Heat-Exchanger Network Synthesis Involving Organic Rankine Cycle

Article pubs.acs.org/IECR

Heat-Exchanger Network Synthesis Involving Organic Rankine Cycle for Waste Heat Recovery Cheng-Liang Chen,* Feng-Yi Chang, Tzu-Hsiang Chao, Hui-Chu Chen, and Jui-Yuan Lee Department of Chemical Engineering National Taiwan University, Taipei 10617, Taiwan, ROC ABSTRACT: This article aims to present a mathematical model for the synthesis of a heat-exchanger network (HEN) which can be integrated with an organic Rankine cycle (ORC) for the recovery of low-grade waste heat from the heat surplus zone of the background process. An ORC-incorporated stagewise superstructure considering all possible heat-exchange matches between process hot/cold streams and the ORC is first presented. On the basis of this superstructure, the model for synthesizing ORCintegrated HENs is formulated as a mixed-integer nonlinear program (MINLP). A two-step solution procedure is proposed to solve the MINLP model. First, a stand-alone HEN is synthesized to minimize the external utility consumption. An ORC is then incorporated into the HEN with the objective of maximizing the work produced from waste heat (in the heat surplus zone below the process pinch) without increasing the use of a hot utility. A literature example is solved to demonstrate the application of the proposed model for industrial waste heat recovery. of steam, dry organic fluid such as n-butane can be used as the working medium circulating in an ORC to convert low-grade heat into mechanical or electrical power. Several researchers have investigated the application and the performance of ORCs. Saleh et al.8 gave a thermodynamic screening of pure components as working fluid for the ORC systems in geothermal power plants. Mago et al.9 examined some typical dry organic fluids used to convert waste heat to power from low-temperature sources. Papadopoulos et al.10 proposed systematic methods for the selection of optimal working fluids for the ORC under various operating conditions. Roy et al.7 developed a computer program to optimize and compare the performance of the waste heat recovery system for different heat sources with different temperatures. Sprouse and Depcik11 reviewed the application of ORCs for waste heat recovery from exhaust of the internal combustion engine. Recently, Papadopoulos et al.12 proposed using the computer-aided molecular design method for the selection of binary working fluid mixtures used in an ORC. Concerning the integration of an ORC with the background process, Desai and Bandyopadhyay13 performed a thorough analysis of ORC application for heat recovery from typical background processes. Therein, the inlet/outlet temperatures of hot and cold process streams were given and the waste heat is taken from the heat surplus zone below the process pinch. The authors presented a feasible method for the integration of an ORC with the background process to generate mechanical work. Valencia et al.14 proposed a superstructure-based modeling and optimization approach for optimal ORC integration in industrial processes. In this work, the design objective was the minimization of the overall capital and

1. INTRODUCTION Industrial processes require heat and power which are mostly generated from fossil fuels. The costs of fossil fuels have risen by nearly three times during the past decade, while the prices of many industrial products remain almost the same because of oversupply or stiff worldwide competition. To face the competitive environment, one of the most important activities in the production process plants is the pursuit of efficient energy utilization. Process integration (PI), defined as a holistic approach to process design, retrofitting, and operation which emphasizes the unity of the process,2 is one of the most widely used techniques for the process industries to enhance energy utilization. Over the last two decades, numerous research works have focused on heat integration between process streams requiring heating and those requiring cooling.3 Furman and Sahinidis4 provided a thorough review of the literature on heat-exchanger network synthesis (HENS) for heat recovery within the process. Milestones and major discoveries in HENS were highlighted. A comprehensive guide to the recent PI research and applications was summarized in a review paper5 and the handbook edited by Klemeš.6 This review also gave tribute to all those pioneers in the development of the insightbased pinch analysis and design methods and the superstructure-based mathematical optimization approaches. However, recovery of waste heat from low temperature range is not considered much in industrial applications. A large portion of heat generated expensively is released to the environment without being utilized. It is reported that industrial low-grade waste heat accounts for nearly 50% or more of total energy input. 7 Therefore, in addition to conventional HENS techniques, it is also desired to develop effective methods for further recovery of waste heat from below the process pinch. This recovered heat may be used to produce work or electricity, thereby increasing the overall energy efficiency. The organic Rankine cycle (ORC) is one of the feasible technologies to recover waste heat from low to medium temperature heat sources and for electricity generation. Instead © XXXX American Chemical Society

Special Issue: Jaime Cerdá Festschrift Received: January 22, 2014 Revised: April 18, 2014 Accepted: April 23, 2014

A

dx.doi.org/10.1021/ie500301s | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

Figure 1. Proposed superstructure for HENS.

3. SUPERSTRUCTURES FOR STAND-ALONE AND ORC-INTEGRATED HENS For optimal HENS, the stagewise superstructure proposed by Yee et al.1 is adopted. This superstructure will then be expanded to include an ORC for waste heat recovery. Reference 1 illustrates a two-stage superstructure with two hot and two cold process streams. Figure 1 shows the proposed generic representation of the HENS superstructure to handle multiple hot and cold process streams. In the proposed superstructure, HEijk is the heat exchanger for the hot process stream i and the cold process stream j at stage k ∈ 2 ; Hhj is the heater for providing extra hot utility h at the end of cold stream j, and Cic is the cooler for removing extra energy at the end of hot process stream i by external cold utility c; rhijk ∈ [0, 1] and rcijk ∈ [0, 1] are the split ratios for the hot and cold process streams at stage k for heat exchanger HEijk, where the exit temperatures are expressed as thijk and tcijk; tik and tjk denote the intermediate temperatures for hot process stream i and cold process stream j at location k ∈ 2 which are the results of nonisothermal mixing of split streams. Figure 2a shows a schematic diagram of an ORC. There are four state points: the low-pressure liquid organic fluid at state point 1 passes through the pump to reach the evaporator pressure. This requires Wp of mechanical work. The highpressure liquid at state point 2 absorbs Qe of waste heat from the process isobarically in the evaporator to form a vapor. The high-pressure vapor at state point 3 generates Wt of electricity or mechanical work in the turbine and reaches the condenser pressure through expansion. The low-pressure vapor, usually superheated in the case of dry fluids,13 is then condensed to saturated liquid (state point 1) by discharging residual heat to the process and external cold utility. Figure 2b is the proposed superstructure for ORC-integrated HENS. PVm and TCm denote the pump and the turbine for working fluid m ∈ 4 in the ORC. Evaporators Eimk and condensers Cmjk are used for exchanging the waste heat between the hot/cold process streams and working fluid m ∈ 4 . te* and tc* are intermediate temperatures around the evaporators or the condensers.

operating costs, including the revenue from the shaft power produced by the ORC. The ORC integration problem was formulated as a mixed-integer nonlinear program (MINLP). Numerical examples showed improved results compared to previous studies because the trade-off between capital and operating costs as well as the sale of electricity produced from waste heat was considered simultaneously. However, the use of external heating utility was usually higher than that for the stand-alone heat exchanger network (HEN) in this work. Recently, Lira-Barragan et al.15 extended their previous analysis to integration of trigeneration systems with HENs for efficient utilization of industrial waste heat. This article aims to present a unified MINLP model for HENS involving an ORC for recovery of low-grade waste heat from the background process. This model is based on a stagewise superstructure describing the integration of hot and cold process streams with an ORC. A literature example is used to demonstrate the effectiveness of the proposed MINLP formulation for maximizing waste heat recovery.

2. PROBLEM STATEMENT The problem of integrating an ORC with the background process can be stated as follows. Within the background process, givens are a set 0 = {i|1, ..., I} of hot process streams to be cooled from their supply temperatures Ti,in to their target values, Ti,out and a set 1 = {j|1, ..., J} of cold process streams to be heated from their supply temperatures Tj,in to their target temperatures, Tj,out. Also givens are the heat capacity flow rates of hot and cold process streams, FCi and FCj, and a set / = {h| 1, ..., H} of available hot utilities for heating and a set * = {c|1, ..., C} of cold utilities for cooling. A set 4 = {m|1, ..., M} of dry organic working fluids may be used to recover low-grade waste heat. The HENS problem is first solved to determine an HEN configuration that maximizes heat recovery and minimizes the consumption of hot and cold utilities. An ORC is then used in the below-pinch heat surplus zone to recover the waste heat from the hot process streams without increasing the use of hot utilities when compared with the stand-alone HEN without an ORC. Having absorbed the waste heat, the high-pressure evaporated working fluid of the ORC can produce electricity or work through a turbine.

4. SUPERSTRUCTURE-BASED MINLP FORMULATION In this article, a two-stage approach is proposed to maximize the recovery of low-grade waste heat through the integration of B



Article

Figure 2. (a) Operating mechanism of an ORC. (b) Proposed ORC-integrated superstructure.

4.1. Stand-Alone HENS for Minimum Utility Consumption. A stand-alone HEN with minimized external utilities can be found by solving problem P1, which is formulated as an MINLP.1 x1 represents the set of variables to be optimized, including binary variables z* (* ∈ {ijk, ic, hj}) representing the HEN configuration and continuous variables for temperatures and heat loads, and Ω1 denotes the feasible searching space defined by constraints for HENS based on the

an ORC with the background process. The ORC is meant for further heat recovery and is not supposed to increase the consumption of external hot utility, compared to the situation without an ORC. In the first stage, stand-alone HENS is considered to target the minimum external utility requirements. The inclusion of an ORC is then considered in the second stage based on the optimized HEN. C



Article

remain the same. However, the supply temperatures of the hot streams and the target temperatures of the cold streams will change. There are three possible cases for the hot streams: Ti,in hot hot > Thot pinch > Ti,out, Ti,in > Ti,out > Tpinch, and Tpinch > Ti,in > Ti,out, in which the supply temperature, also designated by Ti, to simplify hot the formulation, is updated with Tpinch , Ti,out, and Ti,in, respectively. Note that the hot process stream is inactive for ORC-integrated HENS in the second case (Ti,in > Ti,out > Thot pinch) since the updated supply temperature is the same as the target temperature. Similarly, there are three possible cases for cold the cold streams: Tj,out > Tcold pinch > Tj,in, Tj,out > Tj,in > Tpinch, and cold Tpinch > Tj,out > Tj,in, in which the target temperature, also designated by Tj,out, is updated with Tcold pinch, Tj,in, and Tj,out, respectively. The cold stream is inactive in the second case. With the updated hot and cold process stream temperatures, one can model the ORC-integrated superstructure in Figure 2b for maximizing the recovery of low-grade waste heat. Equations 1 and 2 are the overall heat balance for all process streams. The overall enthalpy change of hot process stream i, FCi(Ti,in − Ti,out) where FCi is the heat capacity flow rate, equals the sum of the heat exchanged with cold process stream j (qijk) and working medium m (qimk) at all stages k ∈ 2 , and the external cold utilities c (qic). Similarly, the overall enthalpy change of cold process stream j equals to the sum of all of the heat exchanged with hot process stream i and working medium m at all stages k, and the external hot utilities h at the end of the cold process streams.

superstructure in Figure 1. These constraints include energy balances, mass balances, existence of heat exchangers, and temperature assignments. A detailed description will be given later when the ORC is considered. P1: min j1 = x1∈ Ω1

∑ ∑ qhj+ ∑ ∑ qic h∈/ j∈1

i∈0 c∈*

⎧ z , z , z ; q , q , q ; rh , rc ; th , tc ; ⎪ ijk ic hj ijk ic hj ijk ijk ijk ijk out in x1 ≡ ⎨ tik , t jk ; dtijk , dtijk ; dticin , dticout ; dthjin , dthjout ⎪ ⎪ ∀ i ∈ 0, j ∈ 1, k ∈ 2, h ∈ /, c ∈ * ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ω1 = ⎨x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎫ ⎪ ⎬ ⎪ ⎪ ⎭

⎫ ⎪ ⎪ ⎪ ∑ qhj + ∑ ∑ qijk = FCj(Tj ,out − Tj ,in) ⎪ h∈/ j∈1 k∈2 ⎪ ⎪ ⎪ ∑ qijk = FCi(tik − ti ,k+ 1) ⎪ j∈1 ⎪ ⎪ ∑ qijk = FCj(t jk − t j ,k+ 1) ⎪ i∈0 ⎪ ⎪ ∑ rhijk = 1, qijk = rhijkFCi(tik − thijk) ⎪ j∈1 ⎪ ⎪ ∑ rcijk = 1, qijk = rcijkFCj(tcijk − t j ,k+ 1) ⎪ i∈0 ⎪ ⎪ ∑ qic = FCi(ti ,K + 1 − Ti ,) ⎪ c∈* ⎪ ⎪ ⎪ ∑ qhj = FCj(Tj , − t j1) ⎬ h∈/ ⎪ L U ⎪ Q zijk ≤ qijk ≤ Q zijk ⎪ ⎪ Q Lzic ≤ qic ≤ Q Uzic ⎪ ⎪ L U Q zhj ≤ qhj ≤ Q zhj ⎪ ⎪ ⎪ Ti ,in = ti1 tik ≥ ti , k + 1 ti , K + 1 ≥ Ti ,out ⎪ Tj ,out ≥ t j1 t jk ≥ t j , k + 1 t j , K + 1 = Tj ,in ⎪ ⎪ dtic = Ti ,out − Tc ,in dthj = Th ,in − Tj ,out ⎪ ⎪ ⎪ ΔTmin ≤ dtijk ≤ tik − t jk + Γ(1 − zijk) ⎪ ΔTmin ≤ dtij , k + 1 ≤ ti , k + 1 − t j , k + 1 + Γ(1 − zijk)⎪ ⎪ ⎪ ΔTmin ≤ dticout ≤ ti , K + 1 − Tc , + Γ(1 − zic) ⎪ out ⎪ ΔTmin ≤ dthj ≤ Th , − t j1 + Γ(1 − zhj) ⎪ ⎪ ∀ i ∈ 0, j ∈ 1, k ∈ 2; h ∈ /, c ∈ * ⎭

∑ qic + ∑ ∑

c∈*

qijk = FCi(Ti ,in − Ti ,out)

j∈1 k∈2

∑ ∑ j∈1 k∈2

∑ ∑

qijk +

m∈4 k∈2

qimk + ∑ qic = FCi(Ti ,in − Ti ,out) c∈*

∀i∈0

∑∑ i∈0 k∈2

(1)

qijk +

∑ ∑ m∈4 k∈2

qmjk +

∑ h∈/

qhj = FCi(Ti ,out − Ti ,in)

∀j∈1

(2)

Equations 3 and4 are the total heat exchanged at a certain stage k. The heat balance for hot process stream i at stage k, FCi(tik − ti,k+1), equals the sum of the heat exchanged with cold process stream j and working medium m at this stage. Similarly, the heat balance for cold process stream j equals the sum of all of the heat exchanged with hot process stream i and working medium m at the same stages k.

∑ qijk + ∑

j∈1

m∈4

qimk = FCi(tik − ti , k + 1)

∀ i ∈ 0, k ∈ 2

The solution of P1 gives the HEN configuration and operating conditions, including the external hot and cold utility targets, (Q H U ) P*1 = ∑j ∈ 1 ∑h ∈ / q h*j and (Q C U ) P*1 =

∑ qijk + ∑ i∈0

∑i ∈ 0 ∑c ∈ * qic*. 4.2. HENS Involving an ORC for Maximum Waste Heat Recovery. An ORC can be integrated with a stand-alone HEN to recover low-grade waste heat from the heat surplus zone below the process pinch. The pinch temperatures on the hot cold and cold sides, Thot pinch and Tpinch, can be found in the stand-alone HEN obtained from P1 by calculating the temperature approach of all heat exchangers. The pinch location corresponds to where the temperature difference is equal to the given minimum, ΔTmin. With ORC integration, in the heat surplus zone (below the pinch), the target temperatures of the hot streams and the supply temperatures of the cold streams

m∈4

(3)

qmjk = FCj(t jk − t j , k + 1)

∀ j ∈ 1, k ∈ 2

(4)

Equations 5−7 denote the quantity of heat transferred in each heat exchanger (qijk), evaporator (qimk), and condenser (qmjk). In these equations, rhijk is the split ratio for hot process stream i that exchanges heat with cold process stream j at stage k, its output temperature is designated as thijk. rcijk and tcijk are the split ratio and output temperature for cold process stream j. rwimk and rwmjk are the split ratios for the circulated working fluid m in the evaporator and condenser zones. Note that the split working fluid streams are assumed to be mixing isothermally at each stage for simplifying the formulation D



Article

Table 1. Peng−Robinson Equation of State Parameters for Some Selected Working Fluids for the ORC Antonine equation

liquid

vapor

working fluid

Pc (bar)

Tc (°C)

Am

Bm

Cm

k (kJ/kg K)

n

k (kJ/kg K)

n

n-butane n-pentane n-hexane

38.0 33.7 30.1

151.98 196.55 234.67

4.3281 3.9892 4.0027

1132.108 1070.617 1171.530

0.918 −40.454 −48.784

8.65 10.64 12.63

1.06 1.06 1.06

24.89 29.99 35.39

0.74 0.74 0.74

since the working fluid may have phase change inside the heat exchanger.

∑ ∑ j∈1 k∈2

∀ i ∈ 0, m ∈ 4, k ∈ 2

(6)

+

∀ m ∈ 4, j ∈ 1, k ∈ 2

i∈0

∑ rhijk + ∑

rhimk = 1

∑ rcijk + ∑ i∈0

c c ∑ qmjk = fcmc (tmk − tmc , k + 1) + ∑ λmjk

(18)

Equation 19 means that the total latent heat obtained by the working medium m is equivalent to the summation of individual latent heat in all evaporators, where the heat comes from waste heat of all hot process streams or the complementary hot utilities. The Peng−Robinson equation of state (PR-EOS) is adopted for calculating the latent heat and the enthalpy at various state points for potential working fluids. The calculation details of the PR-EOS are depicted in the Appendix, and the substance specific parameters for some pure components are shown in Table 1. Equation 20 gives similar meaning for the total latent heat discharged to the cold process streams and external cold utilities.

∀ m ∈ 4, k ∈ 2 (10)

∀ m ∈ 4, k ∈ 2 (11)

j∈1

For reaching the final target temperatures, Ti,out and Tj,out, the hot process stream i has to release surplus heat qic to the external cold utilities, and the cold process stream j needs to supplement deficit energy qhj from the external hot utilities. Equations 12 and 13 are the total consumption for the external cold (FCi(ti,K+1 − Ti,out)) and hot (FCj(Tj,out − tj1)) utilities.

∑ qic = FCi(ti ,K + 1 − Ti ,out) qhj = FCj(Tj ,out − t j1)

Λ em =

∀i∈0

Λ mc =

∀j∈1

h∈/

qimk = fcme (Tme ,out − Tme ,in) + Λ em

e λimk

∀m∈4 (19)

∑ ∑

c λmjk +

j∈1 k∈2

(13)

∑ λmcc

∀m∈4 (20)

c∈C

Equations 21 and 22 present a simple method to determine the heat-capacity flow rate of the working medium m in the evaporator and the condenser, fcem and fccm. Here, f m is the mass flow rate of the circulated working fluid m, ΔHm(Te, Pe) and ΔHm(Tc, Pc) are the enthalpy changes in the evaporators and the condensers, Lm(Te, Pe) and Lm(Te, Pe) are the latent heats.

Equation 14 shows the heat balance for the working medium m in an ORC over the evaporators, and eq 15 is the heat balance around the condensers. The total heat transferred from the hot process streams to the working fluid m, with heat capacity flow rate fcme , can increase the working fluid temperature from Tem,in to Tem,out and then evaporate the working fluid into high-pressure vapor, as shown in eq 14. Equation 15 shows similar mechanism around the condensers. Note that the heat involved in the balance of working medium is composed of both sensible and latent heat.

∑∑

∑∑ i∈0 k∈2

(12)

c∈*

∑

∀m∈4

c∈*

c∈C

(9)

i∈0

∑ rwmjk = 1

(17)

∑ qmc = fcmc (tmc , K + 1 − Tmc ,out) + ∑ λmcc

∀ j ∈ 1, k ∈ 2

m∈4

∑ rwimk = 1

j∈1

∀ m ∈ 4, k ∈ 2 (8)

rcmjk = 1

(16)

j∈1

∀ i ∈ 0, k ∈ 2

m∈4

i∈0

∀ m ∈ 4, k ∈ 2

(7)

Also note that those split ratios at each specific location sum to one, as shown in eqs 8−11. j∈1

(15)

e e ∑ qimk = fcme (tmk − tme , k + 1)+ ∑ λimk

c − tmc , k + 1) qmjk = rcmjkFCj(tcmjk − t j , k + 1) = rwmjkfcmc (tmk c λmjk

c∈*

Equation 16 is the heat balance of working medium m at e e stage k, where tmk and tm,k+1 are the input and output e temperatures. λimk denotes the latent heat if its temperature reaches the boiling point. Equations 17 and 18 show the heat balance around the condenser at stage k.

(5)

e qimk = rhimk FCi(tik − thimk ) = rwimkfcme (tmk − tme , k + 1) e + λimk

∑ qmc = fcmc (Tmc ,in − Tmc ,out) + Λcm

∀m∈4

qijk = rhijk FCi(tik − thijk) = rcijkFCj(tcijk − t j , k + 1) ∀ i ∈ 0, j ∈ 1, k ∈ 2

qmjk +

fcme = fm

ΔHm(T e , P e) − Lm(T e , P e) Tme ,out − Tme ,in

∀m∈4 (21)

c

fcmc = fm

∀m∈4

i∈0 k∈2

(14)

c

c

c

ΔHm(T , P ) − Lm(T , P ) Tmc ,in − Tmc ,out

∀m∈4 (22)

E



Article

Equation 34 is the generated work from the turbine by applying the evaporated high-pressure working fluid. Note that the isentropic efficiencies in the pump and the turbine have been included.

On the basis of the proposed ORC-integrated superstructure, working fluid in an ORC is circulated through a series of evaporators and condensers. The working fluid absorbs waste heat from the hot process streams in series. The absorbed energy can first raise the working fluid temperature from the initial value Tem,in to its boiling point Tem,out, and then evaporate the working fluid. Binary variable ye,1 mk is used to designate the occurrence of evaporation in the evaporator at stage k. ye,1 mk = 1 denotes the occurrence of evaporation and the working temperature remains at its boiling point, whereas ye,1 mk = 0 means there is no evaporation and the output temperature should be less than its boiling point. This situation can be represented by the following disjunction, and eqs 23−27 represent the equivalent formulation.16 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Wmt = fm ΔHmt ηmt

e,1 e,1 tmk = (Tme ,out)ymk

∀ m ∈ 4, k ∈ 2

≤

(Tme ,out

− δ)(1 −

≥

Tme ,in(1

e,1 ymk )

e,1 ymk )

t j , k + 1 = Tj ,in tme , K + 1 = Tme ,in tmc 1 = Tmc ,in

−

e e,1 ≤ (Λ em)ymk ∑ λimk

=

+

tmc,2, k + 1

c,1 tmc,1, k + 1 = (Tmc ,out)ymk

∀ m ∈ 4, k ∈ 2

c,1 tmc,2, k + 1 ≥ (Tmc ,out + δ)(1 − ymk )

(23)

Tmt ,in = Tme ,out

∀m∈4

(41)

(24)

Tmt ,out = Tmc ,in

∀m∈4

(42)

Equations 44−50 guarantee the decreasing temperature of hot process streams and the increasing temperature of cold process streams when they flow through the HENs. Similar assignments are also given for the working fluid.

(26)

∑ j∈1

≤

c λmjk

Tmc ,in(1 ≤

−

c,1 ymk )

ti , K + 1 ≥ Ti ,out

(28)

c,1 (Λ mc )ymk

∀i∈0

(43) (44)

t jk ≥ t j , k + 1

∀ j ∈ 1, k ∈ 2

(45)

Tj ,out ≥ t j ,1

∀j∈1

(46)

e tmk ≥ tme , k + 1

∀ m ∈ 4, k ∈ 2

(47)

Tme ,out ≥ tme 1

∀m∈4

(48)

c tmk ≥ tmc , k + 1

∀ m ∈ 4, k ∈ 2

(49)

tmc , K + 1 ≥ Tmc ,out

∀m∈4

(50)

The temperature differences defined in eqs 51−62 ensure that the temperature difference is always positive when a certain match of heat exchange exists. Furthermore, eq 63 is used to guarantee all of the temperature differences will be greater than the given minimum approach temperature ΔTmin.

(29)

∀ m ∈ 4, k ∈ 2

∀ m ∈ 4, k ∈ 2

∀ i ∈ 0, k ∈ 2

tik ≥ ti , k + 1

(30)

tmc,2, k + 1

(38)

(40)

(27)

∀ m ∈ 4, k ∈ 2

∀m∈4

(37)

∀m∈4

c,1 c,1 ⎧ ⎫ ⎧ ⎫ ¬ymk ymk ⎪ ⎪ ⎪ ⎪ c c ⎪ ⎪ tc ⎪ ⎪T c ≥ t c = ≥ T T m,k+1 m ,out ⎪ m ,out ⎪ ⎪ m,k+1 ⎪ m ,in ⎬ ⎨ ⎬∨⎨ c c =0 ∑ λmjk ⎪ ⎪ ∑ λmjk ≥ 0 ⎪ ⎪ ⎪ ⎪ j∈1 ⎪ ⎪ j∈1 ⎪ ⎪ ⎪ ⎪ ⎩ m ∈ 4, k ∈ 2 ⎭ ⎩ m ∈ 4, k ∈ 2 ⎭

Tmc,1, k + 1

∀m∈4

(36)

Tmp ,out = Tme ,in

The following disjunction represents similar situation in the condensers in series, and eqs 28−32 describe the equivalent formulation.

tmc , k + 1

∀j∈1

(35)

(39)

∀ m ∈ 4, k ∈ 2

i∈0

(34)

∀m∈4

∀ m ∈ 4, k ∈ 2

∀ m ∈ 4, k ∈ 2

(33)

Tmp ,in = Tmc ,out

(25) e,2 tmk

∀m∈4

∀i∈0

ti1 = Ti ,in

e,1 ⎫ ⎧ ⎫ ¬ymk ⎪ ⎪ ⎪ e ⎪ ⎪ ⎪ Te ≤ te ≤ Te = Tme ,out tmk mk m ,out ⎪ ⎪ ⎪ m ,in ⎬ ⎬∨⎨ e e ≥0 =0 ∑ λimk ∑ λimk ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i∈0 i∈0 ⎪ ⎪ ⎪ ∀ m ∈ 4, k ∈ 2 ⎭ ⎩ ∀ m ∈ 4, k ∈ 2 ⎭

∀ m ∈ 4, k ∈ 2

∀m∈4

Equations 35−42 assign some equivalent temperatures for the process streams and the circulated working fluids in an ORC.

e,1 ymk

e e,1 e,2 tmk = tmk + tmk

e,2 tmk

Wmp = fm ΔHmp /ηmp

in dtijk ≤ tik − tcijk + Γ(1 − zijk)

(31)

∀ i ∈ 0, j ∈ 1, k ∈ 2

∀ m ∈ 4, k ∈ 2

(51)

out dtijk ≤ thijk − t j , k + 1 + Γ(1 − zijk)

(32)

Equation 33 gives the shaft work Wpm required for compressing the working fluid m from a low-pressure and subcooled liquid to high enough pressure for subsequent evaporation. The term ΔHpm denotes the change in enthalpy due to compression, which can be computed from the operating conditions.

∀ i ∈ 0, j ∈ 1, k ∈ 2

(52)

in e dtimk ≤ tik − tmk + Γ(1 − zimk)

∀ i ∈ 0, m ∈ 4, k ∈ 2 F

(53)



Article

previously determined external cold utility consumption is added as a constraint, as shown in eq 68.

out dtimk ≤ thimk − tme , k + 1 + Γ(1 − zimk)

∀ i ∈ 0, m ∈ 4, k ∈ 2 in dtmjk

≤

tmc , k + 1

(54)

zhj = 0

− t j , k + 1 + Γ(1 − zmjk)

∀ m ∈ 4, j ∈ 1, k ∈ 2

Based on the stand-alone HEN obtained in P1, the problem of ORC-integrated HENS for maximizing the work produced from the recovered low-grade waste heat can be formulated as the following MINLP problem, P2,

(56)

∀ i ∈ 0, c ∈ *

P2: max J2 =

(57)

dticout

≤ ti , K + 1 − Tc ,out + Γ(1 − zic)

x2 ∈ Ω 2

∀ i ∈ 0, c ∈ * ∀ h ∈ /, j ∈ 1 (59)

dthjout ≤ Th ,out − t j1 + Γ(1 − zhj)

∀ h ∈ /, j ∈ 1 (60)

in dtmc

≤

Tmc ,out

− Tc ,in + Γ(1 − zmc)

∀ m ∈ 4, c ∈ * (61)

out dtmc ≤ tmc , K + 1 − Tc ,out + Γ(1 − zmc)

∀ m ∈ 4, c ∈ *

(62)

⎧ ijk , ic , hj , imk , mjk , mc ⎫ ⎪ ⎪ ∀ † ∈ ⎨ ∀ i ∈ 0, j ∈ 1, k ∈ 2,⎬, ⎪ ⎪ ⎩ h ∈ /, c ∈ *, m ∈ 4 ⎭

dt†* ≥ ΔTmin

* ∈ {in, out}

e ⎡ rwimkfcme (tmk − tme , k + 1) ⎤ e e, n ⎥ − tmk ⎢thimk + ) + Γ(1 − ymk rhimk FCi ⎦ ⎣

(64)

c ⎡ rwmjkfcmc (tmk − tmc , k + 1) ⎤ ⎢ ⎥ + Γ(1 − y c, n ) − tcmjk − mk rcmjkFCj ⎢⎣ ⎥⎦

≥ ΔTmin

∀ i ∈ 0, j ∈ 1, m ∈ 4, k ∈ 2 L

(65) U

Equation 66 gives lower and upper bounds, Q and Q , heat exchange loads, where index † is as defined in eq 63. Q ZL† ≤ q† ≤ Q ZU†

∑

Wmp

m∈4

5. NUMERICAL EXAMPLE A literature example adapted from the work of Desai and Bandyopadhyay13 is used to demonstrate the efficiency of the proposed strategy for implementing an ORC-integrated waste heat recovery system on a background process. To solve the MINLP model for HENS, with or without the involvement of an ORC, the General Algebraic Modeling System (GAMS)17 is used as the main solution environment on a Core 2, 2.53 GHz, 1.00 GB RAM processor with BARON as the MINLP solver. This example involves three hot and four cold process streams along with steam and cooling water as the heating and cooling utilities. Given data for the problem, including the supply and target temperatures of the hot and cold process streams, their heat-capacity flow rates, the available utilities, etc., are listed in Table 2 (P1). Suppose ΔTmin is 20 °C, one can solve P1 to find a HEN with minimum hot and cold utilities, such as shown in Figure 3a. The resulting HEN consists of nine process-to-process heat exchangers to recover 1613.7 kW inside the process (755.9/857.8 kW from above/below the pinch), two heaters on C1 and C4, and one cooler on H3 with the targeting hot and cold utilities, (QHU)*P1 = 244.1 kW and (QCU)*P1 = 172.6 kW. The hot and cold pinch temperatures are cold found from Figure 3a, Thot pinch = 244 °C and Tpinch = 224 °C. Note the temperature for the hot process stream below the pinch is still quite high and the surplus heat of 172.6 kW is released to the cooling water. Thus, it is worth recovering part of the waste heat from the heat surplus zone below the process pinch via integration of an ORC. P1 involves 429 constraints,

Note that eqs 51−62 apply to heat exchangers without phase change. However, the working fluid usually involves phase change when absorbing waste heat from hot process streams or discharging heat to cold process streams. The temperature differences inside these heat exchangers need to be further restricted, as shown in eqs 64 and 65.

tmc , k + 1

m∈4

Wmt −

Ω 2 = {x 2|eqs 1−68} (63)

≥ ΔTmin

∑

⎧ z , z , z ; z , z , z ; y e,1 , y c,1 ; Λ e , Λ c ⎫ m m ⎪ ⎪ ijk ic hj imk mjk mc mk mk e c c ⎪ ; λimk ⎪ , λmjk , λmc ; ⎪ ⎪ ⎪ q , q , q ; q , q , q ; f , fc e , fc c ; t , t ⎪ ⎪ ijk ic hj imk mjk mc m m m ik jk ⎪ e c e,1 c,1 ⎪ ; tmk ⎪ , tmk , tmk , tmk ; ⎪ ⎪ ⎪ rh , rh , rc , rc , rw , rw ; th , th , t ⎪ ijk imk ijk imk imk mjk ijk imk ⎬ x2 ≡ ⎨ e,2 c,2 ⎪ cijk , tcmjk ; tmk ⎪ , tmk ; ⎪ ⎪ ⎪ dt in , dt out ; dt in , dt out , dt in , dt out ; dt in , dt out ⎪ imk imk mjk mjk ic ic ⎪ ijk ijk ⎪ in out ⎪ ; dthjin , dthjout ; dtmc ⎪ , dtmc ⎪ ⎪ ⎪ ∀ i ∈ 0, j ∈ 1, k ∈ 2, m ∈ 4, h ∈ /, c ⎪ ⎪ ⎪ ⎩ ∈* ⎭

(58)

dthjin ≤ Th ,in − Tj ,out + Γ(1 − zhj)

(68)

i∈0 c∈*

out c dtmjk ≤ tmk − tcmjk + Γ(1 − zmjk)

dticin ≤ Ti ,out − Tc ,in + Γ(1 − zic)

(67)

∑ ∑ qic ≤ (QCU )*P1

(55)

∀ m ∈ 4, j ∈ 1, k ∈ 2

∀ h ∈ /, j ∈ 1

(66)

Since in this work the integration of an ORC is aimed at further recovering low-grade waste heat from the background process, it should not increase the consumption of external hot utility. Therefore, eq 67 is added to prevent the cold process streams from using additional hot utility. In addition, the G



Article

the formulation and the use of solver BARON for global search, it takes long to find optimal solutions, as shown in Table 3. However, the aim of the present work is mainly to develop a mathematical technique for HENS with an ORC for further recovery of low-temperature waste heat. It is worth to develop effective solution strategies in the future to facilitate problem solving. Note that the net work rate under various condensation temperatures ranges from 36 to 49 kW, and there is an abrupt change around Tcout = 73 °C. The sharp increase in the net work rate is due to the possible recovery of waste thermal energy from the expanded working fluid to the cold process stream C3, as illustrated in the following. Two sets of temperature pairings, Tcout = 70 °C with Teout = 200 °C and Tcout = 73 °C with Tcout = 187 °C, are illustrated, where the resulting ORC-integrated HEN configurations are shown in Figure 3b and c. As the HEN configuration in Figure 3b is for the former scenario, the process-to-process heat exchangers below the pinch have recovered 752.3 kW of thermal energy. The saturated liquid n-hexane with a flow rate of f m = 0.46 kg/s under 17.46 bar recovers additional 257.07 kW surplus heat from hot process stream H3 and then evaporates into saturated vapor at 200 °C. The 17.46 bar nhexane vapor passes through the turbine and generates Wt = 37.83 kW electricity rate with an exit state of superheated vapor under 1.05 bar and 142.31 °C. This low-pressure n-hexane vapor is cooled down to its saturated liquid at 70 °C by delivering 5.3 and 79.2 kW to cold process streams C3 and C4 and releasing 136.63 kW residue heat to cooling water. The liquid n-hexane is then compressed into 17.46 bar again by a pump, with a pumping work 1.87 kW, for next circulation. The resulting net work generated from circulating the n-hexane in the ORC is 37.83 − 1.87 = 35.97 kW, and the thermal efficiency of the ORC is (35.97/257.07) × 100% = 13.99%. For the scenario of Tcout = 73 °C with Teout = 186.5 °C, Table 3 and Figure 3c show that the process-to-process heat recovery rate below the pinch is reduced to 644.3 kW. However, the flow rate of working fluid and the recovered waste energy increase to 0.69 kg/s and 366.11 kW, and the rate of net useful work also augments to 47.97 kW. The working fluid at the turbine exit delivers 5.3 + 82 = 87.3 and 104.8 kW to cold process streams C3 and C4, which relax the energy requirements of C3 and C4 from the hot process stream H3. Thus, more process waste heat can be transferred from H3 to the working fluid to increase the flow rate of working fluid and the generation of net useful work. The dramatic difference between these two scenarios come from the assignment of minimum approach temperature ΔTmin = 20 °C. A lot of condensation and evaporation temperature pairings are investigated furthermore to find the optimal operating condition which can generate the maximal net useful work. It is found that Tcout = 77 °C and Teout = 180 °C provide the maximum net useful work, 48.28 kW, such as shown in Table 3 and Figure 3d. The process-to-process heat recovery rate is reduced furthermore to 618.9 kW. The circulated n-hexane flow rate is increased to 0.77 kg/s under pressure 12.75 bar. The saturated liquid n-hexane can recover 391.48 kW waste heat from H3 and then generate 50.48 kW power through turbine expansion. The 1.30 bar n-hexane vapor provides 99.4 and 119.5 kW thermal energy for heating C3 and C4, which also causes the reduction of cold utility consumption to 124.32 kW. Pumping work of 2.20 kW increases the liquid pressure for further circulation. The net useful work generated from the

Table 2. Original Supply/Target Stream Temperatures (P1) and the Updated Temperatures (P2) for Integration of an ORC (ΔTmin = 20 °C) P1: stand-alone HEN

P2: ORC-involved HEN

stream

FCp (kW/K)

Tin (°C)

Tout (°C)

Tin (°C)

Tout (°C)

H1 H2 H3 C1 C2 C3 C4 steam CW

9.802 2.931 6.161 7.179 0.641 7.627 1.690

353 347 255 224 116 53 40 377 20

313 246 80 340 303 113 293 377 30

313 246 244 224 116 53 40 377 20

313 246 80 224 224 113 224 377 30

295 continuous variables, and 43 binary variables. The computation time is 53 s. From the resulting HEN shown in Figure 3a, two hot (H1 and H2) and one cold (C1) process streams are located in the heat deficit zone above the pinch. These streams above the process pinch will be excluded in the subsequent ORCintegrated HENS. Also note that cold process stream C3 is totally located in the zone below the pinch. Whereas the temperatures of H3, C2, and C4 range across the process pinch. These facts can be used to update the new inlet temperature for the hot process stream H3, and the new outlet temperature for the two cold process streams C2 and C4 for subsequent ORCintegrated HENS, such as those listed in Table 2 (P2). Therein, the supply temperatures for H1 and H2, and the target temperature for C1 have been updated to relax their effective involvement in P2, Ti=1,in = Ti=1,out = 313 °C, Ti=2,in = Ti=2,out = 246 °C, and Tj=1,out = Tj=1,in = 224 °C. Furthermore, for those process streams with temperatures ranging across pinch point, the pinch temperature is assigned as their updated supply/ target values, i.e., Ti=3,in = Thot pinch = 244 °C and Tj=2,out = Tj=4,out = Tcol pinch = 224 °C. Whereas the supply/target temperatures for C3 remain the same as P1. With the updated stream supply/target temperatures in the zone below process pinch, one can solve P2 for synthesizing a modified HEN involving an ORC for maximum recovery of waste heat. Note the supply temperature of the surplus heat source is 244 °C. Thus, n-hexane is selected as the working fluid for implementing the ORC since its critical temperature Tc = 234.7 °C is higher than the cold pinch temperature, 224 °C. In this example, Desai and Bandyopadhyay13 reported that setting the evaporation temperature at 207.5 °C gave the highest efficiency for converting the absorbed waste heat to useful work. However, the design objective in this article is to maximize the net useful work from the recovered waste heat. Some pairings of condensation and evaporation temperatures around these operating ranges are investigated to find the net useful work generated by the ORC-integrated waste-heat recovery system. Both the isentropic efficiencies for the pump and the turbine are assumed to have constant values, ηp = 0.65 and ηt = 0.8. The P2 problem involves 214 constraints, 131 continuous variables, and 28 binary variables. By solving P2 under each assigned condensation temperature, the associated evaporation temperature in the ORC which results in the maximal net useful work is depicted in Table 3, including the operation pressure, mass flow rate of the working fluid, net work from the waste heat, the computation time, etc. Due to nonlinearities of H



Article

Figure 3. continued

I



Article

Figure 3. (a) Synthesized HEN without ORC. ORC-integrated HENs with (b) Tc = process streams when they flow at 70 °C and Te = 200 °C; (c) Tc = 73 °C and Te = 186.5 °C; (d) Tc = 77 °C and Te = 180 °C.

waste heat by using the ORC is 48.28 kW, with a thermal efficiency (48.28/391.48) × 100% = 12.33%.

The optimized result is better than the design reported by Desai and Bandyopadhyay,13 where 42.7 kW net work is J



Article

Table 3. Main Results under Various Condensation and Evaporation Temperatures Tcout

Tein

Teout

Tcin

Pmax

(°C) 50 55 60 65 70 73 75 76 77 78 80 85 90

51.28 56.29 61.27 66.29 71.21 73.97 75.87 76.72 77.87 78.87 80.88 85.73 90.73

207 207 205 205 200 186.5 180 180 180 180 180 170 170

Pmin (bar)

135.74 138.33 139.89 142.35 142.31 136.77 134.21 134.67 135.13 135.59 136.51 133.24 135.46

19.36 19.36 18.80 18.80 17.46 14.17 12.75 12.75 12.75 12.75 12.75 10.76 10.76

Wt

f

Wp

(kg/s) 0.54 0.64 0.76 0.90 1.05 1.16 1.23 1.27 1.30 1.34 1.42 1.64 1.89

0.37 0.38 0.40 0.43 0.46 0.69 0.73 0.75 0.77 0.77 0.77 0.86 0.88

Wnet

(kW) 39.06 38.60 37.86 38.36 37.83 50.20 49.69 50.09 50.48 49.84 48.50 46.16 43.57

(%)

1.64 1.73 1.76 1.88 1.87 2.23 2.11 2.16 2.20 2.20 2.19 1.99 2.00

■

generated under Tcout = 80 °C and Teout = 186.5 °C. However, the generated electricity is much less than the design reported by Valencia et al.14 Therein, the trade-off between the energy and capital cost is considered. In their design, the consumption of hot utility is increased from 244.1 to 297.13 kW, and the generated electricity is raised dramatically to 98.22 kW. This dramatic jump of electricity generation comes from the increased transfer rate of thermal energy from hot process stream H3 to the compressed working fluid liquid (682.11 kW) and from the expanded working fluid vapor to the cold process streams C3 and C4 (457.62 and 128.27 kW). The feasibility of these large amounts of heat transfer should be examined furthermore if one considers the limitation of the minimum approach temperature inside the heat exchanger where the working fluid proceeds with phase change. Note that our approach limits the total demand of hot utility not larger than the stand-alone design and also assigns a minimum approach temperature ΔT = 20 °C. The total hot utility in the design of all scenarios remains the same value since the ORC is restricted to recover waste heat after the execution of process heat integration.

Qe

eff

37.40 36.87 36.11 36.47 35.97 47.97 47.57 47.94 48.28 47.65 46.32 44.16 41.57

QCU

time

135.20 135.73 136.49 136.12 136.63 124.63 125.02 124.66 124.32 124.95 126.28 128.43 131.03

22 698 29 756 57 600 57 600 18 000 57 600 50 400 50 400 57 600 50 400 30 111 4219 21 494

(kW)

16.44 15.87 15.26 14.71 13.99 13.10 12.56 12.45 12.33 12.22 11.99 10.83 10.25

227.50 232.23 236.66 247.99 257.07 366.11 378.80 385.17 391.48 389.92 386.13 407.63 405.44

(s)

APPENDIX The Peng−Robinson equation of state P=

V aV − 2 V−b RT (V + 2bV − b2)

where a = 0.45724 b = 0.0778

(RTc)2 α Pc

RTc Pc

2 ⎡ ⎛ ⎛ T ⎞0.5⎞⎤ ⎢ ⎥ ⎜ ⎟ α = 1 + κ ⎜1 − ⎜ ⎟ ⎟ ⎢ ⎝ Tc ⎠ ⎠⎥⎦ ⎝ ⎣

κ = 0.37464 + 1.54226ω − 0.26992ω 2 ⎡ ⎤ ⎛P ⎞ ⎥−1 ω = ⎢ −log10⎜ s ⎟ ⎢ ⎥ P ⎝ ⎠ c T / T = 0.7 ⎦ ⎣ c

6. CONCLUSION This article aims to propose a systematic procedure to synthesize an ORC-integrated heat recovery system on a background process. A generic stagewise superstructure for HENS adapted from Yee et al.1 was presented first. This supersturcture was then expanded to include an ORC for recovering waste heat from the heat surplus zone below the process pinch. Based on the stand-alone and ORC-integrated superstructures, the HENS problem is formulated as two-stage MINLPs. A stand-alone HEN with minimum external utility consumption is synthesized in the first stage (problem P1). An ORC is then integrated into the below-pinch heat surplus zone for further heat recovery, and this second-stage problem (P2) is solved with the objective of maximizing the work produced from waste heat. In ORC-integrated HENS, the heat exchange matches in the heat deficit zone (above the pinch) remain the same, while the heat recovery configuration in the heat surplus zone changes with updated supply/target temperatures of the hot and cold streams. One literature example was solved to demonstrate the application of the proposed ORC-integrated HENS on an background process for generating useful work from the waste heat.

⎛ Bm ⎞ log10(Ps) = A m − ⎜ ⎟ ⎝ T + Cm ⎠

Therein, Pc and Tc are the critical pressure and temperature, Ps is the saturated pressure, Am, Bm, and Cm are the Antonine equation parameters. One can find the compressibility factor Z = PV/RT by solving the following cubic equation (R is gas constant) 0 = Z3 + (B − 1)Z2 + (A − 3B2 − 2B)Z + (B3 + B2 − AB)

where A=

aP (RT )2

and

B=

bP RT

The enthalpy and latent heat of the working fluid, H(T, P) and L(T, P), can be calculated with the following formulas H(T , P) = ΔHIG + (H − HIG)T , P

where K


Industrial & Engineering Chemistry Research ΔH

IG

∫T

=

T

Cp dT

and

ref

Article

ΔHpm = specific enthalpy difference of medium m in pump

⎛ T1.47 ⎞n Cp = k ⎜ ⎟ ⎝ T + 140 ⎠

Binary Variables

zijk = existence of a unit for the match (ij) in stage k zimk = existence of a unit for the match (im) in stage k zmjk = existence of a unit for the match (mj) in stage k zic = existence of a unit for the match (ic) zhj = existence of a unit for the match (hj) zmc = existence of a unit for the match (mc) e,n ymk = binary variables of evaporator used to model disjunction c,n ymk = binary variables of condenser used to model disjunction

(H − HIG)T , P = RT (Z − 1) T +

( ddTa ) − a ln⎡ Z + (1 + 2 2b

⎢ ⎣ Z + (1 −

2 )B ⎤ ⎥ 2 )B ⎦

⎛ Tc − T ⎞0.38 L(T , P) = Lref ⎜ ⎟ ⎝ Tc − Tref ⎠

In these equations, Cp is the heat capacity of ideal gas, H and HIG are the specific enthalpy of working fluid and the specific enthalpy of ideal gas, Lref is the specific enthalpy of evaporation or condensation at reference state. The numerical values of key parameters for some selected working fluids are given in Table 1.

■

Positive Variables

dtinijk = temperature difference for match (ij) at the hot end of the heat exchanger dtout ijk = temperature difference for match (ij) at the cold end of the heat exchanger dtinimk = temperature difference for match (im) at the hot end of the heat exchanger dtout imk = temperature difference for match (im) at the cold end of the heat exchanger dtinmjk = temperature difference for match (mj) at the cold end of the heat exchanger dtout mjk = temperature difference for match (mj) at the hot end of the heat exchanger dtinic = temperature difference for match (ic) at the cold end of the heat exchanger dtout ic = temperature difference for match (ic) at the hot end of the heat exchanger dtinhj = temperature difference for match (hj) at the hot end of the heat exchanger dtout hj = temperature difference for match (hj) at the cold end of the heat exchanger dtinmc = temperature difference for match (mc) at the cold end of the heat exchanger dtout mc = temperature difference for match (mc) at the hot end of the heat exchanger f m = mass flow rate of medium m fcem = heat capacity flow rate of working medium m in evaporator part fccm = heat capacity flow rate of working medium m in condenser part qijk = heat exchanged between hot steam i and cold steam j in stage k qimk = heat exchanged between hot steam i and working medium m in stage k qmjk = heat exchanged between working medium m and cold steam j in stage k qic = heat exchanged between hot steam i and cold utility qhj = heat exchanged between hot utility and cold steam j qmc = heat exchanged between working medium m and cold utility rhijk = split ratio of hot i that is connected to cold j at stage k rhimk = split ratio of hot i that is connected to medium m in stage k rcijk = split ratio of cold j that is connected to hot i in stage k rcmjk = split ratio of cold j that is connected to medium m in stage k rwimk = split ratio of working fluid m connected to hot i in stage k rwmjk = split ratio of working fluid m connected to cold j in stage k

AUTHOR INFORMATION

Corresponding Author

*Tel.: 886-2-33663039. Fax: 886-2-23623040. E-mail: CCL@ ntu.edu.tw. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank the National Science Council of ROC for supporting this research under Grant NSC102-2221-E-002216-MY3.

■

NOMENCLATURE

Indices, Sets, and Abbreviations

i ∈ 0 = {1, ..., I}, hot process stream below the pinch j ∈ 1 = {1, ..., J}, cold process stream below the pinch m ∈ 4 = {1, ..., M}, working medium k ∈ 2 = {1, ..., K}, stage h ∈ / = {1, ..., H}, hot utility c ∈ * = {1, ..., C}, cold utility in, out = inlet and outlet e, c, t, p = evaporator, condenser, turbine, and pump Parameters

Ti,in, Ti,out, Tj,in, Tj,out = inlet and outlet temperatures Tem,in, Tem,out = inlet and outlet temperatures of medium m in evaporator Tcm,in, Tcm,out = inlet and outlet temperatures of medium m in condenser hot Tcold pinch, Tpinch = the cold and hot pinch temperatures ΔTmin = minimum-approach temperature difference FCi, FCj = heat capacity flow rates K = total number of stages Lcm = specific latent heat difference of medium m in condenser Lem = specific latent heat difference of medium m in evaporator QU, QL = upper and lower bounds for heat exchange Γ = upper bound for temperature difference ΔHme = specific enthalpy difference of medium m in evaporator ΔHmc = specific enthalpy difference of medium m in condenser ΔHtm = specific enthalpy difference of medium m in turbine L



Article

Rankine cycles using molecular design and sensitivity analysis. Ind. Eng. Chem. Res. 2013, 52, 12116. (13) Desai, N. B.; Bandyopadhyay, S. Process integration of organic Rankine cycle. Energy 2009, 34, 1674. (14) Valencia, B. J. H.; Castro, E. R.; Ortega, J. M. P.; Gonzalez, M. S.; Rivera, F. N.; El-Halwagi, M. M. Optimal integration of organic Rankine cycles with industrial processes. Energy Convers. Manage. 2013, 73, 285. (15) Lira-Barragán, L. F.; Ponce-Ortega, J. M.; Serna-González, M.; El-Halwagi, M. M. Sustainable integration of trigeneration systems with heat exchanger networks. Ind. Eng. Chem. Res. 2014, 53, 2732. (16) Ponce-Ortega, J. M.; Jiménez-Gutierrez, A.; Grossmann, I. E. Optimal synthesis of heat exchanger networks involving isothermal process streams. Comput. Chem. Eng. 2008, 32, 1918. (17) GAMS: A User’s Guide; GAMS Development Corp.: Washington, DC, 2008.

tik = temperature of hot stream i at the temperature location k tjk = temperature of hot stream j at the temperature location k thijk = temperature of part of hot i connected to cold j in stage k tcijk = temperature of part of cold j connected to hot i in stage k thimk = temperature of part of hot i connected to medium m in stage k tcmjk = temperature of part of cold j connected to medium m in stage k temk = temperature of working medium m at location k in evaporator part tcmk = temperature of working medium m at location k in condenser part te,n mk = disaggregated variables used to model disjunction in evaporator part tc,n mk = disaggregated variables used to model disjunction in condenser part Wtm = turbine work of medium m Wpm = pump work of medium m λeimk = heat load of m given by i in stage k in evaporator part, where k ∈ 2 λcmc = heat load of m in cold utility of condenser part λcmjk = heat load of m released to j in stage k in condenser part, where k ∈ 2 Λem = overall evaporation heat load of working medium m in evaporator part Λcm = overall condensation heat load of working medium m in condenser part

■

REFERENCES

(1) Yee, T. F.; Grossmann, I. E.; Kravanja, Z. Simultaneous optimization models for heat integration-I. area and energy targeting and modeling of multi-stream exchangers. Comput. Chem. Eng. 1990, 14, 1151. (2) El-Halwagi, M. M. Process integration; Elsevier: Amsterdam, 2006. (3) Smith, R. Chemical process design and integration; John Wiley & Sons, Ltd: England, 2005. (4) Furman, K. C.; Sahinidis, N. V. A critical review and annotated bibliography for heat exchanger network synthesis in the 20th century. Ind. Eng. Chem. Res. 2002, 41, 2335. (5) Klemeš, J. J.; Kravanja, Z. Forty years of heat integration: pinch analysis and mathematical programming. Curr. Opin. Chem. Eng. 2013, 2, 461. (6) Klemeš, J. J., Ed. Handbook of process integration (PI): minimisation of energy and water use, waste and emissions; Woodhead Publishing Series in Energy; Woodhead Publishing, 2013; No. 61. (7) Roy, J. P.; Mishra, M. K.; Misra, A. Parametric optimization and performance analysis of a waste heat recovery system using Organic Rankine Cycle. Energy 2010, 35, 5049. (8) Saleh, B.; Koglbauer, G.; Wendland, M.; Fischer, J. Working fluids for low-temperature organic Rankine cycles. Energy 2007, 32, 1210. (9) Mago, P. J.; Chamra, L. M.; Srinivasan, K.; Somayaji, C. An examination of regenerative organic Rankine cycles using dry fluids. Appl. Therm. Eng. 2008, 28, 998. (10) Papadopoulos, A. I.; Stijepovic, M.; Linke, P. On the systematic design and selection of optimal working fluids for Organic Rankine Cycles. Appl. Therm. Eng. 2010, 30, 760. (11) Sprouse, C., III; Depcik, C. Review of organic Rankine cycles for internal combustion engine exhaust waste heat recovery. Appl. Therm. Eng. 2013, 51, 711. (12) Papadopoulos, A. I.; Stijepovic, M.; Linke, P.; Seferlis, P.; Voutetakis, S. Toward optimum working fluid mixtures for organic M


Heat-Exchanger Network Synthesis Involving Organic Rankine Cycle

Recommend Documents