 64

delta temperature. The subscript HE works in the same way as superscript S which changes in function of pressure level e (Vaporizer for e=High and Condenser if e=Low). The number of tubes of each HEX is given by the next equation:

NT,e =

AHE,e Ltube (π dout ) Ns,e

(92)

where Ltube is the tube length, π is the math constant, dout is the tube external diameter and Ns,e is the number of shell per HEX (for this work equal to 1). After computing NT,e we obtain the cross flow area and the mass flow per area unit, respectively, as follows:

at,e =

NT,e π d2in 4 Ntp

m ˙ t,e G˙ t,e = at,e

(93) (94)

where din is the tube internal diameter, Ntp is the number of passes in tubes and m ˙ t,e is the mass flow in tube-side. Next, the Reynolds number reads as follows:

Ret,e =

din G˙ t,e µSe

(95)

The Prandtl and Nusselt numbers are given by:

P rt,e =

S Cp,e µSe λSe

N ut,e = 0.023 (Ret,e )0.8 (P rt,e )nDB

(96) (97)

S is the heat capacity of the waste-heat source and the exponent nDB is a paramwhere Cp,e

eter of the Dittus-Boelter correlation with a value of 0.3 when the fluid is cooling and 0.4 when is heating. The wall heat transfer coefficient in tube-side is calculated by:

30


Page 31 of 64

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hsp−t,e =

N ut,e λSe Deq,t

(98)

where Deq,t is the equivalent diameter and for the tubes is equal to din . It is important to stress that one of the limitations of this work is the calculation of hsp that is realized considering that there is not phase change. At last, the pressure drop in the tube-side is given by:

∆Pt,e

2 ρSe vt,e = 2

4 ff,e

Ltube Ntp Ns,e + 4 Ntp Ns,e din

(99)

where ff,e is the friction factor and vt,e is the velocity inside the tubes. The friction factor reads as follows:

ff t,e = exp(0.576 − 0.19 ln Ret,e )

(100)

Shell-side Calculations The cross flow area and mass flow per area unit in that order are given by:

as,e =

Ds,e Cs Bs,e P t Nsp

m ˙ s,e G˙ s,e = as,e

(101) (102)

where Ds,e is the shell diameter, Cs is the clearance between tubes, Bs,e is the baffle spacing, P t is the tube pitch, Nsp is the number of shell passes and m ˙ s,e is the mass flow on shell-side. The shell diameter is estimated by next equation:

Ds,e = Db,e + Cs

(103)

where Db,e is the bundle tubes diameter and Cs is the clearance as it was mentioned and 31



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Page 32 of 64

they are given by:

Db,e = dout

NT,e K1

(1/n1)

Cs = P t − dout

(104) (105)

where coefficients K1 and n1 depend on tube pattern and number of tube passes, 48 respectively. To get high turbulence due to the working fluid inside the shell, segmented baffles perpendicular to the flow are required. 49 The baffle spacing is obtained from: Ds,e 2

Bs,e =

(106)

The Reynolds number in the shell-side is given by: Deqs G˙ s,e µ ê

Res,e =

(107)

where Deq,s is the equivalent diameter of shell. This parameter is function of tube pattern and for triangular pattern reads as follows:

Deq,s =

2 P t2

√

3 − π d2out π dout

(108)

Both dimensionless numbers Prandtl and Nusselt, and the wall heat transfer coefficient are given by the next set of equations:

P rs,e =

Cˆp,e µ ê ê λ

N us,e = 0.023 (Res,e )0.8 (P rs,e )nDB hsp−s,e =

32

ê N us,e λ Deq,s


(109) (110) (111)

Page 33 of 64

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Finally, the pressure drop in shell-side is given by:

∆Ps,e =

ff s,e G˙ 2s (NB,e + 1) Ds,e Ns,e Nsp 2 ρê Deq,s

(112)

where NB,e is the number of baffles and is calculated as follows:

NB,e =

Ltube Nsp Bs,e

(113)

Equipment Cost Correlations For cost evaluation of the ORC, the correlation developed by Turton et al. 15 was used:

log10 C0 = K1T + K2T log10 (Y T ) + K3T log10 (Y T )

2

(114)

where C0 is the base purchase cost in USD at atmospheric conditions and carbon steel as material fabrication. The parameter Y T represents the size or capacity of the process equipment. The coefficients K1T , K2T and K3T used for the pump, turbine and heat exchangers of the ORC system are listed in Table 3. Table 3: Y T Units and K1T , K2T and K3T values Equipment YT K1T K2T K3T (Units) Pump (Centrifugal) kW 3.3892 0.0536 0.1538 Vaporizer (Fixed U-tubes) m2 4.3247 -0.303 0.1634 Turbine (Axial) kW 2.7051 1.4398 -0.1776 Condenser (Fixed U-tubes) m2 4.3247 -0.303 0.1634 It is necessary to take into account inflation, construction and raw materials costs. The Chemical Engineering Plant Cost Index (CEPCI) 50 was employed to consider inflation due to materials. The values deployed were 397 as base value and 597 for present value. 51 The sum of the cost of each processing equipment is the Capital Cost (CapCost). On the other hand, the cost of thermal sources must be considered. The estimated prices were 33



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Page 34 of 64

taken from Turton et al. 15 and their inflation was rated by CEPCI method. The sum of these prices is the Utility Cost (UtiCost). It is important to mention that the working fluid cost was neglected due to null affectation on Total Annual Cost (The difference in the cost magnitude was very large). So, the sum of CapCost and UtiCost is the T AC. Finally, the operating period of the ORC was established as one year plant (345 days) for the calculations of energy consumed and generated. Besides, the replacement of the working fluid is neglected because of its short time of use. It is necessary to emphasize that all variables and equations have been defined on the molar basis of the working fluid (100 mol/sec).

Optimization Methodology The development of the thermo-economic optimization model involves the objective function and a set of constraints generally split into modeling equations and operating restrictions. The aim of the objective function is to establish the goal of the design issue. The process modeling equations make reference to the relationships among the process variables (i.e. mathematical relationships). Finally, a set of additional constraints are used to enforce design or operating limitations to which the process variables are subject to.

Multi-objective Optimization There are optimization problems that require to maximize/minimize two or more conflicting objectives functions simultaneously, like the present case study. The optimal Pareto curve is used to solve this kind of competing optimization issues. The Pareto curve is composed of a series of optimal points instead of a single optimal solution in one objective function. 52 Broadly, in the literature there exists some methods to generate the Pareto curve: 53 In this work the ε-constraint method was chosen to solve the competing multiobjective optimization problem due to its wide use and easy application. The method involves optimization of one objective function while the other objectives are 34


Page 35 of 64

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treated as constraints. 52 The multiobjective optimization formulation of this work was developed deploying GAMS (General Algebraic Mathematic System) which resulted in a Non-linear programming (NLP) problem whose numerical solution was sought using the CONOPT solver. In summary, the multiobjective optimization problem formulation to be solved can be cast as follows:

Ω2

(115)

Ω1 ≥ ǫ

(116)

g(x, p) = 0

(117)

Min x

s.t.

where x stands for the decision variables vector, p is the vector of system parameters. Moreover, g(x, p) is the set of process constraints represented by Eqns 4.3-4.114. It is important to stress that the inequality represented in Equation 116 reflects our desire of maximizing the thermal efficiency; in other words, we want that η takes the bigger possible value and not force it to an upper bound.

Problem initialization Because of nonlinearities embedded in the process mathematical model, the way the optimization problem is initialized is important to achieve the solution of the non-linear optimization problems. Only local optimal solutions were sought. For good initialization purposes an equimolar mixture of the components set was employed as initial composition. The initial values for pressure levels were set on 3 bar for low-level and 9 bar for high-level. Moreover, they were chosen arbitrarily but taking into account the results presented by Molina et al.; 10 in addition, the upper bound was set at 20 bar and lower bound was set at 1.01325 bar (1 atm). These values were selected considering operating the cycle 35



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

at low pressures for safety and economic criteria. The upper bound on temperature was set as the lowest pure component critical temperature. The fugacity, activity and compressibility factors were set to 1 (assuming ideal gas). Initial values of bubble and dew points temperatures are set as a composition average over the individual saturation temperatures. Most variables only take positive values. As temperature and pressure upper bounds, critical properties values of pure components were enforced. At last, the upper and lower bounds of variables like residuals and excess properties were estimated from their observed and expected values.

Results and discussion Table 4: Equipment Parameters of ORC Parameter Value Unit ηSP 0.9 ηST 0.9 HS,IN T 363.15 K T CS,IN 303.15 K HS ∆T 20 K ∆T CS 10 K S ∆Tmin 5 K ∆Pmin 1 bar NΨ 3 dimensionless In this work we have taken the equipment parameters from Molina et al. 10 which are listed in Table 4. Table 5 shows the results of multi-objective optimization. The number of decision variables was 8391, while the number of constraints was 8589. The problems were solved in a i5-4300 CPU, 2.50GHz, 8 Gb RAM processor using Windows 10 Pro and GAMS version 24.6.1. The thermal efficiency (η) objective function was set as constraint to generate the Pareto Curve. η was discretized within the (1-10%) interval with 1% step increases. This interval was selected because some published works about Thermoeconomic optimization of ORC systems, deploying similar organic mixtures, present ther36


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Table 5: Multi-objective Optimization Results; Pressure in bar, Energy in kW, Temperature in K and T AC in $KUSD

Var

0.01

0.02

0.03

Epsilon-constraint value of η 0.04 0.05 0.06 0.07

0.08

0.09

0.10

zR-22 zR-134a zR-152a zR-245ca zHC-270 zR-C318 zFC-4-1-12 zR-600 zR-600a zR-601

— — — — 0.095 — — — 0.905 —

— — — — 0.001 — — — 0.999 —

— — — — 0.204 — — — 0.796 —

— — — — 0.001 — — — 0.999 —

— — — — 0.108 — — — 0.892 —

— — — — 0.001 — — — 0.999 —

— — — — 0.001 — — — 0.999 —

0.001 — — — — 0.999 — — — —

0.001 — — — — 0.999 — — — —

0.001 — — — — 0.999 — — — —

PHigh Tˇsub,High Tˇover,High ∆Tˇf PLow Tˇsub,Low Tˇover,Low f ∆TˇLow ∆P

8.86 325.24 331.60 6.36 7.86 325.24 327.80 2.56 1.00

8.50 324.22 331.86 7.64 7.09 324.21 326.34 2.14 1.41

10.01 320.44 334.42 13.98 7.63 320.42 325.42 4.99 2.38

9.28 319.73 335.72 15.99 6.34 319.71 324.16 4.45 2.94

10.41 316.72 338.42 21.70 6.44 317.34 323.03 5.69 3.97

10.23 314.95 340.06 25.12 5.61 314.91 321.89 6.98 4.62

10.77 312.43 342.44 30.01 5.25 312.39 320.71 8.33 5.52

9.78 312.64 339.51 26.87 4.84 312.61 325.16 12.55 4.94

10.24 310.20 341.42 31.22 4.51 310.16 324.73 14.56 5.72

10.77 307.74 343.57 35.83 4.20 307.70 324.38 16.68 6.57

WNet T AC η qIN nHS nCS CPU (s)

23.66 35.56 54.33 75.02 95.75 449.13 472.13 507.20 544.23 582.37 0.014 0.02 0.03 0.04 0.05 1743.4 1778.2 1810.9 1875.5 1915.0 903.1 921.1 938.0 971.5 992.0 2283.5 2313.8 2332.4 2390.7 2415.7 106.3 88.8 157.3 87.6 162.1

High

37

118.80 142.46 190.52 222.80 257.34 623.65 672.97 748.91 803.28 870.99 0.06 0.07 0.08 0.09 0.10 1980.0 2035.1 2381.5 2475.5 2573.4 1025.6 1054.1 1233.6 1282.3 1333.0 2471.3 2513.0 2909.2 2991.2 3075.3 145.5 166.4 139.2 186.8 173.1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

mal efficiency results within this efficiency range. 10,18,21 As it can be observed from Table 5 the thermal efficiency takes the lower limit value in each optimization solution except for the first value. This is caused by the ε-constraint method which forces the algorithm to converge, featuring in most cases the minimum value of thermal efficiency. Another important fact is that all the resulting mixtures are binary. In the first 7 points the binary mixture is made out by n-butane (zR-600a ) and cyclopropane (zHC-270 ). The base of the mixture is zR-600a with values above 79% mole percent. The other resulting binary mixture is made out by Chlorodifluoromethane (zR-22 ) and Octafluorocyclobutane (zR-C318 ). In the three points where this mixture takes place the composition is the same 0.999 mole fraction of zR-C318 and 0.001 mole fraction of zR-22 . Besides, in 7 of 10 mixtures the principal component has a composition of 0.999 per mole, which in practical terms is a pure component. This can be due to several factors. The fluid featuring larger composition is the one who possess the best performance with respect to other components. However, there is no a trend in the composition of the mixture. Because in some points the binary mixture is almost pure but in another ones the component with the smaller composition increases its molar fraction slightly. Also it can be noticed that ∆P in the cycle increments as thermal efficiency does. Nevertheless, there exists a decrease when η is equal to 0.08. The reason why ∆P increases is strongly related to the temperatures of HEX. Namely, in order to obtain high efficiencies it is necessary to operate both pressures and temperatures at high levels. ∆P increases the power required by the pump and raises the temperature of working fluid. This allows a better conversion of heat into power, leading to larger thermal efficiencies. Other issue is the temperature difference ∆Tˇef in the working fluid f on HEX. Increasing ∆TˇHigh also increases the thermal efficiency because a larger amount

of heat (from thermal source) goes into the ORC and consequently it can be turned into power. However, to increase heat input in the cycle, it is also necessary to increase the HS mole flow causing that UtiCost increases. In addition, the temperature difference of f the working fluid in the condenser (∆TˇLow ) also increases, but not in the same magnitude

38


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Page 39 of 64

like in the vaporizer (see Table 5). So, the CS mole flow increases as HS mole flow does because the system is a cycle. This means that the heat input into the system must be retired to return to the initial conditions.

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K Figure 2: Results of the trade-off Pareto Curve The T AC increases in the same way the thermal efficiency does as shown in Figure 2. An important slope change can be noted around the 7-8% range in Pareto Curve. Where the efficiency increases in smaller proportion while the TAC increments its magnitude in larger proportion. Hence, larger efficiencies imply higher costs of the TAC starting from this thermal efficiency point. As shown in this figure, there is not way of increasing η without simultaneously increasingTAC as well. Therefore, the operation of the ORC system lies somewhere in a point along this trade-off curve chosen by the designer. Another way to check how the thermal efficiency impacts process economy is evaluating the equipment sizing. Table 6 shows results of equipment sizes. At first glance, it can be noticed that pump work rises as long as efficiency does. Since pressure level PHigh increases along with thermal efficiency then more work in the pump is needed to reach the high pressures required in the ORC. Likewise, the power produced by the turbine also 39



Table 6: Equipment Sizes of ORC system Epsilon-constraint value of η Var

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

P ump(kW ) 1.23 1.80 2.78 3.70 4.76 5.74 6.82 8.00 9.21 10.50 2 AV ap (m ) 216.84 213.56 233.55 261.47 314.76 367.48 476.65 429.93 538.21 733.91 T urbine (kW) 24.90 37.36 57.11 78.73 100.51 124.54 149.28 198.52 232.00 267.84 ACond (m2 ) 79.76 88.84 130.90 140.73 205.34 262.98 296.34 226.88 327.46 511.56 increases as efficiency does. To get a larger efficiency it is necessary to have larger Net work (WNet ) which is calculated substracting the required pump work to the produced work by the turbine. The heat transfer areas of HEX also increase with thermal efficiency. Nevertheless, when thermal efficiency is equal to 8% the heat transfer areas of both HEX have a significant decrease approximately of 10% for the vaporizer and 24% for the condenser. This issue is relevant for Plant-layout purposes because the TAC does not reduce its value even if heat transfer areas do. Moreover, the heat transfer area of vaporizer is f larger than heat transfer area of condenser in all optimal points. Because ∆TˇHigh remains

7XUELQH :RUN 3URGXFHG N:

f larger than ∆TˇLow in every point then a bigger area in vaporizer is needed.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 64

K

K

(a)

(b)

Figure 3: (a) Effect of η in Heat Transfer Areas, (b) Effect of η in Produced Work by Turbine However, the key issue about equipment sizing is how this affects the process econ40


Page 41 of 64

omy and thermal efficiency. Because η was chosen as ǫ-constraint, the rest of variables in model are selected to achieve the required value of thermal efficiency during the multiobjective optmization. Hence, the equipment sizing depends on η. Figure 3(a) shows the effect of η on the sizes of HEXs. As previously explained, the heat transfer areas increase with η, but they decrease when thermal efficiency is equal around 7%. The reason why heat transfer areas are larger is because a larger amount of heat is transferred from thermal source to ORC getting the required η value. A reason of this decrease could be associated to the change of components in working fluid, because even if still continues being a binary mixture, the thermodynamic properties of components are different and so, the Heat transfer area could be reduced. The effect of thermal efficiency in the turbine size is shown in Figure 3(b). The work produced in ORC increases because ∆P does as efficiency becomes higher. Then, to return the cycle to its initial conditions the turbine produces more power to reduce the pressure. The curve has form of a straight line with a small slope change between the points 7 and 8. Therefore, the relationship between efficiency and the work produced by the turbine is approximately proportional.

3XPS 7XUELQH &RQGHQVHU 9DSRUL]HU

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


K Figure 4: Percentage contribution by equipment to the CapCost 41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

To evaluate the effect of equipment sizing on TAC, a percentage contribution of each equipment to CapCost was developed (see Figure 4). TAC was replaced by CapCost because is the only cost variable that is affected directly by equipment sizing. UtiCost is the cost of both thermal sources, so does not change if the equipment sizes do. The graph shows that Turbine Cost increases from 30% to 60% changing and it is the dominant equipment cost. The vaporizer starts with a given percentage but as η increases its impact on CapCost is decreased. The condenser cost range decreases from 27% to 19.5%. This could be explained if the small increase in the heat transfer area of condenser (regarding the increment in the other equipments like turbine or vaporizer) is considered. The pump was not considered in the analysis because its effect on CapCost is almost null as can be seen on Figure 4. Table 7 shows the results for single-objective optimizations. When T AC is minimized a value of the 451.24 $KUSD is obtained and the thermal efficiency is just 1.25%. If η is maximized a value of 9.3% is obtained but the T AC value raises to 822.7 $KUSD. It should be stressed that the minimization of TAC results in working fluid binary mixtures. Moreover, the maximization of η led to a pure working fluid. These results fully agree with information shown in Table 2 where the same trend is observed: increasing η results in a pure fluid. Figure 5(a) shows the contribution in percentage of both CapCost and UtiCost to the T AC. The UtiCost remains the dominant cost in ORC but the CapCost increases, keeping closer to the UtiCost percentage for large thermal efficiency. Another way to see this effect is by comparing both costs in the same graph (see Figure 5(b)). The meaning of the Figure is clear: increasing thermal efficiency leads to increase both type of costs. The CapCost is affected directly whereas the UtiCost also increases but in smaller magnitude. This is so because for larger efficiency, the heat transfer areas must be larger while molar flows of thermal sources increase slightly without producing a high impact on UtiCost. Finally, the amount of produced WNet in the cycle is shown in Figure 6. It can be noticed the same 42


Page 42 of 64

Page 43 of 64

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Table 7: Single-objective optimization of ORC system Var

Min T AC

Max η

— — — — 0.095 — — — 0.905 — 8.86 325.24 331.60 6.36 7.86 325.24 327.80 2.56 1.00 1.23 216.84 24.90 79.76 23.66 449.13 0.013 1743.4 903.1 2283.5 152.8

— — — 0.001 — 0.999 — — — — 10.08 308.41 340.79 32.38 4.27 308.37 323.47 15.1 5.81 9.29 474.28 242.28 398.43 232.95 822.70 0.093 2512.6 1301.5 3026.9 175.7

zR-22 zR-134a zR-152a zR-245ca zHC-270 zR-C318 zFC-4-1-12 zR-600 zR-600a zR-601 PHigh Tˇsub,High Tˇover,High f ∆TˇHigh PLow Tˇsub,Low Tˇover,Low f ∆TˇLow ∆P Pump (kW) AV ap (m2 ) Turbine (kW) ACond (m2 ) WNet T AC η qIN nHS nCS CPU (s)

43



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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 64

&DS&RVW 8WL&RVW

K

K

(a)

(b)

Figure 5: (a) Comparison between Percentage of CapCost and UtiCost in T AC, (b) Comparison between the cost of CapCost and UtiCost slope change in the optimal points as in Figure 2. This means than both the T AC and WNet keep proportional with thermal efficiency changes. Our results clearly indicate that the best way of designing ORC systems is by considering a true multiobjective optimization approach where the two conflicting design goals (TAC and η) are simultaneously taken into account. It is up to the designer to pick an operating point along the Pareto trade-off curve that best suites his/her target design goals.

Conclusions A thermo-economic optimization model for ORC systems was developed in this work. The composition of the mixture working fluid and operating conditions, were calculated simultaneously to maximize the energy produced and to minimize the T AC in ORC systems. The problem was solved as a multi-objective optimization Non Linear Programming problem using the ε-constraint method. In general, the results show that a binary mixture as working fluid offers the best

44


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3URGXFHG :1HW N:

Page 45 of 64

K Figure 6: Produced WNet in ORC performance of the ORC system around the region of low thermal efficiency. Also we found that there exists a relationship between the two objective functions employed in this work. Moreover, the T AC of the ORC system could be arbitrarily small assuming that we are willing to reduce thermal efficiency and therefore reducing the power of the system. It is important to remark that to achieve high thermal efficiency requires large TAC values, as expected. Another important point was that when T AC doubles its value, the generated power increases 10 times its value. So, it might be easy to enhance the cycle efficiency without incurring in large T AC increase and thus reducing the price of the generated energy. A general conclusion from this work is that optimal designs of conflicting goals require the use of true multiobjective optimization formulations from which the designer can choose the best operating point.

45



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Page 46 of 64

Notation αif e

EOS variable

dimensionless

α ˇf e

EOS mixture variable

dimensionless

αif e

Molar partial α ˆf e

dimensionless

αLat

Latini correlation parameter

dimensionless

αRc,i

Riedel CSP parameter

dimensionless

βif e

Pure EOS variable

dimensionless

βˆf e

Mixture EOS variable

dimensionless

βLat


dimensionless

γif Le

Activity coefficient

dimensionless

C γif pe

Combinatorial activity coefficient

dimensionless

R γif pe

Residual activity coefficient

dimensionless

γLat


dimensionless

γπ ∗

Derivative of Gibbs free energy respecting π ∗

dimensionless

(ln γif pe )′T

ln Activity coefficient temperature derivative

dimensionless

Γkf e

Residual activity contribution for groups of pure i

dimensionless

Γkf pe

Residual activity contribution mixture of groups

dimensionless

Γi

Reduced, inverse thermal conductivity

W [m K]−1

ˆ eLV ∆H

Mixture vaporization-enthalpy change

kJ kmol−1

∆hLV b,i

Normal boiling point pure vaporization enthalpy

kJ kmol−1

∆hLV ie

Pure vaporization-enthalpy change

kJ kmol−1

∆HCS

Heat sink overall enthalpy change

kJ kmol−1

∆HHS

Heat source overall enthalpy change

kJ kmol−1

∆HCS,f

Heat sink enthalpy change

kJ kmol−1

∆HHS,f

Heat source enthalpy change

kJ kmol−1

∆Ps,e

Pressure drop on shell-side

bar

(i)

46


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


∆Pt,e

Pressure drop on tube-side

bar

∆Pmin

Minimum Pressure change of the cycle

bar

∆T CS

Cooling medium exchanger temperature change

K

∆T HS

Heating medium exchanger temperature change

K

∆Tef

Increment/Decrement in working fluid temperature

K

taking from thermal source in HEX S ∆Tmin

Profile minimum temperature difference

K

4 δr,i

Dimensionless dipolar moment

dimensionless

η

Thermal efficiency of cycle based on first law thermodynamic

dimensionless

ηSP

Pump isentropic efficiency

dimensionless

ηST

Turbine isentropic efficiency

dimensionless

θm

UNIFAC area fraction variable for pure i

dimensionless

θnf pe

UNIFAC area fraction variable

dimensionless

κi

Correction factor por high polar components function

dimensionless

λ0,e

Dilute-gas limit of thermal source function

Function dependent

λ1,e

Finite density of thermal source function

Function dependent

λ2,e

Critical enhancement of thermal source function

Function dependent

ê λ

Mixture thermal conductivity

W [m K]−1

λSe

Thermal conductivity of thermal source

W [m K]−1

λi,e

Pure component thermal conductivity

W [m K]−1

λr i,e

Dimensionless reduced thermal conductivity

dimensionless

(λΓ)int, i

Internal energy function of thermal conductivity

W [m K]−1

(λΓ)tr, i

Traslational energy function for thermal conductivity

W [m K]−1

Λij

Interaction factor for mixture thermal conductivity

dimensionless

µ0,e

Dilute-gas limit of viscosity function

Function dependent

µ1,e

Finite density of viscosity function

Function dependent

µ2,e

Critical enhancement of viscosity function

Function dependent

(i)

47



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Page 48 of 64

µ ê

Mixture viscosity

cP

µi,e

Viscosity of pure component

cP

µSe

Viscosity of the thermal source

Kg [ms]−1

νk

UNIFAC group m frequency on molecule i

dimensionless

νsp

Specific volume of the thermal source

m3 Kg−1

π

Mathematic Constant

dimensionless

π∗

Pressure dimensionless IAPWS parameter

dimensionless

ρ¯e

Density dimensionless function

dimensionless

ρê

Mass density of mixture working fluid

kg m−3

ρSe

Mass density of the Thermal source

kg m−3

τ∗

Temperature dimensionless IAPWS parameter

dimensionless

φîf e

Mixture fugacity coefficient

dimensionless

φsat if e

Pure saturation fugacity coefficient

dimensionless

φij

Interaction factor for mixture viscosity

dimensionless

ψbR,i


dimensionless

ψf

Vaporization fraction

dimensionless

Ψkmf e

UNIFAC binary interaction variable

dimensionless

Ω 1 , Ω2

Objective functions

Function dependent

ων,i

Collision integral term in mixture viscosity

dimensionless

ωi

Acentric factor

dimensionless

A1

PSRK parameter

dimensonless

A∗


dimensonless

A+ R,i

Riedel CSP vapour pressure parameter

dimensonless

Acz,if e

Chanz-Zhao polynomial

dimensonless

AHE,e

Heat transfer area of HEX

m2

ALat, i

Function of Latini’s correlation

dimensonless

(i)

List of symbols

48


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AN eu

Constant for Neufeld correlation

dimensonless

a0

Chang-Zhao liquid volume parameter

dimensonless

a1


dimensonless

a2


dimensonless

a3


dimensonless

a4


dimensonless

as,e

Shell-side cross flow area

m2

at,e

Tube-side cross flow area

m2

a ˆ′e

Mixture a temperature derivative

cm6 bar mol−2 K−1

a′ie

Pure a temperature derivative


amn

UNIFAC binary interation paramter

dimensonless

+ BR,i


dimensonless

Bcz,i


dimensonless

BN eu


dimensonless

Bs,e

Baffle spacing

m

ˆb

Mixture EOS variable

cm3 mol−1

bcz,0


dimensonless

bcz,1


dimensonless

bi

EOS parameter

cm3 mol−1

+ CR,i


dimensonless

C0

Base purchase cost

USD

C1,i

Ideal gas capacity function parameter

kJ kmol−1

C2,i


kJ kmol−1

C3,i


K

C4,i


kJ kmol−1

C5,i


K

Ccz


dimensonless

49



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Page 50 of 64

CN eu


dimensonless

IG Cp,S

Source ideal gas heat capacity

kJ kmol−1

IG Cp,i

Pure fluid ideal gas heat capacity

kJ kmol−1

CpL,i

Thermal fluid liquid heat capacity

kJ kmol−1

CpL,S

Source liquid heat capacity S=HS or CS

kJ kmol−1

Cs

Clearence in tubes

m

CapCost

Capital Cost

USD

cω,i

Acentric factor function parameter

dimensonless

+ DR,i


dimensonless

Db,e

Bundle tubes diameter

m2

Dcz


dimensonless

Deq,s

Shell equivalent diameter

m

Deq,t

Tube equivalent diameter

m

DN eu


dimensonless

Ds,e

Shell diameter

m

din

Inner tube diameter

m

dout

Outer tube diameter

m

EN eu


dimensonless

Eps,i

Przezdziecki and Sridhar Parameter

dimensonless

FN eu


dimensonless

FT,e

Temperature factor correction

dimensonless

F ci

Parameter of factor correction for polar substances

Function dependent

ff,e

Friction factor in tube-side

dimensonless

ff s,e

Friction factor in shell-side

dimensonless

(Gee )′T

Gibbs excess temperature derivative


Gef pe

Gibbs excess function

kJ kmol−1 for energy

G˙ s,e

Mass flux in shell-side

kg [m2 s]−1 50


Page 51 of 64

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G˙ t,e

Mass flux in tube-side

kg [m2 s]−1

ˇ de h

Sub-cool liquid enthalpy

kJ kmol−1

ˇ de H

Over-heat vapor enthalpy

kJ kmol−1

ê h

Mixture saturated liquid enthalpy

kJ kmol−1

ê H

Mixture saturated vapor enthalpy

kJ kmol−1

ˆR H e

Mixture residual enthalpy

kJ kmol−1

Hee

Excess enthalpy

kJ kmol−1

Hiw

Coefficients for µ0,e

dimensonless

Hijw

Coefficients for µ1,e

dimensonless

R HHS,f

Heat source residual enthalpy

kJ kmol−1

R HHS,OUT

Heat source outlet residual enthalpy

kJ kmol−1

HieR

Pure residual enthalpy

kJ kmol−1

hsp−s,e

Single phase wall heat transfer coefficient in shell-side

W [m2 K]−1

hsp−t,e

Single phase wall heat transfer coefficient in tube-side

W [m2 K]−1

h0

Reference enthalpy

kJ kmol−1

Ii

Coefficient for derivative of gibbs free energy

dimensonless

Ji


dimensonless

K1T , K2T , K3T

Parameters for Turton’s cost correlation

dimensonless

K1

Coefficient of Bundle tubes diameter correlation

dimensonless

Kif e

Equilibrium ratio

dimensonless

li

UNIFAC parameter

dimensonless

Lw ij

Coefficients for λ1,e

dimensonless

Lw k

Coefficients for λ0,e

dimensonless

Ltube

Tube length

m

Mi

Molecular weight of pure component i

g-mol or kg-kmol

m ˙ t,e

Mass flow in tube-side

kg s−1

M LDTe

Mean logarithmic delta temperature

K

51



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Page 52 of 64

NC

Number of components

dimensonless

NB,e

Number of baffles per HEX

dimensonless

NG

Number of groups

dimensonless

Ns,e

Numbers of shells in series/parallell per HEX

dimensonless

Nsp

Number of shell passes

dimensonless

NT,e

Numbers of tube in HEX

dimensonless

Ntp

Number of tube passes

dimensonless

N us,e

Nusselt number in shell-side

dimensonless

N ut,e

Nusselt number in tube-side

dimensonless

n1

Coefficient of Bundle tubes diameter correlation

dimensonless

nCS

Heat sink fluid mole

mole per working fl

nHS

Heat source fluid mole

mole per working fl

nDB

Exponent parameter for Dittus-Boelter equation

dimensonless

nπi


dimensonless

Pifsate

Saturation Pressure

bar

P0

Reference pressure

bar

Pc,i

Critical pressure

bar

Pe

Pressure

bar

PeS

Pressure of thermal source

bar

P rs,e

Prandtl number in shell-side

dimensonless

P rt,e

Prandtl number in tube-side

dimensonless

Pt

Tube pitch diameter

m

QHE,e

Heat flow in HEX

kJ s−1

Qk

UNIFAC parameter

dimensonless

QR,i


dimensonless

qIN

Input heat

kJ kmol−1

qOUT

Output heat

kJ kmol−1 52


Page 53 of 64

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qi

UNIFAC parameter

dimensonless

Rg

Ideal gas constant

kJ kmol−1 K−1 for en

cm3 bar mol−1 K−1 f Res,e

Reynolds number in shell-side

dimensonless

Ret,e

Reynolds number in tube-side

dimensonless

ri

UNIFAC parameter

dimensonless

Rk

UNIFAC parameter

dimensonless

Sˇde

Sub-cool liquid entropy

kJ kmol−1 K−1

Sˇde

Over-heat vapor entropy

kJ kmol−1 K−1

Sê

Mixture saturated vapor entropy

kJ kmol−1 K−1

See

Excess entropy

kJ kmol−1 K−1

SêR

Mixture residual entropy

kJ kmol−1 K−1

R Sie

Pure residual entropy

kJ kmol−1 K−1

SˆS

Mixture saturated liquid entropy

kJ kmol−1 K−1

Ti∗

Reduced temperature function

dimensonless

T CS,IN

Cold-fluid inlet temperature

K

T CS,OUT

Cold-fluid outlet temperature

K

TfCS

Heat sink fluid temperature

K

T HS,IN

Hot-fluid inlet temperature

K

T HS,OUT

Hot-fluid outlet temperature

K

TfHS

Heat source fluid temperature

K

T0

Reference temperature

K

Tb,i

Normal boiling temperature

K

Tˇde

Over-heat/sub-cool Temperature

K

T¯e

Reduced dimensionless temperature of

dimensonless

thermal source based on T mSe avg

Te

Avarage temperature

K 53



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Page 54 of 64

Tf e

Working fluid temperature

K

Tif p

Freezing point temperature

K

T mipe

Mean temperature of pure component

K

Tr,if e

Reduced temperature

dimensonless

Trm,ipe

Reduced temperature based on T mipe

dimensonless

T AC

Total Annual Cost

USD

T me

Mean arithmetic temperature of working fluid

K

T mSe

Mean arithmetic temperature of thermal source

K

Uc,e

Global heat transfer coefficient

W [m2 K]−1

UtiCost

Utility Cost

USD

vc,i

Critical volume

cm3 mol−1

vL,if e (P )

Liquid molar volume

cm3 mol−1

sat vL,if e

Saturation liquid volume

cm3 mol−1

Vps,i

Przezdziecki and Sridhar Parameter

dimensonless

vt,e

Fluid velocity inside tubes

m s−1

Vˆm

Mixture molar volume

cm3 mol−1

V mi

Pure component molar volume

cm3 mol−1

V mfi p

Pure molar volume at freezing point

cm3 mol−1

Wnet

Net work

kW

wi

Mass fraction

dimensonless

xif pe

Liquid-phase mole fraction

dimensonless

Xmf e

Group mole fraction

dimensonless

Xm

Group mole fraction for pure i

dimensonless

YT

Size or capacity of process equipment for cost correlation

m2 or kW

yi

Gas-phase mole fraction

dimensonless

Zc,i

Critical Compressibility factor

dimensonless

Zˆf e

Mixture compressibility factor

dimensonless

(i)

54


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Zie

Pure compressibility factor

dimensonless

zi

Global composition

dimensonless

55



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References 1. Hung, T.; Shai, T.; Wang, S. A review of organic Rankine cycles (ORCs) for the recovery of low-grade waste heat. Energy 1997, 22, 661–667. 2. Hung, T. Waste heat recovery of organic Rankine cycle using dry fluids. Energy Conversion and Management 2001, 42, 539–553. 3. Kalina, A. I. Combined cycle and waste heat recovery power systems based on a novel thermodynamic energy cycle utilizing low-temperature heat for power generation. 1983 Joint Power Generation Conference: GT Papers. 1983; pp V001T02A003– V001T02A003. 4. Chen, H.; Goswami, D. Y.; Stefanakos, E. K. A review of thermodynamic cycles and working fluids for the conversion of low-grade heat. Renewable and sustainable energy reviews 2010, 14, 3059–3067. ˘ Zhaeseleer, ´ 5. Walraven, D.; Laenen, B.; DâA W. Comparison of thermodynamic cycles for power production from low-temperature geothermal heat sources. Energy Conversion and Management 2013, 66, 220–233. 6. Quoilin, S.; Van Den Broek, M.; Declaye, S.; Dewallef, P.; Lemort, V. Techno-economic survey of Organic Rankine Cycle (ORC) systems. Renewable and Sustainable Energy Reviews 2013, 22, 168–186. 7. Guo, T.; Wang, H.; Zhang, S. Fluids and parameters optimization for a novel cogeneration system driven by low-temperature geothermal sources. Energy 2011, 36, 2639– 2649. 8. Wang, Z.; Zhou, N.; Guo, J.; Wang, X. Fluid selection and parametric optimization of organic Rankine cycle using low temperature waste heat. Energy 2012, 40, 107–115.

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enhancement for a low-grade heat source. International Journal of Energy Research 2013, 37, 857–867. 18. Li, S.; Dai, Y. Thermo-Economic Analysis of Waste Heat Recovery ORC Using Zeotropic Mixtures. Journal of Energy Engineering 2014, 141, 04014050. 19. Rusev, T. Comparative Study of Different Organic Rankine Cycle Models: Simulations and Thermo-Economic Analysis for a Gas Engine Waste Heat Recovery Application. 2015. 20. Thierry, D. M.; Flores-Tlacuahuac, A.; Grossmann, I. E. Simultaneous optimal design of multi-stage organic Rankine cycles and working fluid mixtures for lowtemperature heat sources. Computers & Chemical Engineering 2016, 89, 106–126. 21. Lecompte, S.; Huisseune, H.; van den Broek, M.; De Schampheleire, S.; De Paepe, M. Part load based thermo-economic optimization of the Organic Rankine Cycle (ORC) applied to a combined heat and power (CHP) system. Applied Energy 2013, 111, 871– 881. 22. Dimitrova, Z.; Lourdais, P.; Maréchal, F. Performance and economic optimization of an organic rankine cycle for a gasoline hybrid pneumatic powertrain. Energy 2015, 86, 574–588. 23. Wang, J.; Yan, Z.; Wang, M.; Li, M.; Dai, Y. Multi-objective optimization of an organic Rankine cycle (ORC) for low grade waste heat recovery using evolutionary algorithm. Energy Conversion and Management 2013, 71, 146–158. 24. Feng, Y.; Zhang, Y.; Li, B.; Yang, J.; Shi, Y. Sensitivity analysis and thermoeconomic comparison of ORCs (organic Rankine cycles) for low temperature waste heat recovery. Energy 2015, 82, 664–677.

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25. Imran, M.; Park, B. S.; Kim, H. J.; Lee, D. H.; Usman, M.; Heo, M. Thermo-economic optimization of Regenerative Organic Rankine Cycle for waste heat recovery applications. Energy Conversion and Management 2014, 87, 107–118. 26. Quoilin, S.; Declaye, S.; Tchanche, B. F.; Lemort, V. Thermo-economic optimization of waste heat recovery Organic Rankine Cycles. Applied thermal engineering 2011, 31, 2885–2893. 27. Schuster, A.; Karellas, S.; Kakaras, E.; Spliethoff, H. Energetic and economic investigation of Organic Rankine Cycle applications. Applied thermal engineering 2009, 29, 1809–1817. 28. Yu, G.; Shu, G.; Tian, H.; Wei, H.; Liu, L. Simulation and thermodynamic analysis of a bottoming Organic Rankine Cycle (ORC) of diesel engine (DE). Energy 2013, 51, 281–290. 29. Feng, Y.; Hung, T.; Greg, K.; Zhang, Y.; Li, B.; Yang, J. Thermoeconomic comparison between pure and mixture working fluids of organic Rankine cycles (ORCs) for low temperature waste heat recovery. Energy Conversion and Management 2015, 106, 859– 872. 30. Li, Y.-R.; Du, M.-T.; Wu, C.-M.; Wu, S.-Y.; Liu, C. Potential of organic Rankine cycle using zeotropic mixtures as working fluids for waste heat recovery. Energy 2014, 77, 509–519. 31. Heberle, F.; Preißinger, M.; Brüggemann, D. Zeotropic mixtures as working fluids in Organic Rankine Cycles for low-enthalpy geothermal resources. Renewable Energy 2012, 37, 364–370. 32. Calise, F.; Capuozzo, C.; Carotenuto, A.; Vanoli, L. Thermoeconomic analysis and offdesign performance of an organic Rankine cycle powered by medium-temperature heat sources. Solar Energy 2014, 103, 595–609. 59



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44. Riedel, L. Liquid density in the saturated state. extension of the theorem of corresponding states II. Chem.-Ing.-Tech 1954, 26, 259–264. 45. Cooper, J.; Dooley, R. Revised release on the IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam. The International Association for the Properties of Water and Steam, 2007. 46. Cooper, J.; Dooley, R. Release of the IAPWS formulation 2008 for the viscosity of ordinary water substance. 2008. 47. Daucik, K.; Dooley, R. Release on the IAPWS Formulation 2011 for the Thermal Conductivity of ordinary water substance. 2011. 48. Green, D.; Perry, R. Perry’s Chemical Engineers’ Handbook; McGraw-Hill New York, 2007; Vol. 8. 49. Kern, D. Q. Process heat transfer; Tata McGraw-Hill Education, 1950. 50. Jenkins, S. Economic indicators: CEPCI. Chemical Engineering Essentials for the CPI Professional.< http://www. chemengonline. com/economic-indicators-cepci 2015, 51. Heberle, F.; Brüggemann, D. Thermo-economic analysis of zeotropic mixtures and pure working fluids in organic Rankine cycles for waste heat recovery. Energies 2016, 9, 226. 52. Caramia, M.; Dell’Olmo, P. Multi-objective management in freight logistics: Increasing capacity, service level and safety with optimization algorithms; Springer Science & Business Media, 2008. 53. Deb, K. Multiobjective Optimization using evolutionary algorithms; Wiley, 2001.

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Page 62 of 64

Appendix of Tables This appendix shows all the coefficients, parameters, and Constants used by transport properties methods in sections and . Table 9: Constants for collision integral term AN eu 1.16145

BN eu 0.14874

CN eu 0.52487

DN eu 0.77320

EN eu 2.16178

FN eu 2.43787

Table 10: Coefficients Ii , Ji and nπi for γπ∗ i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Ii 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2

Ji -2 -1 0 1 2 3 4 5 -9 -7 -1 0 1 3 -3 0 1

nπi 1.4632971213167E-01 -8.4548187169114E-01 -3.7563603672040 3.3855169168385 -9.5791963387872E-01 1.5772038513228E-01 -1.6616417199501E-02 8.1214629983568E-04 2.8319080123804E-04 -6.0706301565874E-04 -1.8990068218419E-02 -3.2529748770505E-02 -2.1841717175414E-02 -5.2838357969930E-05 -4.7184321073267E-04 -3.0001780793026E-04 4.7661393906987E-05

i 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Ii 2 2 3 3 3 4 4 4 5 8 8 21 23 29 30 31 32

Ji 3 17 -4 0 6 -5 -2 10 -8 -11 -6 -29 -31 -38 -39 -40 -41

Table 11: Coefficients HiW for µ0,e i 0 1 2 3

HiW 1.6775200 2.2046200 0.6366564 -0.241605

62


nπi -4.4141845330846E-06 -7.2694996297594E-16 -3.1679644845054E-05 -2.8270797985312E-06 -8.5205128120103E-10 -2.2425281908000E-06 -6.5171222895601E-07 -1.4341729937924E-13 -4.0516996860117E-07 -1.2734301741641E-09 -1.7424871230634E-10 -6.8762131295531E-19 1.4478307828521E-20 2.6335781662795E-23 -1.1947622640071E-23 1.8228094581404E-24 -9.3537087292458E-26

Page 63 of 64

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 12: Coefficients HijW for µ1 i 0 1 2 3 0 1 2 3 5 0 1 2 3 4 0 1 0 3 4 3 5

j 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 4 4 5 6 6

HijW 0.520094 0.0850895 -1.08374 -0.289555 0.222531 0.999115 1.88797 1.26613 0.120573 -0.281378 -0.906851 -0.772479 -0.489837 -0.257040 0.161913 0.257399 -0.0325372 0.0698452 8.72102E-3 -4.35673E-3 -5.93264E-4

Table 13: Coefficients LW k for λ0 k 0 1 2 3 4

LW k 0.002443221 0.01323095 0.006770357 -0.003454586 0.0004096266

Table 14: Coefficients LW ij for λ1 j

0

1

i 0 1.603973570 -0.646013523 1 2.33771842 -2.78843778 2 2.19650529 -4.54580785 ´ 3 âLŠ1.21051378 1.60812989 4 â´LŠ2.7203370 4.57586331

2

3

0.111443906 1.53616167 3.55777244 -0.62117814 -3.18369245

0.102997357 -0.463045512 -1.40944978 0.071637322 1.1168348

63


4

5

-0.0504123634 0.00609859258 0.083282702 -0.007192012 0.275418278 -0.020593882 0 0 -0.19268305 0.012913842


Table of Contents Graphic

7$& . 86'

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

K

64


Page 64 of 64

Thermo-Economic Multiobjective Optimization of a LOW Temperature

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