Simultaneous Optimal Design of Organic Mixtures and Rankine

Article pubs.acs.org/IECR

Simultaneous Optimal Design of Organic Mixtures and Rankine Cycles for Low-Temperature Energy Recovery David Paul Molina-Thierry and Antonio Flores-Tlacuahuac* Departamento de Ingeniería y Ciencias Químicas, Universidad Iberoamericana, Prolongación Paseo de la Reforma 880, México D.F. 01219, México ABSTRACT: Because of pollution issues and a forecasted scarcity of fossil fuels, there exists a strong need to consider alternative and sustainable energy sources. Among all the types of alternative energies, normally heat recovery from lowtemperature sources is not considered seriously for this purpose because of the poor performance of the conversion to power method. The conventional Rankine cycle that uses water as the working fluid is the most simple way for heat to power conversion. But for low-temperature applications, the conventional Rankine cycle will not achieve good results. In this work, we propose to replace water with a proper optimal combination of a priori selected set of organic fluids as the working fluid. Therefore, the determination of the right type of organic components and the composition of the mixture become decision variables. Moreover, because strong interactions exist between the selection of the type and composition of the organic components and the operating conditions of the Rankine cycle for a given fixed process flowsheet configuration, a simultaneous solution approach will be sought. Hence, by approaching the design problem in a simultaneous rather than in a sequential manner, improved optimal solutions were achieved. The sequential design problem is posed as a nonlinear optimization problem. The proposed methodology is illustrated using three case studies.

1. INTRODUCTION Energy production around the world has left a footprint in the environment that cannot be easily removed nor mitigated. Because energy demand will only grow in future years, this issue is of great importance for the scientific community. As a result, present research on energy utilization and generation focuses on the development and utilization of alternative energy sources that are available at both low cost and environmental impact. On this matter, low-temperature heat sources are an optional source of energy. Traditionally, low-temperature heat sources are included in waste industrial processing streams, solar radiation streams, and geothermal streams. Roughly speaking, we can consider a processing stream as a lowtemperature heat source if its operating temperature happens to be less than 200 °C. Most of these streams are considered to be wasted energy sources because the amount of heat that can be recovered and transformed into power by traditional methods is not large. The preferred way of recovering heat from low-temperate heat sources is through the Rankine cycle, where a low-temperature stream releases heat to a working fluid. Once vaporized, the working fluid is used for power production in a turbine. Due to economic factors, water is commonly one of the most widely used working fluids. This is true especially for processing streams well above 200 °C. However, when the temperature of the heat source decreases below this temperature target, the power production gradually becomes inefficient.1,2 Consequently, modifications to the conventional Rankine cycle are required in order to increase the power generation performance. The first logical modification consists in using a set of alternative working fluids featuring better energy recovery characteristics from lowtemperature heat sources. Ideally, this set of working fluids should display target physical properties such as low-boiling point and high vaporization enthalpy. Among the set of © 2015 American Chemical Society

compounds that can be used for low-temperature energy recovery, organic fluids had shown some of the better performance characteristics. In fact, when an organic fluid is used as the working fluid in the Rankine cycle, the process has been called an Organic Rankine Cycle (ORC).3,4 In this regard, the selection of a proper organic fluid plays an important role on the performance of the cycle,5 as well as the operating parameters.6 Furthermore, the effect of both decisions (organic fluid selection and operating conditions) has been acknowledged.7−13 Moreover, the selection of the working fluid for the ORC by means of computer-aided molecular design and group contribution methods in an optimization model has been recently addressed.14,15 Another way of increasing the efficiency of the Rankine cycle is through the implementation of a supercritical cycle,16,17 where the working fluid is compressed above its critical pressure, and the consecutive heating in the vaporizer takes place without the constant temperature boiling section, allowing the resulting temperature profiles of the heat source and the working fluid to be better matched. However, the supercritical cycle requires operating at high pressure, which may lead to difficulties in operation18 and can result in high operating costs. Likewise, the use of a multicomponent mixture as the working fluid19 is another possibility for improving the efficiency of the Rankine cycle for energy recovery from low-temperature heat sources. Angelino et al.20 explored the theoretical application of multicomponent mixtures for the Rankine cycle. When a constant-pressure phase change of a pure fluid takes place, Received: Revised: Accepted: Published: 3367

September 17, 2014 March 12, 2015 March 12, 2015 March 12, 2015 DOI: 10.1021/ie503675v Ind. Eng. Chem. Res. 2015, 54, 3367−3383

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Industrial & Engineering Chemistry Research

Figure 1. Rankine cycle flowsheet.

works acknowledged the importance of taking into account simultaneously the operating conditions of the Rankine cycle and even the optimization of the mixture composition as a way to obtain improved optimal solutions. However, the modeling and optimization of an organic mixture-based Rankine cycle that simultaneously accounts for the thermodynamic description of the mixture and the operating conditions have not been addressed. The contributions of this work are as follows: (a) determination of the type of components in the mixture from a previously chosen set of organic compounds, (b) determination of the optimum amount of each component in the mixture (i.e., optimum composition of the mixture), and (c) determination of the optimum processing conditions of the Rankine cycle. Moreover, the above decisions will be approached using a simultaneous framework, meaning that the past three items will exploit the natural interactions among such problems to improve optimality conditions. It should be stressed that because in this work we assume that the chemical nature of the components has been previously chosen no computer-aided molecular design (CAMD) techniques will be used. The use of CAMD for product design can be consulted elsewhere.14,15

the temperature is also constant; whereas for a mixture, phase change occurs at a range of temperatures and compositions. This nonisothermal phase change of the mixture allows a better match between the temperature profiles of the working fluid and the heat source, and it strongly depends on the composition of the mixture. Many studies have been done on the performance of mixtures in Rankine cycles accordingly.21−29 As a general conclusion, the mixture-based organic cycle always shows an improvement of the performance over the cycle that uses a pure component, and this depends on variables like composition, phase equilibrium temperature gradients, pressure ratio, and superheating degree. These works also recognize the potential of organic mixtures featuring optimal composition on the performance of the Rankine cycle. Along this line, Herbele et al.30 concluded that the nonisothermal phase change of the mixtures increases the efficiency and decreases the irreversibilities of the system, making the temperature gradient one of the most valuable operating parameters in the performance of the Rankine cycle. However, Chys et al.31 states that the maximization of phase equilibrium temperature gradients will not necessarily achieve the optimal thermal match with the heat source/sink temperature profiles, and thus, efficiency can decrease. In summary, a suitable temperature gradient on a particular heat source has the potential to achieve the maximum efficiency of the cycle. Following the idea of mixture design, Papadopoulos32 formulated and solved a two-stage optimization problem that sequentially identifies candidates (from a previously selected set of functional groups) for a mixture and their optimal concentration using a stochastic optimization approach (simulated annealing). However, the deployment of mixtures for lowtemperature energy recovery leads to some technical issues. Among them, we can mention the variation of the global composition of the mixture by means of leakages or composition shifts due to the two phase holdup in the heat exchangers.33 Most of the previously addressed published works account for the characterization of mixtures and focus on determining the composition of binary and even ternary mixtures. Some of these

2. PROBLEM DEFINITION For power generation from a heat source, the Rankine cycle is a conceptually simple process that relies on the use of a fluid that undergoes a series of thermodynamic state changes. As shown in Figures 1 and 2, the Rankine cycle, whether a pure substance or a mixture is used as the working fluid, can be summarized in four main steps: (1) compression of the working liquid up to a highpressure level (high), (2) constant pressure heating to a superheated vapor state, (3) expansion of the vapor to a lowpressure level (low), and (4) constant pressure cooling of the gas until condensation conditions are reached. Phase change takes place only in steps 2 and 4 (represented by the squares and triangles in Figure 2). The expansion of the vapor in the turbine results in work output that is used in a generator to produce 3368

DOI: 10.1021/ie503675v Ind. Eng. Chem. Res. 2015, 54, 3367−3383

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Figure 2. Rankine cycle thermodynamic diagrams. (a, b): Pure component as working fluid. (c, d): Mixture as working fluid.

power. As the pressure difference increases, the work production will rise, but this comes with an increment in the work input of the compression process (1). Moreover, for every constant-pressure heat transfer stage in the cycle, the two-phase region is crossed. When a pure substance is used as the working fluid (Figure 2a,b), this process starts and finishes at the saturation temperature (marked with squares). In the case of a mixture (Figure 2c,d), these points differ from each other giving rise to the bubble (square) and dew (triangle) temperature points. Beyond the dew temperature (saturation temperature for a pure component), successive heating will lie on the vapor-only or superheated region. Whether or not operation in this region occurs will impact the amount of heat required and the overall work production. When no superheating takes place and point 3 lies at the same place of the saturation/dew point, the cycle is called saturated. Additionally, in Figure 2b, the temperature profiles in the segments 2−4 and 1−4 show a horizontal section that is characteristic for a pure fluid constant-pressure vaporization. However, this is not generally true for a mixture of substances as

shown in Figure 2d, where constant-pressure vaporization process takes place within a range of compositions and temperatures. This nonisothermal vaporization property will allow a better thermal match with the hot fluid inside the heat exchanger. Furthermore, the profile and thermodynamic properties depend on the composition, pressure and temperature of the processing system (the cycle). This leads to the problem of simultaneously finding the optimal values of these variables, such that the performance of the Rankine cycle is maximized by means of a model that accounts for the calculation of the thermodynamic properties and cycle constraints. In summary, the problem to be addressed in the present work can be formulated as follows: “For a given set of low-temperature heat source and heat sink processing streams, and given a set of organic working fluids previously chosen, the problem is to select the operating conditions of a Rankine cycle and the composition of a mixture of the organic working fluids simultaneously, such that optimal performance is achieved holding the thermodynamics constraints of the cycle.” 3369


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Figure 3. Thermodynamic-associated sets graphical representation.

(1) The first objective function consists in the maximization of the change of the enthalpy of vaporization at the highpressure level of the cycle. The aim of this objective is to increase the heat requirement for vaporization that in consequence will also increase the overall amount of heat absorbed by the mixture.15 This function will deal mainly with the equilibrium properties of the mixture:

3. PROBLEM FORMULATION Sets. Pure species associated i,j = 1,2,...,NC; Mixture components k,m,n = 1,2,...,NG; UNIFAC functional group Thermodynamic associated f = B,1,2,...,Nψ,D; Vaporization level p = L,V; Phase (liquid,vapor) e = low,high; Pressure level d = sub,subS,over,overS; Superheat(Overheat)/subcool point nonisentropic, isentropic The sets are the indices for which the equations and variables are defined. They can be classified in two groups. One group belongs specifically to the pure species in the mixture: component i and the UNIFAC method functional groups k,m,n. The other group belongs to the sets that are associated with a point inside the thermodynamic cycle (Figure 3): pressure level e, compression/ expansion points d, and phase p and equilibrium internal points f. Note that when an equation is defined over set d the phase equilibrium sets (f,p) are overridden because set d is exclusively for single-phase calculations or properties. Objective Functions. In order to analyze how the optimum mixture-based Rankine cycle is affected both by the nature and proportion of the organic compounds and by the design parameters, five objective functions were addressed. All of them are based on a performance variable or a property of the mixture on the basis of 1 mol of mixture. It also should be noted that no economic objective was considered for this work because it requires the sizing of the equipment. Nevertheless, it will be addressed in future work.

LV ̂ max Ω1 = ΔĤHigh = ĤHigh − hHigh x

(1)

where x stands for the freedom degrees of the model. The change of vaporization enthalpy is calculated from the difference between the enthalpy of the saturated vapor Ĥ e (at the dew temperature TDe) and the enthalpy of the saturated liquid ĥe (at the bubble temperature TBe), both at the pressure level e of the cycle and global molar composition zi. (2) The second objective function concerns the specific net work output of the cycle, i.e., the specific work produced by the turbine minus the work input in the pump; both of them calculated with the nonisentropic pressure change. This is an obvious choice of objective because it is desired to obtain as much net work as possible for a given cycle: max Ω 2 = Wnet x

̌ ̌ = (Ȟ over,High − Ȟ over,Low ) − (hsub,High − hsub,Low ) (2)

where the term in the first parentheses is the turbine work output, and the second one is the pump work requirement. The symbol Ȟ over,e denotes the gas mixture enthalpy of the nonisentropic expansion process of the pressure levels e = High, Low at superheat temperature Ť over,e, and ȟsub,e is 3370


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Figure 4. Mixture-based Rankine cycle temperature−enthalpy profiles for the case C.

⎛ T avg ⎞ ⎟ ηCarnot = ⎜⎜1 − Low avg ⎟ THigh ⎠ ⎝

the liquid mixture enthalpy at the subcooling temperature Ť sub,e. Note that those enthalpy change arrangements were manipulated so a positive value is ensured (instead of Ȟ over,Low−Ȟ over,High); this is also applied in the following efficiency equations. (3) The third objective function is the maximization of the efficiency of the cycle by the first law of thermodynamics. The maximization of this objective will generate the best net output to input ratio of energy. This means that the amount of heat absorbed is taken into account in contrast to objective number 3 that makes no regard to this situation: max Ω3 = ηI

where each average temperature at pressure level e is defined with the absolute heat input/output and the entropy change from those processes: ̌ Ȟ over, e − hsub, e Teavg = ̌ Sover, e − ssub, ̌ e (8) (5) In this case, the aim consists in the minimization of the area inbetween profiles (temperature−enthalpy) of the working fluid and heat source or sink on the heat exchanger (Figure 4, areas formed between HS-High P and CS-Low P). In other words, we intend to use the maximum energy content of the hot source. The objective function Ω5 reads as

(3)

x

where ηI is the first law efficiency defined in terms of the enthalpy changes at the subcooled and superheated points in the cycle: ηI =

̌ ̌ (Ȟ over,High − Ȟ over,Low) − (hsub,High ) − hsub,Low (Ȟ ) − ȟ over,High

min Ω5 = x

sub,High

(4)

x

2 CS 2 ∑ (Tf ,High − T HS f ) + ∑ (Tf ,Low − T f ) f

f

̌ + (Tover,High − T HS,IN)2

where the numerator is the specific net work, and the term in the denominator is the overall heat input to the cycle. This is an indicator of the useful energy output to energy input ratio.34 (4) The fourth objective function is the maximization of the second law efficiency, i.e., the ratio of the first law and the Carnot efficiency of the cycle. However, as observed in ref 35, there are many ways of defining this efficiency, some of them by means of exergy. The approach used in this work was taken from ref 20.

max Ω4 = ηII

(7)

Tavg e

̌ + (Tsub,High − T HS,OUT)2 ̌ ̌ + (T CS,OUT − Tover,Low )2 + (T CS,IN − Tsub,Low )2 (9) CS where THS f and Tf stand for the temperatures of the heat and sink sources, respectively. TS,IN and TS,OUT are the temperature of the heat source (or sink) at the inlet and outlet of the heat exchanger, respectively. Ť de and Tfe are the temperatures of the working fluid at the nonphase equilibrium stages and the temperature at phase equilibrium, respectively. Constraints. (1). Mole Fraction Consistency. The composition of the mixture zi is the composition that would be found on a given nonphase equilibrium section (namely, turbine and pump) of the cycle. For consistency, the summation of the mole fraction ought to be equal to 1:

(5)

The second law efficiency is the ratio of the first law and the Carnot efficiencies of the cycle: ηI ηII = ηCarnot (6)

NC

∑ zi = 1

where ηCarnot is the average Carnot efficiency. Usually this index requires constant temperature levels, but for the Rankine cycle it can be defined from average cycle temperatures as follows:

i

(10)

For the sections of the cycle where phase equilibrium takes place (i.e., vaporizer and condenser), the composition turns out to be a 3371


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Industrial & Engineering Chemistry Research decision variable over the whole vaporization process. With this in mind, any internal two-phase point has its own composition (and so does every phase p). Therefore, the consistency of the mole fractions for every phase, point, and pressure level must be enforced.

li =

R ln γifpe =

NC

∑ xifpe = 1,

∀ f ,p ,e

i

zi(1 − K ife) 1 + ψf (K ife − 1)

= 0,

1 + ψf (K ife − 1)

,

θm(i) =

∀ f ,e

θmfpe =

X m(i) =

∀ i ,f ,e (14)

Pe vL , ife(P)

Pifesat R g Tfe

(15)

C R ln γifpe = ln γifpe + ln γifpe ,

ln

−

= ln

ri N

∑ j C rjxjfpe

ri N

∑ j C rjxjfpe

∑ νk(i)R k k

10 + q ln 2 i

N qi ∑ j C rjxjfpe N ri ∑ j C qjxjfpe

(16a)

∑ xjfpelj ,

∀ i ,f ,p ,e

j

∑ νk(i)Q k , k

m

⎤

θmfpe Ψkmfe ⎥, NG ∑n θnfpe Ψnmfe ⎥⎦

∀ k ,f ,p ,e

N

∑n G Q nX n(i)

∀ i ,m;

,

Q mX mfpe N

∑n G Q nX nfpe

∀ m ,f ,p ,e

,

(16h)

νm(i) N

∑n G νn(i)

,

∀ m ,n ,f ,p ,e (16i)

∀ i ,m;

∑i C xifpeνm(i) N

N

∑i C (xifpe ∑n G νn(i))

,

∀ m ,f ,p ,e (16j)

0.08664 Pe , Tr , ife Pc , i

αife =

0.42748 1 2 [1 + cω , i(1 − Tr0.5 , ife)] , 0.08664 Tr , ife

∀ i ,f ,e (17)

∀ i ,f ,e (18)

(16b)

where Tr,ife is the reduced temperature, Pc,i is the critical pressure of i, and cω,i is a parameter in terms of the acentric factor ωi:

NG

qi =

NG

∑

βife = + li

NC

NG

ri =

∀ i ,f ,p ,e

∀ i ,k ,f ,e

The activity coefficient natural logarithm ln γifpe is calculated from contributions due to a configurational term ln γCif pe and a interaction term ln γRifpe. The parameters Rk and Qk for each group k and also two amn for each couple of groups mn are required. It has to be noted that only eqs 16j-1, 16h-1, and 16f do not depend on the composition. Equations 16e through 16g and 16i are temperature dependent; this means that only the residual part features the temperature derivative. The main limitation of the model is the availability of the amn parameters, so only the substances that can be described readily by the current set of data of UNIFAC were included in the Results section. (5). Fugacity Coefficient. The calculation of the mixture fugacity coefficient was carried out using a cubic equation of state (EOS). The Predictive Soave−Redlich−Kwong (PSRK) EOS39 was chosen because it represents well the vapor-phase behavior. PSRK consists of the combination of SRK−EOS with a Gibbs excess model like UNIFAC.40 The dimensionless parameters for the standard SRK equation for a pure species i are

∀ i ,f ,e

,

m

θm(i)Ψkmfe ⎤ ⎥, NG (i) ∑n θn Ψnmfe ⎥⎦

N

X mfpe =

dP

ϕifê Pe

Q mX m(i)

⎛ a ⎞ Ψmnfe = exp⎜⎜ − mn ⎟⎟ , ⎝ Tfe ⎠

where γif Le is the activity coefficient for the i species at the liquid sat phase, Psat ife is the vapor pressure, ϕife is the fugacity coefficients for the pure species at the saturation pressure, and ϕ̂ ife is the mixture fugacity coefficient. The exponential term is the Pointing factor, where vL,ife is the molar volume of the pure liquid as a function of the pressure. (4). Activity Coefficients: UNIFAC. The activity coefficient has to be calculated with a model, general enough to be used with as many different systems of compounds as possible. The UNIFAC model for predicting the activity coefficient is suitable for this application. This model is based on a group contribution calculation and the UNIQUAC activity coefficients model,38 and it can be summarized in the following group of equations:

C γifpe

(16e)

(16g)

(3). K Values. Most of the systems addressed in this work lead to nonideal mixtures. Hence, the form of the phase equilibrium ratio Ki was chosen to be the Gamma-Phi formulation:37 γifLePifesatϕifesat exp ∫

NG

∑

⎡ NG ln Γkfpe = Q k ⎢1 − ln(∑ θmfpe Ψmkfe) − ⎢⎣ m

(12)

ziK ife

K ife =

∀ i ,f ,p ,e

(16f)

where ψf is the vapor fraction that is set to 0 for f = B and 1 for f = D. Any internal point other than the bubble or dew point has a value of ψf between 0 and 1. Kife is the phase equilibrium ratio. The subsequent mole fractions of the liquid and vapor phase are calculated as follows: zi , ∀ i ,f ,e xifLe = 1 + ψf (K ife − 1) (13) xifVe =

(16d)

ln Γ(kfei)),

⎡ NG ln Γ(kfei) = Q k ⎢1 − ln(∑ θm(i)Ψmkfe) − ⎢⎣ m

(11)

(2). Equilibrium at Bubble and Dew Point. As the working fluid is gradually heated/cooled, it evaporates/condensates and enters the two-phase equilibrium region. The vaporization level is characterized by the vapor to feed ratio ψf (coupled with the Rachford−Rice equation) and provides a way to handle all the vapor−liquid equilibrium conditions. The Rachford−Rice equation reads as follows:36 NC

NG

∑ νk(i)( ln Γkfpe −

∀i

k

i

∑

10 (ri − qi) − (ri − 1), 2

∀i

cω , i = (0.48 + 1.574ωi − 0.176ωi2)

(16c) 3372

(19)


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at the phase equilibrium region. This also must be true for the points outside the two-phase region, which happens to be bounded by the input and output temperatures of the heating and cooling sources:

The reduced temperature is defined with the critical temperature as

Tfe

Tr , ife =

Tc , i

,

∀ i ,f ,e (20)

For the mixture, β̂fe is calculated with the mole fractions of the mixture components: βfê =

S ̌ + ΔTmin ≤ T HS,OUT Tsub,High

(28a)

S ̌ − ΔTmin ≥ T CS,IN Tsub,Low

(28b)

S ̌ + ΔTmin ≤ T HS,IN Tover,High

(28c)

S ̌ − ΔTmin ≥ T CS,OUT Tover,Low

(28d)

NC

∑ xifVeβife ,

∀ f ,e (21)

i

Whereas the mixture parameter α̂ fe is calculated as a function of the composition and a reference Gibbs excess energy. The PSRK model states that this reference Gibbs excess energy is calculated at an hypothetic liquid state with the composition of the vapor phase. In general, the Gibbs excess property can be obtained using the UNIFAC activity coefficients model: E Gfpe

R gTfe

For simplicity of the model, the rigorous calculation of the mixture true critical points is not performed. Nevertheless, to avoid operating beyond the mixture critical point, the operating temperature should not approach the pure components critical points. For that reason the next constraint is enforced:

NC

=

∑ xifpe ln γifpe ,

Tfe < Tc , i ,

∀ f ,p ,e (22)

i

⎛ E 1 ⎜ G fVe + 0.64663 ⎜⎝ R gTfe

NC

∑ xifVe ln i

βfê ⎞ ⎟+ βife ⎟⎠

NC

∑ xifVeαife ,

∀ f ,e

PHigh ≥ PLow + ΔPmin

i

With the previous PSRK mixture parameters, the equation for compressibility factor is a cubic polynomial. Because only the vapor phase root is required, the special implicit arrangement of the cubic polynomial that follows is required: 1 Zfê = 1 + βfê − αfê βfê , Zfê (Zfê + βfê )

ln Pifesat = AR+, i −

∀ f ,e

ln ϕifê =

βfê

(Zfê − 1) − ln(Zfê − βfe) + αife ̅ ln

Zfê

,

AR+, i = − 35Q R , i ,

+ CR+, i ln Tr , ife + DR+, iTr6, ife ,

BR+, i = − 36Q R , i ,

DR+, i = − Q R , i ,

∀ i ,f ,e

CR+, i = 42Q R , i + αRc , i ,

Q R , i = 0.0838(3.758 − αRc , i)

(32)

The parameter αRc,i is calculated from critical parameters:

∀ i ,f ,e

(25)

αRc , i =

where α̅ ife is the molar partial EOS variable α̂ fe, i.e., the composition partial derivative of α̂ fe, and it can be obtained from eq 23 ⎛ ⎞ βfê βife 1 ⎜ ln γ + ln + − 1⎟⎟ + αife , αife ̅ = A1 ⎜⎝ ifVe βife βfê ⎠

Tr , ife

with the coefficients given by

Using this equation and a guess value of 1 for Ẑ fe will lead to the vapor phase root.41 Finally, the vapor phase fugacity coefficient reads as follows: βife

BR+, i

(31)

(24)

Zfê + βfê

(30)

(8). Vapor Pressure: Riedel CSP. The Riedel corresponding states method uses critical and normal properties to calculate the vapor pressure.42 It was chosen because it requires few parameters. The correlation reads as follows:

(23)

Zfê − βfê

(29)

(7). Pressure Levels. For power generation in a turbine, the pressure must be consistent with the pressure levels (i.e., the high pressure level must lie above the low pressure level):

And so, taking GEf Ve for the vapor phase, α̂ fe is given as follows: αfê = −

∀ i ,f ,e

ψbR , i = −35 + 36

(26)

where ln γif Ve is the natural logarithm of the activity coefficient for the hypothetical liquid state with the composition of the vapor phase. (6). Temperature Levels. There must exist a minimum temperature difference between the working fluid and the thermal sources in order to make the heat transfer inside the exchangers feasible. So for the phase equilibrium region, the following constraints apply: ∀f

(27a)

S Tf ,Low − ΔTmin ≥ T CS f ,

∀f

(27b)

0.0838ψbR , i − ln Tb , i /Tc , i

(33)

and the parameter ψbR,i is computed using information regarding the normal boiling temperature Tb,i:

∀ i ,f ,e

S Tf ,High + ΔTmin ≤ T HS f ,

0.3149204ψbR , i + ln(Pc , i/1.0135)

Tc , i Tb , i

+ 42 ln

Tb , i Tc , i

⎛ Tb , i ⎞6 − ⎜⎜ ⎟⎟ ⎝ Tc , i ⎠

(34)

(9). Vapor Pointing Factor: Compressed Volume. The Pointing factor requires the liquid volume as a function of the pressure; usually this is done by means of a reference state that is also estimated from a reasonable correlation or experimental data. The Chang and Zhao43 equation was chosen as the compressed volume function. It is defined in terms of the liquid volume at saturation vsat L,ife, two parameters, and two polynomials. The equation of Chang and Zhao (as a function of the pressure) reads as follows: Bcz , i

where Δ TSmin is the minimum temperature approach, and CS THS f and Tf are the temperatures of the heating/cooling source

vL , ife(P) = 3373

vLsat, ife

Acz , ife Pc , i + Ccz(Dcz − Tr ,ife) (P − Pifesat) Acz , ife Pc , i + Ccz(P − Pifesat)

(35)


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Industrial & Engineering Chemistry Research Table 1. Parameters for Compressed Volume a0

a1

a2

a3

a4

bcz,0

bcz,1

Ccz

Dcz

−170.335

−28.578

124.809

−55.539

130.010

0.165

−0.091

2.718

1.006

with constants Ccz and Dcz and the polynomials Acz,ife and Bcz,i given by a Acz , ife = a0 + a1Tr , ife + a 2Tr3, ife + a3Tr6, ife + 4 , ∀ i ,f ,e Tr , ife

temperature, taking the normal boiling point as reference state. The Watson equation is given by ⎛ 1 − Tr , iBe ⎞0.375 ⎟⎟ , ΔhieLV = ΔhbLV ⎜ ,i ⎜ ⎝ 1 − Tb , i /Tc , i ⎠

(36)

Bcz , i = bcz ,0 + ωibcz ,1

(1 − Tr , ife)2/7

where is the vaporization enthalpy at the normal boiling point, which is calculated as47

(37)

,

∀ i ,f ,e

ΔhbLV , i = 1.093R g Tc , iTb , i

NC

∑ zi(∫ i

TBe

T0

CpL , i(T )dT +

∫P

Pe

vL , iBe(P)dP) + HeE ,

∀e

0

(39)

where h0 is the enthalpy at a reference state, CpL i is the pure component liquid heat capacity as a function of temperature, and HEe is the excess enthalpy for the liquid mixture at TBe and composition zi. The reference state was chosen to be zero enthalpy for the pure components at PLow and mixture bubble temperature TB Low. Next, the enthalpy of the mixture at the saturated vapor state is calculated as follows: Ĥe = hê − HeE +

NC

∑ zi(ΔhieLV − HieR + ∫ i

TDe

TBe

R CpIG, i (T )dT ) + Ĥe ,

a′ie = − 0.42748

bi = 0.08664

0.93 − Tb , i /Tc , i

(44)

R g2Tc , i Pc , i

−0.5 [cω , i + cω2 , i(1 − Tr0.5 , iBe)]Tr , iBe ,

∀ i ,e

(46)

R gTc , i Pc , i

(47)

Zie is given in the same manner as Ẑ fe in eq 24 but using pure component parameters at the bubble point temperature:

∀e

(40)

Zie = 1 + βiBe − αiBeβiBe

ΔhLV ie

where is the vaporization enthalpy change for the pure species i and bubble temperature TBe, HRie is the enthalpy departure from ideal gas for a pure component i at a bubble temperature TBe and pressure Pe, and Ĥ Re is the enthalpy departure for the vapor phase mixture at TDe,Pe and composition zi. Note that this is not an unique way of calculating enthalpies; it is possible to choose other routes.37,41 The ideal gas heat capacity45 for pure component i as a function of temperature was calculated from the following equation:

1 Zie − βiBe , Zie (Zie + βiBe)

Ĥ Re , the

∀ i ,e (48)

HRie apply, and

Regarding same rules for the calculation of also, the temperature derivative of the Gibbs excess energy can obtained from the UNIFAC method. This yields R ⎛ ⎞ ̂ Ĥe a′̂ ZDe ̂ − 1, ̂ − e ⎟⎟ ln = ⎜⎜αDe + ZDe ̂ ̂ + β̂ R gTDe ⎝ R gb ⎠ ZDe e

∀e (49)

with b̂ and â′e are obtained from the parameters from the PSRK model and by multiplying eq 23 by RgTfe and then by differentiation, respectively:

⎛ C3, i/T ⎞2 ⎛ ⎞2 C5, i/T ⎟⎟ + C4, i⎜⎜ ⎟⎟ CpIG, i(T ) = C1, i + C 2, i⎜⎜ ⎝ sinh(C3, i/T ) ⎠ ⎝ cosh(C5, i/T ) ⎠ (41)

b̂ =

where C1,i through C4,i are parameters for the i pure component. The liquid heat capacity CpL i(T) can be estimated using the corresponding states method as follows:

NC

∑ zibi

(50)

i

⎡ 1 a′̂ e = b⎢̂ (GeE)′T + ⎢⎣ A1

0.49 CpL , i(T ) = CpIG, i (T ) + 1.586R g + R g 1 − T /Tc , i ⎡ ⎤ 6.3(1 − T /Tc , i)1/3 0.4355 ⎥ + R gωi⎢4.2775 + + ⎢⎣ T /Tc , i 1 − T /Tc , i ⎥⎦

ln Pc , i − 1.013

The departure HRie is the difference between the real pure gas and the ideal gas for i species at bubble temperature, and Ĥ Rs is the difference of the real gas mixture and the ideal solution of ideal gases at the dew temperature. Both are defined from the EOS and SRK for pure components and PSRK for the mixture. The pure component residuals are defined as follows:where Zie is the pure compressibility factor at the bubble temperature, a′ie a temperature derivative parameter, and bi (related to βife) also a parameter both from the SRK equation. The generalized form of the residuals for both entropy and enthalpy and the SRK variables definition was taken from ref 37. The equations for a′ie and bi read as follows:

(38)

(10). Enthalpy Calculations. Assuming that the liquid is incompressible, the enthalpy of the saturated liquid mixture is calculated as follows: hê = h0 +

(43)

ΔhLV b,i

where a0 through a4, bcz,0, and bcz,1 are constants given in Table 1. The saturation liquid volume is calculated using critical properties from the Rackett equation:44 vLsat, ife = vc , iZc , i

∀ i ,e

NC

∑ i

Rg zi a′ie + bi A1

NC

∑ zi ln i

̂ ⎤ βDe ⎥, βiDe ⎥⎦

∀e (51)

(GEe )T′

where is the Gibbs excess energy temperature derivative and is calculated by differentiation of 22 follows:

(42)

where ωi is the acentric factor for the i species. The calculation of the vaporization enthalpy change ΔhLV ie is carried out through the application of the Watson equation46 at the operating

NC

(GeE)′T = R gTDe ∑ zi( ln γiDVe)′T + i

3374

E GDVe , TDe

∀e (52)


Article

Industrial & Engineering Chemistry Research Finally, the liquid excess enthalpy HEe is the enthalpy difference between the real liquid mixture and the ideal solution enthalpies both at bubble point temperature. Its calculation is carried out through the temperature derivative of the natural logarithm of the activity coefficient at the bubble point temperature for the liquid phase given by UNIFAC as follows:41 −

HeE 2 R gTBe

̌ ̌ SoverS,High = SoverS,Low

The nonisentropic properties are calculated with the isentropic efficiencies for both the pump and the turbine: (Ȟ over,High − Ȟ over,Low ) = ηST(Ȟ overS,High − Ȟ overS,Low ) (61)

NC

=

∑ zi( ln γiBLe)′T ,

∀e

̌ ̌ (hsub,High )= − hsub,Low

(53)

i

Every enthalpy that lies at lower temperature will have a lower value at the same pressure; this also holds for enthalpies at the same temperature but lower pressure. As absolute changes of enthalpy are required, these calculations were simplified by changing the position of the final state so a positive number is obtained. This was done in order to avoid the use of the absolute value |x| because it has a noncontinuous first derivative and leads to calculations issues when using optimization solvers. (11). Entropy Calculations. To carry out entropy calculations, the assumptions stated for enthalpy calculations were also used: incompressible liquids, the same reference state (pure components at the mixture bubble point temperature and the pressure level e = Low). The mixture saturated liquid entropy reads as follows: ⎛ sê = s0 + ∑ zi⎜⎜ ⎝ i NC

∫T

TBs

CpL , i(T ) T

0

⎞ dT ⎟⎟ + SeE , ⎠

̌ qOUT = Ȟ over,Low − hsub,Low

(54)

where is the excess entropy and is calculated from the excess properties as follows:

NC

nHSΔHHS = qIN

(55)

⎛ Δh LV

∑ zi⎜⎜ i

ie

⎝ TBe

− SieR +

∫T

TDe

Be

CpIG, i (T ) T

⎞ R dT ⎟⎟ + Sê , ⎠

nCSΔHCS = qOUT

(64)

where nHS and nCS are the molar ratios of the heat source and sink of the working fluid, and ΔHHS and ΔHCS are their respective molar enthalpy changes. It is important to note that all the variables and equations defined in this work deploy one mole of the mixture (working fluid) as a basis for computations. Thus, the molar ratios of heat source and sink are the moles required by one mole of working fluid (mol/mol). Next, the two-phase enthalpy locus are given by the energy balances:

∀e

(56)

The residuals are calculated as follows: SieR a′ Zie = ln(Zie − βiBe) − ie ln , Rg R gbi Zie + βiBe

(63)

Then, qIN and qOUT (both on a molar basis) are related to the mole ratio of heat source and sink with their respective overall enthalpy changes:

This leads to the calculation of the saturated vapor entropy, which requires the pure residuals at bubble temperature and the mixture residual at dew temperature: Sê = sê − SeE +

∀ i, e (57)

̌ ̂ ) + ĥ , nHSΔHHS, f + hsub,High = ψf (ĤHigh − hHigh High

and R

̂ Sê a′ ZDe ̂ − β ̂) − ̂ e ln = ln(ZDe , e ̂ ̂ + β̂ Rg ZDe R gb e

̌ ̂ ) + ĥ , nCSΔHCS, f + hsub,Low = ψf (ĤLow − hLow Low

∀e

∀f ∀f (65)

(58)

where ΔHHS,f and ΔHCS,f are the two-phase point enthalpy change for the heat source/sink. Regarding the enthalpy change of the heat source and sink, there is a limitation on the phase and temperature. These must be known beforehand, and the temperature change must be chosen at some range that guarantees single phase because there is not a continuous enthalpy model that accounts for all phases. Nevertheless, when the nature of the heating and sink fluid is defined, the equations of enthalpy are as follows. For liquid phase fluid:

(12). Overheated and Subcooled Operation. The operation of the Rankine cycle can lead to a liquid below its bubble point (subcooling) and vapor over its dew point (superheating). Those regions are characteristic of the operating region of both the pump and turbine. All of the PSRK, enthalpy, and entropy equations are consequently required again but with slight modifications. The temperature for these regions is renamed as Ť de, and because only nonphase equilibrium is attained, the composition variables are substituted by zi. The isentropic operation of both the pump and the turbine requires the next set of constraints: ssubS,High = ssubS,Low ̌ ̌

(62)

̌ qIN = Ȟ over,High − hsub,High

∀e

E HeE − GBLe TBe

1 ̌ ̌ (hsubS,High − hsubS,Low ) ηSP

where ηST and ηSP are the isentropic efficiencies for the turbine and pump (13). Heating/Cooling Temperature Profiles. The thermal sources profiles (heating/sink fluid) are important because they generate the operation temperature boundaries of the cycle. These profiles can be found with the enthalpy change along the exchangers such as vaporizer and condenser, both at counterflow. First, the heat coming from the heat source is quantified with the enthalpy change at the vaporizer, as is heat rejected to the heat sink with the change at the condenser. Only positive values of the heat were computed (even so the actual heat flow and enthalpy changes must have a sign indicating the direction of the flow). Therefore, the following enthalpy changes are defined:

SEe

SeE =

(60)

ΔHHS, f =

(59)

∫T

T HS f

C (T )dT ; HS,OUT pL ,S

ΔHCS, f =

∫T

T CS f CS,IN

CpL ,CS(T )dT

∀f

(66) 3375


Article


conditions of the Rankine cycle are addressed. The aim being to select both components and their amounts to optimize the objective functions and the corresponding constraints related to processing conditions discussed in section 3. An initial selection of the addressed organic components was made based in the compounds proposed in ref 18. The thermodynamic parameters of the pure components, namely, the ideal gas heat capacity function parameters (CIG p,i ) and the critical and normal parameters, were taken from the PURE28 database of the Aspen plus V8.4 simulator.48 Parameters for the cases of study A and B are listed in Table 2, which include the isentropic efficiencies, temperature changes of vaporizer and condenser, inlet temperature of the heat sink at the condenser and the minimum pressure difference, ideal gas constant, minimum temperature approach, and internal phase equilibrium points. The UNIFAC parameters for refrigerant species were taken from ref 49. We should stress that the i−j component featuring a given functional group m−n whose binary information is not available in the UNIFAC method, such as the i−j binary pair, is not taken into account for the purpose stated above. We also have noted that allowing components with wide critical temperature differences tends to give rise to convergence problems. In addition, as stated in section 3, the resulting operating temperature ought to be below the mixture critical temperatures. Even when we do not rigorously compute the mixture critical temperature, we have checked that the optimal

Table 2. Parameters for the Model parameter

value

ηSP ηST Δ TCS,c ΔTHS THS,IN TCS,IN ΔPminc Rgc ΔTSmin Nψ a

units   K K K (°C) K (°C) bar kJ kmol−1 K−1 cm3 bar mol−1 K−1 K 

0.9 0.9 10 20 363.15 (90) 293.15 (20) 1 8.314a 83.314b 5 3

For energy. bFor EOS calculations. cValid for all three cases.

where CpL,S(T) is the liquid heat capacity function (eq 42). If vapor is selected (only heat source), then: R ΔHHS, f = − HHS,OUT +

∫T

T HS f

R CpIG,HS(T )dT + HHS, f

∀f

(67)

HS,OUT

HRHS,f

where the residuals can be calculated in a way similar to the one shown in section 3 for a pure component.

4. RESULTS In this section, three cases of study regarding the choice of components, their compositions, and the optimal processing Table 3. Results for Case A (Part 1)a Ω1

obj. funct. var

n-C4

Ω2 i-C4

n-C5

mix

n-C4

i-C4

n-C5

0.000 0.133 0.867

1.000  

 1.000 

  1.000

0.000 0.130 0.870

1.000  

 1.000 

  1.000

TB,High TD,High Ť over,High

2.026 298.158 319.979 327.405 358.150

4.120 306.078 315.838 315.838 358.150

5.403 305.914 313.511 313.511 358.150

2.013 309.159 330.760 330.760 330.760

3.734 298.171 342.993 349.475 358.150

8.217 298.190 343.390 343.390 358.150

11.123 298.206 343.936 343.936 358.150

2.837 298.164 342.898 342.898 358.150

PLow Ť sub,Low TB,Low TD,Low Ť over,Low

1.026 298.150 298.150 306.395 342.760

3.120 306.070 306.070 306.070 350.615

4.403 305.906 305.906 305.906 352.383

1.013 309.152 309.220 309.220 316.632

1.020 298.150 298.150 306.284 328.356

3.148 298.150 306.372 306.372 331.170

4.451 298.150 306.305 306.305 330.956

1.013 298.150 309.220 309.220 336.585

ΔĤ LV High Wnet

25167.536 1849.608

19875.111 664.938

17999.539 476.675

24542.104 1568.862

23549.262 3351.633

17823.658 2123.228

15552.808 1929.884

23715.060 2532.200

0.056 0.062 0.910

0.026 0.028 0.907

0.020 0.022 0.906

0.055 0.061 0.901

0.103 0.113 0.906

0.080 0.089 0.901

0.080 0.089 0.898

0.075 0.082 0.906

33010.449 21.830 43.832

25914.091 17.137 34.409

23892.035 15.800 31.724

28419.596 18.794 37.736

32649.561 21.591 43.352

26401.291 17.459 35.056

24130.886 15.958 32.041

33902.325 22.419 45.016

2089 2185 3.261

1027 1097 0.359

1027 1097 0.187

1027 1097 0.187

2089 2185 1.687

1027 1097 0.203

1027 1097 0.141

1027 1097 0.125

zn‑C4 zi‑C4 zn‑C5 PHigh Ť sub,High

ηI ηCarnot ηII qIN nHS nCS Nvar Neqn CPU (s) a

mix

Pressure in bar, temperature in K, and energy in kJ kmol−1. 3376


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Industrial & Engineering Chemistry Research Table 4. Results for Case A (Part 2)a Ω3

obj. funct. var

n-C4

Ω4 i-C4

n-C5

mix

n-C4

i-C4

n-C5

0.000 0.137 0.863

1.000  

 1.000 

  1.000

0.000 0.231 0.769

1.000  

 1.000 

  1.000

PHigh Ť sub,High TB,High TD,High Ť over,High

3.797 298.171 343.196 349.929 349.929

8.124 306.929 342.896 342.896 342.896

11.011 306.897 343.468 343.468 343.468

2.782 309.233 342.178 342.178 342.178

2.289 298.158 317.256 328.592 358.150

4.120 305.998 315.839 315.839 358.150

5.415 298.158 313.600 313.600 358.150

2.013 298.158 330.760 330.760 358.150


1.037 298.150 298.150 306.589 320.038

3.196 306.890 306.890 306.890 316.660

4.518 306.842 306.842 306.842 316.829

1.013 309.220 309.220 309.220 321.098

1.289 298.150 298.150 310.392 344.494

3.120 305.991 306.071 306.071 350.615

4.415 298.150 306.010 306.010 352.396

1.013 298.150 309.220 309.220 343.936

ΔĤ LV High Wnet

23517.906 3259.443

17864.026 1954.099

15595.425 1773.448

23765.442 2348.745

25059.829 1595.091

19875.057 664.922

17993.133 475.522

24542.104 1729.654

ηI ηCarnot ηII qIN nHS nCS Nvar Neqn CPU (s)

0.104 0.115 0.904 31453.568 20.800 41.764 2089 2185 1.063

0.083 0.093 0.897 23421.393 15.488 31.099 1027 1097 0.172

0.083 0.093 0.894 21244.943 14.049 28.209 1027 1097 0.203

0.079 0.088 0.902 29759.728 19.680 39.515 1027 1097 0.063

0.050 0.055 0.910 32137.650 21.253 42.673 2089 2185 1.281

0.026 0.028 0.907 25925.756 17.145 34.425 1027 1097 0.234

0.019 0.021 0.906 25006.824 16.537 33.204 1027 1097 0.125

0.051 0.056 0.908 34100.228 22.550 45.279 1027 1097 0.156

zn‑C4 zi‑C4 zn‑C5

a

mix

Pressure in bar, temperature in K, and energy in kJ kmol−1.

operating temperatures are below such critical temperature. Because the optimization problems approached in this work are nonconvex, the CONOPT50 solver available in GAMS was used. The way the initialization of the optimization problem was approached has an important effect on the solution obtained due to the nonlinearities of the problem. For proper initialization, an equimolar mixture of the available components was used as the initial composition with weighted averages of saturation temperatures as the bubble and dew point temperatures. Pressure levels were initialized at 1 bar as the low level and 3 bar as the high level. Variables like activity, fugacity, and compressibility factors were initialized at 1 (i.e., assuming ideal gases). The composition of each component in a mixture was naturally bounded between 0 and 1. Other variables only take positive values like the temperature or pressure with an upper bound set by the critical properties of the pure components. Finally, upper and lower values of free variables, like residuals and excess properties, were estimated from their observed and expected values. All the optimization problems were solved using an Intel(R) Core(TM) i5-3230 M CPU @ 2.60 GHz processor and 8 GB of RAM. Moreover, it should be stressed that the lower and upper values of all the decision variables, defined in the List of Symbols section, are provided after the name of the given variable within brackets, with the first value being the lower bound and the second value being the corresponding upper bound. 4.1. Case A: n-Butane, Isobutane, n-Pentane (n-C4, i-C4, n-C5) Mixture and Pure Species Optimum Cycles. This system is representative of components featuring close pure

component critical temperatures. Moreover, they also show similar chemical structures. In this case study, we obtain both the optimal composition of the mixture and the optimal processing parameters of the cycle. To compare the performance of such a cycle, we have also considered the same problem as before, assuming that only a given organic pure component is used in a cycle. The heat and sink mediums are available at 90 and 20 °C, respectively, and 1 bar (Table 2). In Tables 3 and 4, the results of the optimal values of the decision variables for the mixture- and pure component-based cycles along the objective function values are displayed. As shown in Table 3 for the first objective function, even when we started with a mixture featuring three components, the optimal solution reduces to a binary mixture formed by the i-C4 and n-C5 components. In fact, the mixture contains a large proportion of n-C5. In this case, the optimal mixture leads to better results in terms of the energy recovered from the lowtemperature heat source. However, the efficiency of the cycle diminishes because a large amount of heat is rejected to the heat sink. As for the second objective function, the results shown in Table 3 indicate that a larger amount of work is generated. However, this improvement requires larger operating pressures and molar ratios of the heat source. It is interesting to note that the optimal mixture composition using any of the first three objective functions is nearly the same. So, improvement of the results also comes from selecting proper operating conditions. The optimum results using as objective function the maximization of the first law efficiency (Table 4) show almost the same mixture composition compared to the second objective 3377


Article


law efficiency. Moreover, smaller operating pressures are required for the binary mixture. Hence, for this case of study, no better optimal results were obtained when considering a ternary mixture. We found that when setting the lower bound on zi at zero that in most of the cases only binary mixtures were obtained. On the other hand, setting such a lower bound to a value slightly larger than zero allows us to obtain mixtures featuring larger number of components. This the explanation of the lower bound on zi shown in Table 5. Finally, in Tables 3, 4, and 5, problem statistics indicate that the CPU time for computing the reported optimal solutions is low. 4.2. Case B: Optimization of Organic Mixture Cycle Efficiency. In this case study, a mixture composed of the three alkanes used in the first case study and eight refrigerants defined in ref 18 were considered. Because the results of the first case study indicate that the best optimal results were obtained when the third objective function (maximization of the first law efficiency) was used, in the present case of study only this objective function was taken into account. For addressing the performance of the cycle, three low temperature heat sources at 90, 120, and 150 °C and 1 bar were analyzed. It should be remarked that the temperature difference of the hot stream was set at 20 K. As lshown in Table 6, when the hot source temperature is set at 90 °C, all the components embedded in the mixture were considered because Tc,i turned out to be greater than 90 °C for any component i. However, at higher temperatures, only the components meeting the constraint THS,IN < Tc,i were considered. This is the reason why not all components were taken into account in columns 2−5 of Table 6. In addition, at columns labeled as THS,IN∗, only the component featuring the larger proportion of the optimal mixture was considered. As shown in Table 6, the best optimal results (measured in terms of ηI) correspond to a hot source fluid at 150 °C, which is not surprising because the hot source temperature has been increased in comparison to a hot source fluid at 90 °C. It should be stressed that larger operating pressures are required when considering hot source fluids at larger temperatures. We would like to remark that the best optimal results were obtained for a binary mixture (150 °C column in Table 6) and that the fourcomponent mixture (90 °C column) leads to a smaller ηI efficiency. In summary, the results obtained in this case study turn out to be better than the ones discussed in the first case study. This is so because a larger temperature hot source fluid was considered. Moreover, we conclude that using pure components as working fluids did not perform better than any organic mixture for energy recovery from low-temperature sources using a typical Rankine cycle. As shown in Table 6, although the CPU time for obtaining the reported optimal solutions is larger than in the past case study, it is always below 1 min except for the case when the size problem is larger. 4.3. Case C: Thermal Match Optimization. In this case study, we examine the optimal design of the cycle using as the objective function the temperature approach along the cycle. The calculations were carried out in a specific scenario taken from ref 51 where a given geothermal fluid is available as heat source. The parameters for this specific case are listed in Table 7. The geothermal fluid consists mainly of water with dissolved salts and gases, but for the sake of simplicity, only pure water is assumed to be the heating source. The plots in Figure 4 show the temperature−enthalpy diagrams for the exchangers (vaporizer and condenser) of the cycle. The curves on the upper and lower positions are the profiles for

function. The amount of required heat is smaller, and so is the net work. Because in most of the cases the heating medium is limited, this solution could be considered to be better than the previous solutions. Also, no overheating in the high-pressure zone of the cycle is required. The final objective function, the maximization of the second law efficiency, states how the operation of the cycle approaches the average Carnot cycle efficiency. The results show that the organic mixture features the best second law efficiency at low pressures in part due to the nonisothermal phase change, which in general provides a better thermal match with the heat source and sink. However, it should be remarked that this time the optimal composition of the mixture turns out to be different in comparison to the previous objective functions. In conclusion, the results obtained from this case study indicate that the second and third objective functions lead to the maximum amount of recovered energy. It should be stressed that the cycle efficiency (measured in terms of ηI) obtained from these objective functions is similar. However, the amount of heat required (qIN) for the third objective function turns out to be smaller in comparison to the second objective function. Globally speaking, this statement indicates that the best results for this case study are obtained when the third objective function was considered. Furthermore, after concluding that the best optimal results are obtained using the third objective function, we decided to explore if better optimal results can be achieved considering the three organic compounds initially proposed. The results of Table 5 indicate that the optimal ternary mixture results (columns 2−4) are similar to the results previously discussed using the third objective function measured in terms of the first Table 5. Case A: Efficiency Maximization and Sensitivity of Lower Bound on zi to Number of Components in the Mixture zi lower bound var zn‑C4 zi‑C4 zn‑C5

0.000

0.010

0.050

0.100

0.000 0.137 0.863

0.010 0.131 0.859

0.050 0.111 0.839

0.100 0.100 0.800

TB,High TD,High Ť over,High

3.797 298.171 343.196 349.929 349.929

3.810 298.171 343.196 349.949 349.949

3.864 298.172 343.194 350.028 350.028

4.037 298.172 343.191 350.654 350.654


1.037 298.150 298.150 306.589 320.038

1.040 298.150 298.150 306.589 320.043

1.055 298.150 298.150 306.589 320.057

1.112 298.150 298.150 307.220 320.561

ΔĤ LV High Wnet

23517.906 3259.443

23503.319 3257.173

23443.449 3247.917

23319.149 3226.912

ηI ηCarnot ηII qIN nHS nCS Nvar Neqn CPU (s)

0.103627 0.115 0.904 31453.568 20.800 41.764 2089 2185 1.063

0.103622 0.115 0.904 31433.165 20.787 41.737 2089 2185 1.250

0.103602 0.115 0.904 31349.882 20.732 41.627 2089 2185 1.687

0.103501 0.114 0.904 31177.616 20.618 41.398 2089 2185 1.719

PHigh Ť sub,High

3378


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Industrial & Engineering Chemistry Research Table 6. Results for the Case B heat source temperature THS,IN (°C) var

90

90

120a

120

150a

150

zR‑22 zR‑134a zR‑152a zR‑245ca zHC‑270 zR‑C318 zR‑3‑1‑10 zFC‑4‑1‑12 zn‑C4 zi‑C4 zn‑C5

0.000 0.000 0.018 0.000 0.000 0.692 0.000 0.033 0.000 0.000 0.258

     1.000     

   0.549 0.000 0.076  0.000 0.000 0.000 0.376

   1.000       

   0.197    0.000 0.000  0.803

          1.000


11.138 298.202 345.779 353.373 353.373

11.524 305.711 346.509 346.509 346.509

14.067 298.233 377.159 384.118 384.118

13.581 306.078 376.859 376.859 376.859

17.167 298.268 408.759 417.023 417.024

12.312 309.305 408.632 408.632 408.633


3.106 298.150 298.150 308.123 327.914

3.941 305.664 305.664 305.664 324.604

1.687 298.150 298.150 305.870 333.282

1.953 306.005 306.005 306.005 326.984

1.024 298.150 298.150 304.829 350.728

1.013 309.220 309.220 309.220 351.455

ΔĤ LV High Wnet

18980.614 3671.042

15558.230 2944.316

19374.910 5470.485

18158.841 4353.993

18282.555 7443.139

18031.485 6346.104

0.120 0.131 0.914

0.110 0.121 0.911

0.155 0.171 0.907

0.135 0.151 0.900

0.184 0.203 0.909

0.168 0.185 0.907

30685.206 20.292 40.744

26730.683 17.677 35.493

35224.766 51.234 46.772

32136.738 46.743 42.672

40373.453 58.346 53.608

37843.753 54.690 50.249

8161 8361 161.765

795 865 0.218

4930 5078 31.906

1384 1454 1.000

3323 3432 17.610

1082 1152 0.328

ηI ηCarnot ηII qIN nHF nCF Nvar Neqn CPU (s) a

a

Pure-based Rankine cycle.

Hfe = ψf(Ĥ e − hê ) + ĥe, where e is the pressure level and ψf is the vaporization fraction. Note that the minimum temperature approach (5 K) was always enforced. Also, the temperature profiles for Ω5 for the high-pressure region and the heat source are closer than the corresponding ones when using Ω3. From the results shown in Table 8, we note that using the Ω5 objective function does not improve the efficiency of the cycle. However, using the Ω5 objective function leads to selection of a binary mixture. On the other hand, the Ω3 objective function results in a complex five-component mixture. Moreover, for this case study, it is also important to take into account the magnitude of the flow rate of the hot source (βHS) to generate 1 MW of power.51 This can be calculated from eq 68 (in kg s−1): n βHS = 1000 HS MW,HS Wnet (68)

Table 7. Parameters for Case C parameter ηSP ηST ΔTCS ΔTHS THS,IN TCS,IN Nψ

value 0.85 0.85 10 60 403.15 (130) 313.15 (40) 8

units   K K K (°C) K (°C) 

the heat source and the heat sink (labeled as HS and CS), respectively. The inbetween curves stand for the working fluid at the pressure levels (upper-high pressure and lower-low pressure). The two extrema points of these curves are the nonphase equilibrium points (i.e., at the inlet and outlet of the exchangers); whereas the 10 points in the middle are the two-phase points (bubble, eight internal, and dew point; seel section 3 for details). The calculation of the internal enthalpy of these points is

where MW,HS is the molecular weight of the heat source. From the results shown in Table 8, it turns out that the Ω3 objective function demands the minimum flow rate of hot source fluids. 3379


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Industrial & Engineering Chemistry Research Table 8. Results for Case Ca var

Ω3

zR‑245ca zHC‑270 zFC‑4‑1‑12 zn‑C4 zi‑C4 zn‑C5

0.000 0.000 0.873 0.127 0.000 0.000

0.160 0.000 0.323 0.183 0.167 0.167


8.630 308.224 363.802 373.741 374.601

11.630 308.248 360.195 365.492 365.492


2.050 308.150 308.150 319.706 353.780

3.063 308.150 308.150 315.842 341.159

ΔĤ LV High Wnet

21477.262 3598.888

19366.793 3142.627

0.095 0.110 0.858

0.102 0.120 0.855

qIN nHF nCF

38056.817 6.619 50.532

30737.356 5.346 40.813

Nvar Neqn CPU (s) Ω5 βHS

7030 7255 40.000 7690.859 31.928

7030 7255 48.406 9321.026 29.385

ηI ηCarnot ηII

a

Ω5

sources, and working fluid was addressed. The results obtained using the proposed objective function show some potential but could not overperform the maximization of the first law efficiency. This approach has merits on the creation of multicomponent mixtures as working fluids for the Rankine cycle that to a certain degree of complexity can improve its overall performance when lowtemperature heat sources are taken into account. In summary, by proper selection of the optimal composition of given organic fluids and taking into account the performance of a typical Rankine cycle, this work has shown that heat recovery from low-temperature sources can be improved leading to a rational and efficient use of alternative energy sources. In future work, we will explore how uncertainty affects the amount of energy that can be recovered. Moreover, the extension to a network of coupled Rankine cycles is also a way to increase heat recovery. Finally, optimal control of the considered heat recovery flowsheets also needs to be addressed.

6. LIST OF SYMBOLS 6.1. Degrees of Freedom [The first and second values within the brackets correspond to the lower and upper bounds, respectively]. αife [0,15] EOS variable, dimensionless EOS mixture variable, dimensionless α̌ fe [0,15] α̅ ife [−100,100] Molar partial α̂ fe, dimensionless βife [0,10] Pure EOS variable, dimensionless β̂fe [0,10] Mixture EOS variable, dimensionless 5 ΔhLV Pure vaporization−enthalpy change, ie [0,10 ] kJ kmol−1 5 ΔHHS,f [0,10 ] Heat source enthalpy change, kJ kmol−1 5 ΔHCS,f [0,10 ] Heat sink enthalpy change, kJ kmol−1 5 ΔĤ LV [0,10 ] Mixture vaporization−enthalpy change, e kJ kmol−1 ηCarnot [0,1] Average Carnot efficiency, dimensionless ηII [0,1] Second law efficiency, dimensionless ηI [0,1] First law efficiency, dimensionless γRifpe [0,exp(10)] Residual activity coefficient, dimensionless γif Le [0,exp(10)] Activity coefficient, dimensionless γCifpe[0,exp(10)] Combinatorial activity coefficient, dimensionless Γ(i) Residual activity contribution for kfe [0,exp(10)] groups of pure i, dimensionless Γkf pe [0,exp(10)] Residual activity contribution mixture of groups, dimensionless x Freedom degrees,  (ln γif pe)T′[−10,10] ln Activity coefficient temperature derivative, K−1 sat ϕife [0,exp(5)] Pure saturation fugacity coefficient, dimensionless ϕ̂ ife [0,exp(5)] Mixture fugacity coefficient dimensionless Ψkmfe [0,10] UNIFAC binary interaction variable, dimensionless θ(i) UNIFAC area fraction variable for pure m [0,1] i, dimensionless θnfpe [0,1] UNIFAC area fraction variable, dimensionless Acz,ife [0,102] Chanz−Zhao polynomial, dimensionless aie′ [−105,105] Pure a temperature derivative, cm6 bar mol−2 K−1 5 5 â′e [−10 ,10 ] Mixture a temperature derivative, cm6 bar mol−2 K−1 ̂b [0,103] Mixture EOS variable, cm3 mol−1

βHS in kg s−1.

This point can be important because the availability of the hot source fluid is limited. In summary, we conclude that the new Ω5 objective function does not improve the energy recovery issue. Also in this case study, as shown in Table 8, the CPU times are lower than 1 min.

5. CONCLUSIONS In this work, we have addressed the simultaneous computation of both the composition of a multicomponent mixture of organic working fluids and the operating conditions of a typical Rankine cycle to maximize energy recovery from low-temperature heat sources meeting a set of process constraints. Afterward, the optimization formulation was applied to three operating cases. The first case was based on the individual implementation of four objective functions, where it was found that the best performing optimal mixture-based cycle was obtained when the maximization of the first law efficiency was accounted for. For the second case, the effect of the temperature of the heating source on the optimal mixture-based cycle was addressed. It was found that the optimal composition of the mixture shifts as the conditions of the cycle change leading to increasing the first law efficiency as the temperature increases. In the final case study, the minimization of the temperature differences of the heat, cold 3380


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Industrial & Engineering Chemistry Research Gibbs excess function, kJ kmol−1 for energy, cm3 bar mol−1 for EOS 3 3 E (Ge )T′ [−10 ,10 ] Gibbs excess temperature derivative, cm3 bar mol−1 K−1 HEe [−103,103] Excess enthalpy, kJ kmol−1 3 3 R Hie [−10 ,10 ] Pure residual enthalpy, kJ kmol−1 R 3 3 HHS,f [−10 ,10 ] Heat source residual enthalpy, kJ kmol−1 ̂he [0,105] Mixture saturated liquid enthalpy, kJ kmol−1 5 Ĥ e [0,10 ] Mixture saturated vapor enthalpy, kJ kmol−1 R 3 3 ̂ He [−10 ,10 ] Mixture residual enthalpy, kJ kmol−1 ȟde [−105,105] Subcool liquid enthalpy, kJ kmol−1 5 Ȟ de [0,10 ] Overheat vapor enthalpy, kJ kmol−1 Kife [0,100] Equilibrium ratio, dimensionless Heat source fluid mole to working fluid nHS [0,102] ratio, dimensionless nCS [0,102] Heat sink fluid mole to working fluid ratio, dimensionless Pe [0,20] Pressure, kJ kmol−1 Psat [0,P ] Saturation pressure, bar ife c,i qIN [0,105] Input heat, kJ kmol−1 qOUT [0,105] Output heat, kJ kmol−1 E Se [−100,100] Excess entropy, kJ kmol−1 K−1 SRie [−100,100] Pure residual entropy, kJ kmol−1 K−1 4 ŝs [0,10 ] Mixture saturated liquid entropy, kJ kmol−1 K−1 Ŝe [0,104] Mixture saturated vapor entropy, kJ kmol−1 K−1 R Ŝe [−100,100] Mixture residual entropy, kJ kmol−1 K−1 4 šde [0,10 ] Subcool liquid entropy, kJ kmol−1 K−1 ̌Sde [0,104] Overheat vapor entropy, kJ kmol−1 K−1 THS [depends] Heat source fluid temperature, K f TCS [depends] Heat sink fluid temperature, K f Tfe [0, min(Tc,i)] Working fluid temperature, K 3 Tavg Avarage temperature, K e [0,10 ] Tr,ife [0, min(Tc,i) /Tc,i] Reduced temperature, K Ť de [0, min(Tc,i)] Overheat/subcool temperature, K vsat Saturation liquid volume, cm3 mol−1 L,ife [0,1000] 5 Wnet [0,10 ] Net work, kJ kmol−1 xifpe [0,1] Mole fraction, dimensionless Xmfe [0,1] Group mole fraction, dimensionless zi [0,1] Global composition, dimensionless Zie [0,10] Pure compressibility factor, dimensionless Ẑ fe [0,10] Mixture compressibility factor, dimensionless 6.2. Parameters. αRc,i Riedel CSP parameter, dimensionless βHS Heat source mole flow rate for 1 MW of power, kg s−1 LV Δhb,i Normal boiling point pure vaporization enthalpy, kJ kmol−1 ΔHHS Heat source overall enthalpy change, kJ kmol−1 ΔHCS Heat sink overall enthalpy change, kJ kmol−1 ΔPmin Minimum pressure change of the cycle, bar ΔTCS Cooling medium exchanger temperature change, K ΔTHS Heating medium exchanger temperature change, K ΔTSmin Profile minimum temperature difference, K ηSP Pump isentropic efficiency, dimensionless ηST Turbine isentropic efficiency, dimensionless ν(i) UNIFAC group m frequency on molecule i, dimensionless k ψbR,i Riedel CSP parameter, dimensionless ψf Vaporization fraction, dimensionless ωi Acentric factor, dimensionless GEfpe [−105,105]

A1 A+R,i a0 a1 a2 a3 a4 amn B+R,i bcz,0 bcz,1 Bcz,i bi C+R,i cω,i C1,i C2,i C3,i C4,i C5,i Ccz D+R,i Dcz h0 HRHS,OUT li NC NG Nvar Neqn P0 Pc,i qi Qk QR,i Rg ri Rk TCS,OUT TCS,IN THS,IN THS,OUT T0 Tb,i vc,i X(i) m Zc,i

PSRK parameter, dimensionless Riedel CSP vapor pressure parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless UNIFAC binary interation paramter, K−1 Riedel CSP vapor pressure parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless EOS parameter, cm3 mol−1 Riedel CSP vapor pressure parameter, dimensionless Acentric factor function parameter, dimensionless Ideal gas capacity function parameter, kJ kmol−1 Ideal gas capacity function parameter, kJ kmol−1 Ideal gas capacity function parameter, K Ideal gas capacity function parameter, kJ kmol−1 Ideal gas capacity function parameter, K Chang−Zhao liquid volume parameter, dimensionless Riedel CSP vapor pressure parameter, dimensionless Chang−Zhao liquid volume parameter, dimensionless Reference enthalpy, kJ kmol−1 Heat source outlet residual enthalpy, kJ kmol−1 UNIFAC parameter, dimensionless Number of components, dimensionless Number of groups, dimensionless Number of variables, dimensionlessl Number of equations, dimensionless Reference pressure, bar Critical pressure, bar UNIFAC parameter, dimensionless UNIFAC parameter, dimensionless Riedel CSP parameter, dimensionless Ideal gas constant, kJ kmol−1 K−1 for energy, cm3 bar mol−1 K−1 for EOS UNIFAC parameter, dimensionless UNIFAC parameter, dimensionless Cold fluid outlet temperature, K Cold fluid inlet temperature, K Hot fluid inlet temperature, K Hot fluid outlet temperature, K Reference temperature, K Normal boiling temperature, K Critical volume, cm3 mol−1 Group mole fraction for pure i, dimensionless Critical compressibility factor, dimensionless

6.3. Functions. −1 CIG p,S (T) Source ideal gas heat capacity function, kJ kmol IG Cp,i (T) Pure fluid ideal gas heat capacity function, kJ kmol−1 CpL,i (T) Thermal fluid liquid heat capacity function, kJ kmol−1 CpL,S (T) Source liquid heat capacity function, S = HS or CS, kJ kmol−1 vL,ife (P) liquid volume function, cm3 mol−1

■

AUTHOR INFORMATION

Corresponding Author

*Phone/fax: +52(55)59504074. E-mail: antonio.fl[email protected]. Notes

The authors declare no competing financial interest. 3381


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Simultaneous Optimal Design of Organic Mixtures and Rankine

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