Design and Optimization of Indirect Energy Storage Systems for Batch

Ind. Eng. Chem. Res. 2008, 47, 4817–4829

4817

Design and Optimization of Indirect Energy Storage Systems for Batch Process Plants Cheng-Liang Chen* and Ying-Jyuan Ciou Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, ROC

One of the main difficulties for handling heat integration problems in batch plants is the time-dependent existence of hot/cold process streams during time period p. This article aims at proposing a generic method for synthesizing an indirect heat exchange network and its associated thermal storage policy for targeting the external utility in a batch plant. Therein, a heat transfer medium (HTM), originally staying in a cold tank, is used to absorb surplus heat from hot process streams, and the HTM with elevated temperature is temporarily stored in a hot tank. The accumulated hot HTM is then utilized to heat subsequent cold process streams, and the cooled HTM recirculates back into its source cold tank. The limitation of heat interchange between timedependent hot/cold process streams can be relaxed by applying the recirculated heat transfer medium to carry out indirect thermal storage. A superstructure is proposed and the mathematical programming approach is applied for investigating the indirect heat exchange policy. Not using any heuristics that are based on the concepts of pinch limitations, the proposed superstructure-based representation for synthesis of indirect heat exchange networks (HEN’s) for batch processes is formulated as a mixed-integer nonlinear program (MINLP). Numerical examples are explored to demonstrate the applicability of proposed indirect HEN synthesis method for batch processes. This method is verified by two examples with different numbers of storage tanks. 1. Introduction Batch production processes have invoked more and more attention during the past decade because of the growing requirements for small amount but high-value-added products. The potential of energy saving in batch plants is thus appealing to academia and industrial people in the area of process systems engineering. However, one of the main challenges for handling heat integration problems in batch plants is the time-dependent existence of process hot/cold streams. The major ways to implement heat integration in batch process plants can be divided into direct and indirect heat interchange methods. In the direct heat exchange methods, those coexisting hot and cold process streams are synthesized for maximizing heat recovery. All existing methods for heat exchange network synthesis can be directly applied. Whereas in the indirect heat interchange method, a recirculated heat transfer medium (HTM) is used to absorb surplus heat from hot streams and then releases energy to latter cold streams. Some reservoirs are used to temporarily store recirculated cold or hot HTM to relax the limitation of heat interchange between timedependent streams. Recently, De Boer et al.,1 evaluated the technical and economical feasibility of different heat storage methods in industrial processes. Because of the novel progress of technologies for thermal storage, the energy storage can play a significant role in shifting cooling or heating loads in industries by reducing external utilities. For handling direct heat integration problem in batch plants, a time-dependent energy flow analysis, also called cascade analysis, was proposed by Kemp et al.2,3 Therein, the process streams were divided into several time intervals in problem table analysis. The cascade analysis can be used for targeting the external utilities for direct heat exchange. Zhao et al.4 presented a mathematical formulation based on cascade analysis for the production scheduling and the direct heat exchange network synthesis for each operating period. Since the hot and cold * To whom correspondence should be addressed. Tel.: 886-223636194. Fax: 886-2-23623040. E-mail: [email protected].

process streams are rarely simultaneous in a batch plant, some types of rescheduling are emphasized by some authors to achieve minimum energy targets by direct heat exchange. VaklievaBantcheva5 presented a formulation to consider of the scheduling complications with direct energy integration between different products in the same campaign. Corominas et al.6 considered the problem of designing a heat exchange strategy by means of shifting the batch processing time based on maximal heat recovery for multiproduct plants operating in campaign mode. Lee and Reklaitis7,8 developed a novel scheduling model for maximizing heat recovery between batch streams which involves countercurrent and cocurrent types contacting units and was later extended to finite heat exchange time problem. Recently, Kemp9 summarized a user guide on process integration for efficient use of energy in batch plants, where studies on some industrial cases are also emphasized. The indirect heat exchange method, with a heat transfer media (HTM) as the thermal storage conveyer, is much less schedulesensitive than the direct heat exchange method and can provide a great deal of operating flexibility. Methods of designing an indirect heat exchange system have been already proposed by several authors. Sadr-Kazemi and Polley10 proposed a concept of indirect heat storage system with targeted external utility by a pinch analysis approach. A storage line between hot and cold composite curves is used to determine reservoirs temperatures at first and then to calculate the tanks size based on the given temperatures of storage tanks. Later, Krummenacher and Favrat11 followed Sadr-Kazemi’s approach and discussed the problem of minimal number of tanks. At the same time, Krummenacher12 also proposed the heat exchange network with close and open storage tanks by applying genetic algorithm optimization method in problem solving. Papageorgiou et al.13 proposed a mathematical programming framework to calculate the variation of mass and energy holdups of heat transfer media over time under known operating policy. Georgiadis and Papageorgiou14 further extended the original method to considering fouling problem during the heat integration in multi-

10.1021/ie0710667 CCC: $40.75  2008 American Chemical Society Published on Web 06/13/2008

4818 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008

Figure 1. Conceptual structure of an indirect heat exchange network with two hot and one cold process streams and one recirculated heat transfer medium.

Figure 2. Configuration of heat exchange units: series type (a) and parallel type (b) for absorption; series type (c) and parallel type (d) for rejection; and energy reservoir (e).

purpose plants. The above two studies are based on several important assumptions, and the operating policy for each operation (i.e., heat integrated) is fixed and known a priori. Most of the previous works on the synthesis of batch HEN’s with indirect heat interchange, however, have focused on pinch analysis or known heat integrated policy. Therein, configuration of the indirect heat exchange network and its associated storage policy are usually determined sequentially. To solve the general indirect heat integration problem in batch plants, this article aims at simultaneously studying the problem of indirect heat exchange network synthesis and its associated thermal storage policy for recirculated hot/cold HTM. A novel superstructure which considers possible configurations of indirect heat exchange network and the operating policy for recirculated heat transfer medium is proposed. The mathematical programming approach is applied for investigating the indirect heat exchange policy. Not using any heuristics that are based on the concepts of pinch limitations, the proposed superstructure-based representation for synthesis of indirect heat exchange networks (HEN’s) for batch processes is formulated as a mixed-integer nonlinear program (MINLP). One numerical example from the literature will be

employed to demonstrate the applicability of proposed model. The effect of the number of energy storage tanks on the efficiency of heat recovery is also investigated. The rest of this paper is organized as follows. The problem statement of indirect heat integration in batch plants is first elucidated in the second section. Next, the concept of indirect thermal storage is introduced. Then, a superstructure based on the concept is therefore proposed for considering possible network configurations with storage tanks. A MINLP formulation follows for modeling the synthesis work. Numerical example is thereafter illustrated for demonstrating the proposed design procedure. Finally, a conclusion is made for summary. 2. Problem Statement The problem of indirect heat exchange network synthesis with thermal storage system in batch plants addressed in this paper can be stated as follows. The givens are a set of hot process streams i ∈ HP, a set of cold process streams j ∈ CP and the start and ending times of each stream, therein subsets of them HPS ⊆ HP and CPS ⊆ CP are managed by series type heat

Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4819

exchange units, and other subsets HPP ⊆ HP and CPP ⊆ CP are applying parallel type units for heat interchange; a set of periods p ∈ TP defining the operating scenarios according to the existence of process streams; and a set of reservoirs k ∈ ST for temporarily storing heat transfer medium under various operating temperatures. Also givens are input and output temperatures of hot and cold process streams; latent and sensible heats or heat capacity flow rate of each stream; existence of streams at different operating periods; minimum temperature approach for transferring heat between two streams; and properties of heat transfer medium such as density and heat capacity. The objective then is to determine the heat integration strategy in batch plants with associated indirect thermal storage system for periodically dependent inlet flows which targets the external utilities. The solution associates with the proposed indirect thermal storage system in batch plants by providing the following: the number of thermal storage reservoirs and their temperatures, the connections between the recirculated heat transfer medium (HTM) and process streams, the heat capacity flow rates of HTM, the heat flow rates of hot process streams absorbed by cold HTM and cold utilities, the heat flow rates of cold process streams supplied by hot HTM and hot utilities, the outlet temperatures of hot and cold process streams of each match in series type heat exchange units. Then, the overall heat exchange network with thermal storages is constructed on the basis of these information. 3. Conceptual Structure of Indirect Thermal Storage Systems Figure 1 illustrates the conceptual idea of the indirect thermal storage system in a batch plant with two hot and one cold process streams having different start and finish times. A recirculated heat transfer medium (HTM) is applied to absorb surplus energy from hot process streams and the resulting hot HTM is temporarily stored in a hot tank. The accumulated thermo inertia is released to latter cold process stream(s) and the incurring cold HTM is sent back to a cold reservoir. In Figure 1, the term absorption describes the process of delivering energy from hot process streams to a cold heat transferring medium and cold external utilities in series or in parallel.11 The term rejection reveals the process of releasing energy to cold process streams from the hot heat transfer medium and hot utilities. By applying energy absorption and rejection through the use of recirculated heat transfer medium, and by using thermal storage to relax the limitation of time-dependent operations in batch plants, the potential of heat integration can reduce significantly the external utility consumption. The heat exchange operations in the indirect HEN are conducted in the series type units (S) or the parallel type units (P) according to the configuration of matching process streams with the recirculated HTM and external utilities. In a series type unit, a hot process stream releases heat to the cold HTM at first and then is followed by a supplementary cooler using external cold utilities. Similarly a cold process stream receives heat from the hot HTM and then is followed by a supplementary heater using external hot utilities. Whereas, a process stream in the parallel unit interchanges energy with HTM and complementary hot/cold utilities simultaneously. For example, a multistream heat exchaner (M) is one typical parallel unit and a jacket heat exchanger with a coil inside (C) is another typical parallel unit, where the HTM flows through the coil and the external utility flows through jacket. Heat exchange between the process stream and both the HTM and external utility proceeds simultaneously. Notably, options for matching series/parallel heat exchange units should be assigned in advance.

The HTM interchanges heat by its temperature difference, also called sensible heat. Therefore, several selection factors of HTM should be considered, such as a wide temperature range to maintain the liquid phase, high heat capacity, high density, low volatility, low corrossiveness, and low heat loss, etc. Especially, the higher heat capacity and density will cause a smaller size of storage tanks. 4. Superstructures and Model Formulation In order to model the problem of synthesizing a heat exchange network with indirect thermal storage for utility targeting, superstructures that can reflect the generic configurations of the indirect heat interchange strategy are presented as shown in Figure 2 for series/parallel type heat exchange units and thermal storage reservoirs. Therein, the indices and sets, parameters, and variables relevant to model formulation are defined in the Nomenclature section. In Figure 2a, the case of two binary (T2H) variables with unity values, zak′i ) za(H2T) ) 1, denotes that ik the cold HTM is flowing from cold tank k′ to hot reservoir k via heat exchange unit(s) for cooling hot process stream i. Similarly, the case of binary variables with unity values in Figure (C2T) 2b, zr(T2C) ) zrjk′ ) 1, indicates that the HTM comes from kj hot tank k and flows into cold reservoir k′ via a heat exchange unit for heating cold stream j. The overall cyclic operating time is divided into several operating periods p ∈ TP, according to the existence of time-dependent hot/cold process streams. With these definitions and superstructures, we can formulate an optimization model for synthesizing the indirect heat interchange system in a batch process. To simplify the formulation, it is assumed in this article that the HTM in each reservoir remains constant input/output temperatures. This tight assumption implies that all inlet (outlet) streams of one reservoir will have the same temperature, which is independent of time periods, and the time varying remainder in a storage tank will also remain at such a constant temperature. It is also assumed that all heat integrated tasks are proceeded indirectly by applying HTM. These assumptions will be relaxed in a subsequent article. 4.1. Overall Heat Balance on the Recirculated Heat Transfer Medium. The heat transfer medium (HTM) is utilized to pass energy from hot to cold process streams by way of an indirect thermal storage system. The energy absorbed by the HTM from hot process streams is temporarily reserved in a hot tank. The accumulated energy is released to latter cold process streams, and the medium will return to a cold reservoir with a reduced temperature. The HTM is recirculated around cold and hot tanks in a closed loop, thus total heat balance can be established for the recirculated HTM during heating/cooling periods, as follows.

∑

i∈HP

qai

(∑

p∈TP

)

tpZ(hot) ) ip

∑

j∈CP

qrj

(∑

p∈TP

tpZ(cold) ip

)

(1)

(hot) Here, tp is the elapsed time of operating period p; Zip and (cold) Zjp denote the predefined existence of hot i and cold j streams during time period p; qai and qrj represent the heat absorption rate of cold HTM from hot i and the heat rejection rate of hot HTM to cold j streams, respectively. The left/right-hand sides of eq 1 denote the total energy occurring between the heat transfer medium and all hot/cold process streams. 4.2. Heat Balance around Series Type Heat Exchange Units. The overall heat transfer rate of hot stream i consists of latent heat and sensible heat, expressed in the first two terms in the left-hand side of eq 2. This overall heat transfer can be achieved in series by the HTM (with a constant rate qai) and by the cold utility (with a rate of qcui), respectively. Equation 3 reflects


the similar relation for cold process stream j, where the transferred heat comes from hot HTM followed by hot utility in series. LHi

∑

(out) + (TH(in) )FHi ) qai + qcui i - THi

tpZ(cold) jp

∀ j ∈ CPS

k

1

(T2H)

p∈TP

(2) LCi

∑

Table 1. Illustration for Ordering of Reservoirs, Where Tank 3 Supplies Cold HTM for Hot Process Stream i and Tank 1 Receives the Resulting Hot HTM and Tank 1 Supplies Hot HTM for Cold Process Stream j and Tank 3 Receives the Resulting Cold HTM

+ (TC(out) - TC(in) j j )FCj ) qrj + qhuj

tpZ(cold) jp

∀ j ∈ CPS

zaki zaik(H2T) zrkj(T2C) zrjk(C2T)

0 1 1 0

2

3

0 0 0 0

1 0 0 1

4

k (T2H)

k

Σl ) 1 zali Σlk ) 1 zail(H2T) Σlk ) 1 zrlj(T2C) Σlk ) 1 zrjl(C2T)

0 0 0 0

1

2

3

4

0 1 1 0

0 1 1 0

1 1 1 1

1 1 1 1

Table 2. Illustration of Equation 21 in Three Cases zqkp

p∈TP

(3) It is reasonable to assume that for a hot process stream i (or a cold process stream j) with phase change, i.e., the heat flow rate of stream i removed (stream j added) by the HTM, qai (qrj) must be greater than its latent heat flow rate if the heat exchange between the HTM and the process stream is existent, zai ) 1 (or zrj ) 1). Otherwise, the total heat including latent heat of stream i is assumed to be removed by the utility only, i.e., zai ) 0 (or zrj ) 0). For hot process stream i, the heat flow rates removed by HTM and cold utility, respectively, are formulated below to calculate interval temperature thi between the heat exchanger and the cooler. Therein if the binary variable denoting heat exchange with HTM equals to one, zai ) 1, the latent heat and a part of sensible heat are removed by the cold HTM and the remaining sensible heat is removed by the cold utility. On the other hand, the latent heat and a part of sensible heat are removed by cold utility and remaining sensible heat is removed by the intermediate medium. LHizai

∑

tpZ(hot) ip

∑

tpZ(hot) ip

∑

+(

∀ i ∈ HPS

)FHi ) qcui

+ (tcj - TC(in) j )FCj ) qrj

∀ j ∈ CPS

p∈TP

LCj(1 - zrj)

∑

tpZ(cold) jp

+ (TC(out) - tcj)FCj ) qhuj j

∀ j ∈ CPS

p∈TP

(5) 4.3. Heat Balance around Parallel Type Heat Exchange Units. In parallel type heat exchange units, the total heat including stream i’s latent heat LHi and sensible heat SHi is removed by the recirculated HTM and the cold utility simultaneously, as depicted in eq 6. Equation 7 represents similar relation for supplying heat to cold process stream j. LHi + SHi

∑

tpZ(hot) ip

) qai + qcui

∀ i ∈ HPP

(6)

ztk

1 2 3

1 0 1

0 0 1

1 0 1

2 0 0

1 0 ×

1 denotes when the HTM is circulated from hot process stream (C2T) i to hot tank k. Binary variables zr(T2C) and zrjk′ give similar kj representations for cold stream j. These binary variables provide information about flowing paths of the recirculated HTM. Therefore, the heat removed by the medium can be calculated according to the temperature difference of various tanks, Tk Tk′, in the following equations where the recirculating paths have been decided. These restraints are relaxed if the HTM is not circulated in a closed loop. (T2H) (H2T) (Tk - Tk’)Fai - qai e q(2 - zak’i - zaik )

∀i ∈ HP, k ∈ ST (T2H) T T Fa qai g - q(2 - zak’i - za(H2T) ) ( k k’) i ik

LCi + SCi tpZ(cold) jp

) qrj + qhuj

∀ j ∈ CPP

(8)

(7)

p∈TP

4.4. Heat Balance on Recirculated HTM. Binary variable (T2H) with unity value, zak′i ) 1, is used to describe the condition that the HTM is circulated from a cold reservoir k′ to absorb surplus heat of hot process stream i. Binary variable za(H2T) ) ik

(C2T) - zrjk’ )

(9)

∀j ∈ CP, k ∈ ST 4.5. Calculation of Approach Temperature. A large enough temperature driving force is needed to guarantee feasible heat transfer between HTM and the hot/cold process streams. This restraint can be relaxed if HTM does not flow through this path (T2H) (C2T) where zak′i ) 0. The terms za(H2T) , zr(T2C) , and zrjk′ play ik kj similar roles, as shown below for series and parallel type heat exchange units. Notice the constraints on the jacket exchanger with a coil, where the closest approaching temperature during the whole operating period should be considered. For series type units (S): (T2H) thi - Tk′ + Γ(1 - zak′i ) g ∆Tmin

TH(in) i - Tk + Γ

∀i ∈ HPS, k ′ ∈ ST

1 - za(H2T) ik

(

) g ∆Tmin

Tk - tcj + Γ(1 - zr(T2C) ) g ∆Tmin kj (C2T) Tk′ - TC(in) j + Γ(1 - zrjk′ ) g ∆Tmin

∀i ∈ HPS, k ∈ ST (10) ∀ j ∈ CPS, k ∈ ST ∀ j ∈ CPS, k′ ∈ ST (11)

For parallel type units (M and C): (T2H) TH(out) - Tk′ + Γ(1 - zak′i ) g ∆Tmin i

p∈TP

∑

Σp∈TP- (zqkp - zqk,p+1)2

(T2C)

thi - TH(out) i

(4)

tpZ(cold) jp

p)3

(Tk - Tk’)Frj - qrj g -q(2 - zrkj

p∈TP

LCjzrj

p)2

∀j ∈ CP, k ∈ ST

p∈TP

LHi(1 - zai)

p)1

∀i ∈ HP, k ∈ ST (C2T) T T Fr qrj e q(2 - zr(T2C) - zrjk’ ) ( k k’) j kj

∀ i ∈ HPS

+ (TH(in) i - thi)FHi ) qai

case

Tk - TC(out) +Γ j

1 - zr(T2C) kj

(

) g ∆Tmin

∀i ∈ HPP, k′ ∈ ST ∀j ∈ CPP, k ∈ ST (12)

For multistream exchanger (M): (H2T) TH(in) ) g ∆Tmin i - Tk + Γ (1 - zaik

∀ i ∈ HPM, k ∈ ST

(C2T) ) g ∆Tmin Tk′ - TC(in) j + Γ(1 - zrkj

∀ j ∈ CPM, k′ ∈ ST (13)


For jacket exchanger with a coil (C):

Table 3. Illustration of Equation 22

∀ i ∈ HPC, k ∈ ST

- Tk + Γ (1 - za(H2T) ) g ∆Tmin TH(out) i ik (C2T) + Γ(1 - zrjk′ ) g ∆Tmin Tk′ - TC(out) j

∀ j ∈ CPC, k′ ∈ ST (14)

4.6. Maximal Number of Tanks (MNT). Binary variable with unity value, ztk ) 1, denotes the existence of storage tank k. Due to the assumption of constant input/output temperature in storage tanks, at least two reservoirs are needed for circulating HTM in an indirect thermal storage system. The maximal number of tanks can be restricted by adding an upper bound, in the following constraint. 2e

∑

ztk e MNT

(zt1, zt2, zt3) with eq 22

(1, 1, 1)

without eq 22

(1, 1, 1)

Σ∀k∈ST ztk

3

Qk,p+1 )

k

k

∑ za

(H2T) il

∀ i ∈ HP, k ∈ ST

∑

∑

zr(C2T) jl

∀ j ∈ CP, k ∈ ST

(T2H) e li

l)1 k

l)1

l)1 k

zr(T2C) g lj

(16)

l)1

Table 1 gives an illustration for a case with four reservoirs, where tank 3 supplies cold HTM for hot process stream i and (T2H) (H2T) tank 1 receives the resulting hot HTM (za3i ) 1, zai1 ) 1); tank 1 supplies hot HTM for cold process stream j and tank (T2C) (C2T) 3 receives the resulting cold HTM (zr1i ) 1, zrj3 ) 1). The right part in Table 1 is the result of implementing eq 16. Multiple reservoirs may be required for storing HTM with various temperatures. As pointed out previously, these reservoirs are ordered according to the decreasing temperature. One further assumption is that there is a minimum temperature difference, DT, between adjacent reservoirs k and k + 1, as depicted below. Tk+1 + Γztk g Tk g Tk+1 + DTztk

∀ k ∈ ST-

(17)

Positive upper and lower bounds are placed on temperature of storage tanks which can be selected as the boiling and freezing points of the HTM. T _ e Tk e T

∀ k ∈ ST

(18)

4.8. Remains in Thermal Storage Tanks. The remaining HTM of each storage tank k at the end of each period p is relevant for designing suitable tank size. The remaining mass of storage k at the start of period p + 1, Qk, p+1, is equal to the tank’s initial mass at period p, Qkp, plus the input mass and minus the outlet mass. For a cyclic operation scenario, the initial

[∑ ( [∑ ( ∀i∈HP

∀j∈CP

k∈ST

∑ za

1, 1, 0, 1,

0) 0) 1) 1)

(1, (1, (0, (0, 1

0, 0, 1, 0,

0) 0) 0) 1)

remains will be equivalent to the condition at the end of the whole cycle, as given in the following.

(15)

4.7. Temperature Order of Tanks. It has been assumed previously that the reservoirs are arranged in a descending temperature order. This restraint holds if one can guarantee that the index of the reservoir that supplies cold HTM to a hot process stream i is larger than the index of the reservoir that receives hot HTM from the same stream i and that the index of the reservoir that supplies hot HTM to a cold process stream j is smaller than the index of the reservoir that receives cold HTM from the same stream j. For cooling hot process stream i by using recirculated HTM, the sum of supplying tank variable, zali(T2H), from l ) 1 to l ) k should be less than the sum of receiving tank variable, zail(H2T), from l ) 1 to l ) k, for all k ∈ ST. Similar relations can be formulated for heat rejection units, that is, the sum of supplying tank variable, zrlj(T2C), from l ) 1 to l ) k, should be greater than the sum of receiving tank variable, zrjl(C2T), from l ) 1 to l ) k, for all k ∈ ST, as shown below.

(1, (1, (1, (0, 2

Qk1 )

za(H2T) - za(T2H) ) ik ki

zr(C2T) - zr(T2C) ) jk kj

Cp Frj

Cp(HTM)

+

Z(cold) tp + Qkp jp

za(H2T) - za(T2H) ) ik ki

Fai

]

(hot) ZiN tN + Cp(HTM) P P Frj (cold) - zr(T2C) ZjN tNP + QkNP (zr(C2T) ) (HTM) jk kj P Cp

∀i∈HP

∀j∈CP

Z(hot)tp (HTM) ip

∀ k ∈ ST, p ∈ TP-

[∑ (

[∑

] ]

Fai

]

∀ k ∈ ST (19)

4.9. Logical Constraints on Remains in Tanks. Binary variable, ztk) 1, has been defined to depict the existence of tank k; and binary variable, zqkp, is used to denote whether the remains in tank k are greater (zqkp ) 1) or smaller (zqkp ) 0) than its lower bound Q. Q_ ztk e Qkp e Qztk ∀ k ∈ ST, p ∈ TP (σ + Q _ )zqkp e Qkp e Q _ (1 - zqkp) + Qzqkp

∀ k ∈ ST, p ∈ TP (20) The effective constraints for three conditions on ztk and zqkp are demonstrated in the following. σ+Q _ e Qkp e Q for ztk ) 1, zqkp ) 1 Q _ for ztk ) 1, zqkp ) 0 ∀ k ∈ ST, p ∈ TP _ e Qkp e Q 0 e Qkp e 0 for ztk ) 0, zqkp ) 0 For keeping minimum tanks size, the remains in each existing reservoir should be equal to its lower bound for at least one period. Equation 21 guarantees the minimal tanks size since at least one zqkp value will become zero for each existing reservoir k (ztk ) 1). The unallowable case that the remaining quantity in tank k stays over its lower bound at all time points, as shown in case 3 of Table 2, is avoided by including the constraint of eq 21. Thus, eq 21 helps determine the minimum tank size and the remains in each tank are equal to its lower bound over at least one period.

∑

(zqkp′ - zqk,p′+1)2 g ztk g zqkp

∀ k ∈ ST, p ∈ TP

(21)

∀p′∈TP-

Equation 22 is also needed to guarantee that required reservoirs are ordered successively. Suppose the maximal number of tanks is assigned as three. Then, the resulting tank numbers in the final design might be three or two. The ordered tanks with and without applying the following equation (eq 22) are shown in Table 3. ztk g ztk+1

∀ k ∈ ST-

(22)

4.10. Logical Constraints on Heat Exchange Units. Binary variable, zai, is used to describe whether or not hot process stream i releases heat to cold HTM. zai ) 0 denotes that hot process stream i is cooled by using external utilities only and

4822 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 Table 4. Stream Data of Simple Examplea hot stream

THi(in) (°C)

THi(out) (°C)

t(s) (h)

t(f) (h)

LHi (MJ)

SHi (MJ)

FHi (MJ/(h °C))

type

170 150

60 30

0.3 0.0

0.9 0.5

0 0

330 180

5 3

S S

H1 H2 cold stream

TCj(in) (°C)

TCj(out) (°C)

t(s) (h)

t(f) (h)

LCj (MJ)

SCj (MJ)

FCj (MJ/(h °C))

type

20 80

135 140

0.3 0.5

0.5 1.0

0 0

230 240

10 8

S S

C1 C2

glycerol 18-290 °C Cp(HTM) ) 0.0024 MJ/(kg °C), F(HTM) ) 1.2578 kg/L

HTM a

∆Tmin ) 10 °C, DT ) 10 °C.

∑ ∑

Table 5. Existence of Streams in the Simple Example period

1

2

3

4

tp (h)

0.3

0.2

0.4

0.1

∀k∈ST

(hot) Zip

H1 H2

0 1

1 1

1 0

0 0

1 0

0 1

∑ ∑

∀k∈HP

0 1

1

2

3

tp (min)

20

10

40

4

5

6

7

65

90

90

20

Zip 1 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1 0

1 1 0 1

1 1 0 0

0 0 0

0 1 0

0 1 0

(cold) Zjp

C1 C2 C3

0 1 0

0 0 0

1 0 0

0 0 1

that the HTM is not involved. Meanwhile, zai ) 1 means that HTM is applied for indirect heat transfer, regardless of whether auxiliary external utilities are employed or not. Binary variable, zrj, plays a similar role for cold process stream j. zcui and zhuj are applied to depict the existence of a cooler for process stream i or a heater for process stream j, respectively. The constraints on heat exchange rates are given below where q and qj are positive lower and upper bounds of the heat exchange rate. qzai e qai e qzai ∀i ∈ HP qzcui e qcui e qzcui ∀ i ∈ HP qzrj e qrj e qzrj ∀ j ∈ CP qzhuj e qhuj e qzhuj ∀ j ∈ CP

(23)

Furthermore, positive upper and lower bounds, F and F, are placed on the heat capacity flow rate of the HTM, Fai or Frj, for each existing exchanger, such as, Fzai e Fai e Fzai Fzrj e Frj e Fzrj

∀ i ∈ HP ∀ j ∈ CP

∀k∈ST

za(H2T) ik

∀ i ∈ HP

zr(C2T) jk

∀ i ∈ CP

(25)

∑ + ∑

∀j∈CP ∀j∈CP

4.11. Logical Constraints on the Inlet/Outlet of Tanks. To simplify the configuration of the resulting indirect heat exchange network, it is assumed that exactly one inlet and one outlet connection are allowed for each existing exchanger that applies HTM, as shown in the following.

∀ k ∈ ST

zr(T2C) g ztk kj

∀ k ∈ ST

+ za(H2T) e ztk za(T2H) ki ik

∀ i ∈ HP, k ∈ ST

zr(T2C) + zr(C2T) e ztk kj jk

∀ i ∈ CP, k ∈ ST

(26)

(27)

4.12. MINLP Formulation for Targeting Utility. A quite common objective for network design is to minimize the total annual cost, including the costs of the utility, heat exchange units, storage tanks, and the recirculated HTM. However, the external hot/cold utility is targeted here for comparing the MINLP approach to pinch design. The utility targeting problem can be formulated as the following mixed-integer nonlinear program (P1). The objective J1 is the total utility; x1 and Ω1 denote the set of all design variables and the feasible searching space defined by all constraints, eqs 1–27. P1: min

x1∈Ω1

J1 )

∑ ∑

∀i∈HP ∀p∈TP

Z(hot) ip tpqcui +

∑ ∑

∀j∈CP ∀p∈TP

Z(cold) tpqhuj jp

Ω1 ) {x1|set of constraints, eqs 1-27} 4.13. MINLP Formulation for Minimizing Storage Tank Size. The objective J1 is used for targeting external utility consumption. However, it is found that there may exist multiple solutions for storage tank size. On the basis of the targeted utility requirement, a second objective J2 is thus further applied to guarantee the minimal total reservoirs volume. Two additional constrains are appended: eq 28 is for unit transformation from kilogram to liter and eq 29 is applied to restrict the utility consumption on its minimal target of J1. Vk g Qkp/F(HTM)

(24)

zr(C2T) g ztk jk

It is also quite obvious that one stream cannot receive HTM from one reservoir and return it to the same storage tank. Equation 27 reflects such a constraint.

(hot)

H1 H2 H3 H4

j

za(H2T) + ik za(T2H) ki

∀k∈HP

Table 6. Existence of Streams in the Single Product Batch Plant Example period

zr(T2C) kj

∀k∈ST

There is at least one inlet and one outlet stream for one existing reservoir, as shown in eq 26.

0 0

(cold) Zjp

C1 C2

∑ ) zr ) ∑

za(T2H) ) zai ) ki

∀k∈ST

∑ ∑

∀i∈HP ∀p∈TP

Z(hot) ip tpqcui +

∀ k ∈ ST, p ∈ TP

∑ ∑

∀j∈CP ∀p∈TP

(28)

Z(cold) tpqhuj e J/1 jp (29)

The minimal total reservoirs volume can be determined by solving the following MINLP (P2), where x2 and Ω2 denote the design variables and the feasible searching space defined


Figure 3. Pinch diagrams to implement indirect heat exchange for simple example.

by all constraints, eqs 1–29. P2: min J2 ) ∑ ∀k∈ST VkΩ2 ) {x2|set of constraints, eqs 1 - 29}

x2∈Ω2

For solving the MINLP formulations for the design of thermal storage system in batch process plants, the general algebraic modeling system (GAMS)15 is used as the main solution tool. All the computation is done at an Intel Core2 CPU E6300 1.86 GHz personal computer with BARON as the global MINLP solver. 5. Illustrative Examples Two examples are used in the following for demonstrating the applicability and advantages of proposed MINLP approach for indirect heat integration problems. The first example is quite simple for comparing the proposed MINLP method to typical techniques, such as the pinch design. A single product batch plant example is then used to further investigate the influence of the number of storage tanks on the indirect thermal storage system. 5.1. Simple Problem for Comparing Design Methods. Methods of designing an indirect heat integration system have been already proposed by several authors. Sadr-Kazemi and Polley10 proposed a concept of indirect heat storage system with targeted external utility by a pinch analysis approach. A storage line between hot/cold load composite curves is used to determine reservoir temperatures at first and then to calculate the tank size based on the given temperatures of storage tanks. Consider one simple problem with two hot/two cold streams, as listed in Table 4. There are four time periods which are defined according to the discontinuous existence of the streams as shown in Table 5. By applying the pinch targeting method, Figure 3 shows the hot/cold load composites and the storage line of the HTM, where the storage temperatures are determined by considering the streams supply/target temperatures and the given minimum approach temperature (10 °C). Figure 3 also shows that the maximal heat recovery is 285 MJ/cycle and the total utility consumption is 410 MJ/cycle (225/185 MJ for cold/ hot utility). Note that there may be multiple network configurations for implementing the indirect heat recovery problem. According to the information of Figure 3, one quite natural solution is to heat both two cold streams into 107.5 °C by the hot HTM (140 °C) and to cool the two hot streams into 100 °C

Figure 4. Indirect heat exchanger network of simple example with two storage tanks by (a) the indirect pinch method and (b) the proposed MINLP approach.

by the cold HTM (90 °C). The heaters and coolers are then appended for reaching the targeted temperatures. The resulting network is shown in Figure 4a. The 90 °C HTM flowing from the cold tank to the hot tank (140 °C) removes 210 MJ () 350 MJ/h × 0.6 h) from H1 over 0.3-0.9 h, which is exactly the same operating time period of H1. One cooler is applied for removing 120 MJ heat load (200 MJ/h × 0.6 h). Similarly, the cold HTM removes heat 75 MJ () 150 MJ/h × 0.5 h) from H2 and its remaining heat load 105 MJ () 210 MJ/h × 0.5 h) is removed by a cooler over 0-0.5 h. The heat load (285 MJ) transferred from two hot streams to the cold HTM is then totally released to cold streams (875 MJ/h × 0.2 h ) 175 MJ to C1 and 220 MJ/h × 0.5 h ) 110 MJ to C2) during 0.3-0.5 and 0.5-1.0 h, respectively. The maximal heat recovery is 285 MJ/ cycle, and the total utility usage is 410 MJ/cycle (200 MJ/h × 0.6 h + 210 MJ/h × 0.5 h ) 225 MJ/cycle for cold utility, and 275 MJ/h × 0.2 h + 260 MJ/h × 0.5 h ) 185 MJ/cycle for hot utility). The remains in storage tanks are given in parts a and b of Figure 5, and the volume of both storage tanks is 497 L. The resulting network configuration by the proposed MINLP approach is shown in Figure 4b, where the heat recovery (285 MJ/cycle), total utility usage (410 MJ/cycle), and the integrated network between the two hot streams and the cold HTM are


Figure 5. Remains in the thermal reservoirs and the recirculated HTM rates to corresponding streams when two tanks are applied by (a and b) the indirect pinch method and (c and d) the proposed MINLP approach.

the same as the indirect pinch method. However, the distribution of the recovered heat to cold streams is a little different to the previous design. Therein C1 and C2 are respectively allocated 100 MJ (500 MJ/h × 0.2 h) and 185 MJ (370 MJ/h × 0.5 h) from the hot HTM. The supplemented heat provided by heaters are thus 130 and 55 MJ for C1 and C2, respectively. Note that the different exchanged heat load also results in different heat capacity flow rate of the hot HTM, the output temperature of the exchanger between the hot HTM and the two cold streams, and the tank size. The remains in storage tanks are given in Figure 5c and d. The volume of both storage tanks is 298 L, which is much smaller than that of indirect pinch method. The solutions of P1 (151 equations with 29 continuous and 32 binary variables) and P2 (160 equations with 31 continuous and 32 binary variables) are obtained in 2.7 and 0.7 s of CPU time, respectively. It is worth noting that though the indirect pinch method and the MINLP approach result in the same utility usage for this simple example, the tank size by the proposed MINLP approach is smaller than that of the indirect pinch method. This result elucidates the importance of objective J2sto ensure a minimal tank size under the minimal utility constraint determined by solving objective J1 at first. Also noted that the indirect pinch method can only deal with flowing streams with series units under sequential design procedure. The MINLP approach, however, can handle various stream types and heat exchange units and can provide the utility target and final network configuration simultaneously, which is further demonstrated in the following example.

Figure 6. Flowsheet of the single product batch plant example.16

5.2. More Complex Single Product Batch Plant. This wellknown example considers a simple batch operation problem originally studied by Gremouti.16 It concerns a single product batch plant comprising two batch reactors separated by a batch distillation, as depicted in Figure 6. The process operation procedures include the following steps.16 Raw material (feed A) available at 10 °C is fed into stirred jacket reactor R1 and is further heated to 60 °C before an exothermic reaction starts to commence. Without any further heat addition the reactor


Figure 7. Gantt chart of the single product batch plant example in repeated operation with a 335min batch cycle time.16

content rises to a temperature of 100 °C. The product of reactor R1 is later discharged to the batch distillation column D1. The still operates at a temperature of 120 °C. The condenser at the top of the column operates at a temperature of 110 °C. Distillate is subsequently cooled from 110 to 50 °C and accumulates in an overhead receiver M. The product from receiver M and the second feed material (feed B) at a temperature of 15 °C are charged into reactor R2 which is equipped with a reflux condenser. The mixture has to be heated to a temperature of 95 °C before reaction starts. This reaction is again exothermic and the reactor temperature quickly rises to 135 °C where the heat of reaction is removed by allowing solvent to boil-off. The solvent then returns to the reactor after being condensed. When the reaction completes, the product is further cooled from 140 to 35 °C before going for downstream treatment. The tasks achieved by the main pieces of equipment of the batch plant (R1, D1, and R2) as a function of time are represented in the Gantt chart of Figure 7. The overall elapsed time needed to process one batch (i.e., to achieve all the operations described above) is called the batch processing time, which is 690 min in this example. To maximize the production capacity of the plant, a new batch may be started before the end of the previous batch, provided that the required equipment units are available when needed. The time separating the start of two successive batches is called batch cycle time. The smallest batch cycle time is determined by the processing unit featuring the longest processing time, R2. Observing that the cycle times of R1 and D1 are significantly shorter, R2 is clearly the time bottleneck which limits the production capacity of the plant. The hot and cold streams within a batch cycle time are shown in Figure 8. There are seven time periods which are defined according to the discontinuous existence of the streams as shown in Table 6. The stream data are given in Table 7. According to these data (original design), the total utility will be 21 225.1 MJ/cycle (Σi(LHi + SHi) + Σj(LCj + SCj)) if one does not consider the possibility of heat integration and all streams heating and cooling loads are supplied by external steam and cooling water. 5.2.1. Heat Integration with Two Storage Tanks. Suppose the heat exchange for all streams can be proceeded in seriestype (S) or parallel-type units including jacket with a coil (C) or multistream heat exchanger (M), as indicated in Table 7, and at most two reservoirs can be used for temporarily storing recirculated hot/cold HTM; an indirect heat exchange network with minimal utility can then be drawn, as shown in Figure 9. The cooling works for H1, H2, and H4 and the heating load for C2 remain as they were in the original design, i.e., HTM is not applied and heat is exchanged by using external utilities only. Streams H3, C1, and C3 are allowed to use recirculated HTM in parallel exchange units. The process flowsheet with two storage tanks is also shown in Figure 10.

Figure 8. Gantt chart of the single product batch plant example in a batch cycle.16

As shown in Figures 9 and 10, the temperature of hot HTM in tank 1 is 125 °C, which is used to provide thermal energy to cold process streams C1 and C3 over 30-70 and 70-135 min, respectively. The cooled HTM (105 °C) then flows into cold tank 2 for temporal storage. The heat capacity flow rates of HTM are 1.38 (to C1) and 2.48 (to C3) MJ/(min °C), which are also marked on the connecting lines in Figure 9. During 135-225 min, tank 2 provides the cold HTM to remove energy from hot process stream H3, and the heated HTM (125 °C) is then flowed into tank 1. Notably, there are two numerical values on those parallel units. One is the heat transfer rate (qai or qrj (MJ/min)) removed or supplied by cold or hot HTM, and the other is the rate removed (provided) by external cold (hot) utilities. Such an indirect heat integration technique with two thermal reservoirs can significantly reduce the external utility to 12 590.1 MJ/cycle (59.3% of the original design). Figure 11 shows the time-dependent remains in the two reservoirs (the upper part) and the flow rates of recirculated HTM which exchanges heat with corresponding streams (the lower part). For example, there is one input flow rate of HTM from H3 (H3T) and two output flow rates from hot tank 1 to process streams C1 (TC1) and C3 (TC3). The required volume of both storage tanks is 71 512 L. The solutions of P1 (241 equations with 41 continuous and 56 binary variables) and P2 (256 equations with 43 continuous and 56 binary variables) are obtained in 4.6 and 1.2 s of CPU time, respectively. From the heat integration network shown in Figures 9 and 10, it is easily found that the potential temperature range of hot tank 1 is between 125 and 105 °C, which is limited by the inlet temperature of H3 (135 °C - ∆Tmin) and the outlet of C1 (60 °C + ∆Tmin) and C3 (95 °C + ∆Tmin). With the same reason, the permissible temperature of cold tank 2 (124-105 °C) should be greater than the outlet temperature of C1 (60 °C + ∆Tmin) and C3 (95 °C + ∆Tmin) and lower than the outlet of H3 (134 °C - ∆Tmin). The greatest temperature difference of the two tanks will cause the smallest flow rate of the HTM and the size of storage tanks, thus the optimal temperature of the two storage tanks are finally located at 125 and 105 °C, respectively. According to these analyses, cold tank 2 is restricted by the outlet temperature of cold stream C3; otherwise, the cold tank has the opportunity to reach lower temperature. Hence, heat integration with three storage tanks is further discussed in the next section.


Figure 9. Final heat interchange strategy for the single product batch plant example when two thermal reservoirs are available, where unit for H3 is a multistream heat exchanger.

exchange is investigated. The optimal solution of the three-tank case is the same as the two-tank solution with zero storage for the third tank. This implies a surprising fact that the additional third tank cannot provide further possibility of heat recovery for this application example. The penalty of artificially applying the third tank is further probed by adapting eq 15 as follows.

∑

ztk ) 3

(30)

k∈ST

The resulting total consumption of external utility is 19 025.1 MJ/cycle (89.6% of the original design), which is higher than that of the two-tank case. The solutions of P1 (373 equations with 50 continuous and 77 binary variables) and P2 (395 equations with 53 continuous and 77 binary variables) are obtained in 25.6 and 7.8 s of CPU time, respectively. The associated heat exchange network and process flowsheet are shown in Figures 12 and 13, respectively.These figures reveals that the HTM of middle tank 2 (105 °C) discharges to cold tank 3 (70 °C) for providing heat to cold stream C1 over 30-70 min. The hot HTM (125 °C) releases heat to cold stream C3 from 70 to 135 min and is charged into middle tank 2. During the period from 135 to 225 min, the cold HTM (70 °C) from tank 3 absorbs the heat of hot stream H3 and flows into hot

Figure 10. Final flowsheet for the single product batch plant example when two thermal reservoirs are available.

5.2.2. Heat Integration with Three Storage Tanks. The scenario of applying three storage tanks for indirect heat Table 7. Stream Data of the Single Product Batch Plant Examplea hot stream

THi(in) (°C)

THi(out) (°C)

t(s) (min)

t(f) (min)

LHi (MJ)

H1

111

110

110

50

H3 H4

135 140

134 35

20 335 20 335 225 315

3149.9

H2

0 225 0 225 135 225

0 4955.4 0

TCj(in) (°C)

TCj(out) (°C)

t(s) (min)

t(f) (min)

LCj (MJ)

C1 C2

10 119

60 120

20

95

70 20 335 135

0 3529.5

C3

30 0 225 70

cold stream

HTM a

∆Tmin ) 10 °C, DT ) 10 °C.

0

SHi (MJ)

FHi (MJ/(min °C))

0 358.8

M 0.046

0 4914 SCj (MJ)

type

S M C

FCj (MJ/(min °C))

type

1100 0

C M

3217.5

C

glycerol 18-290 °C Cp(HTM) ) 0.0024 MJ/(kg °C), F(HTM) ) 1.2578 kg/L


Figure 11. Remains in the thermal reservoirs and the recirculated HTM rates to corresponding streams when two tanks are applied.

Figure 12. Final heat interchange strategy when three thermal reservoirs are available, where unit for H3 is a multistream heat exchanger.

tank 1 with an elevated temperature of 125 °C. Figure 14 shows the time-dependent remains in the three-tank case and the flow rates of recirculated HTM which exchanges heat with corresponding streams. The required volumes of the three storage vessels are all 6626 L. Figures 12 and 13 also show that the integrated streams, H3, C1, and C3, are the same as the case of using two storage tanks. If energy balance is the unique factor of concern, the recovered heat by the recirculated HTM should be the same as that in the two-tank case. The recovered heat by the recirculated HTM is less than that of the two-tank case due to the tight restriction on the amount of recirculated HTM in each tank for maintaining cyclic operation of batch processes. In this three-tank case, each storage tank has only one input and one output stream and the process is operated cyclically. Therefore, the total input/output amounts of the recirculated HTM for these tanks should be the same as the amounts of output. For example, the circulated input/ output amounts of HTM (in kilogram per cycle) are (225 135)Fa3/Cp and (135 - 70)Fr3/Cp for tank 1, (135 - 70)Fr3/ Cp and (70 - 30)Fr1/Cp for tank 2, and (70 - 30)Fr1/Cp and (225 - 135)Fa3/Cp for tank 3, respectively. Rearranging these

Figure 13. Final flowsheet for the single product batch plant example when three thermal reservoirs are available.


Figure 14. Remains in the thermal reservoirs and the recirculated HTM rates to corresponding streams when three tanks are applied.

equivalents results in (225 - 135)Fa3/Cp ) (135 - 70)Fr3/Cp ) (70 - 30)Fr1/Cp (kg HTM/cycle). Now suppose all of the required heating load for C1, 27.50 MJ/min, is supplied by the hot HTM from tank 1 such as in the two-tank case. For keeping the total energy balance on the recirculated HTM, the heat transfer rate from H3 to HTM should be raised from 12.22 to 12.22 + {(27.5 - 17.5) × (70 - 30)}/{(225 - 135)} ) 16.66 MJ/min. Furthermore, with the elevated heat transfer rate for HTM, the heat capacity flow rates of HTM through C1 (Fr1) and H3 (Fa3) will be increased to 27.50/(105 - 70) ) 0.79 and 16.66/(125 -70) ) 0.30 MJ/(min °C), respectively. Nevertheless, these heat capacity flow rates cannot satisfy the constraint on cyclic operation for tank 3, (225 - 135)0.30/Cp < (70 30)0.79/Cp. That is, the total amount of HTM leaving tank 3 is less than its total input amount for each batch cycle, and the HTM will be accumulated in tank 3 after several batches. Therefore, it is clear that the total mass rates of HTM through three storage tanks are not only influenced by energy balance but are also restricted by the cyclic operation constraint on the recirculated HTM.

streams and the accumulated energy is then released to latter cold process streams. By way of the recirculated HTM for implementing indirect heat exchange, the limitation of timedependent process streams is thus relaxed. A novel superstructure is presented for modeling the time-dependent heat exchange operations. A mathematical programming approach based on the superstructure is adopted for indirect thermal storage system design, where the design problem is formulated as a mixedinteger nonlinear program (MINLP) to minimize the consumption of external utility. Numerical examples are supplied to demonstrate the applicability of the proposed method for heat integration in batch plants. Design scenarios also show that using additional storage tanks does not guarantee improved heat recovery efficiency.

6. Conclusion

Nomenclature

One challenge problem for heat integration in batch plants is that the available inlet streams are time-dependent. The design of an indirect thermal storage system for implementing heat integration in batch plants is studied in this article. In an indirect thermal storage system, a recirculated heat transfer medium (HTM) is used to absorb surplus energy from hot process

Indices i ) index for hot process streams, i ) 1,..., NH j ) index for cold process streams, j ) 1,..., NC k ) index for storage tanks, 1,..., NS p ) index for time periods, 1,..., NP, and time points, 1,..., NP + 1

Acknowledgment Financial support of the Ministry of Economic Affairs (under grant 96-EC-17-A-09-S1-019) and the National Science Council ofROC(undergrantNSC96-2221-E-002-151-MY3)isappreciated.

Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4829 Sets HP ) {i|i is a hot process stream, i ) 1,..., NH} HPS ) {i|i is a hot process stream in a series unit} HPP ) {i|i is a hot process stream in a parallel unit} ) HPM ∪ HPC HPM ) {i|i is a hot process stream of a multistream exchanger} HPC ) {i|i is a hot process stream of a jacket exchanger with a coil} CP ) {j|j is a cold process stream, j ) 1,..., NC} CPS ) {j|j is a cold process stream a in series unit} CPP ) {j|j is a cold process stream in a parallel unit} ) CPM ∪ CPC CPM ) {j|j is a cold process stream of a multistream exchanger} CPC ) {j|j is a cold process stream of a jacket exchanger with a coil} ST ) {k|k is a storage tank in a superstructure, k ) 1,..., NS} ST- ) {k|k is a storage tank in superstructure, k ) 1,..., NS - 1} TP ) {p|p is a time period, p ) 1,..., Np} TP- ) {p|p is a time period, p ) 1,..., Np - 1} Parameters Cp(HTM) ) heat capacity of HTM DT ) minimum difference temperature of two neighbor tanks F, F ) upper/lower bound of heat capacity flow rate FCj ) heat capacity flow rate of cold stream j in series unit FHi ) heat capacity flow rate of hot stream i in series unit q, q ) upper/lower bound of heat flow rate LCj ) total latent heat of cold stream j (MJ) LHi ) total latent heat of hot stream i (MJ) T, T ) upper/lower bound of temperature THi(out) ) outlet temperature of hot stream i MNT ) maximum number of storage tanks Q, Q ) upper/lower bound of quality SCj ) total sensible heat of cold stream j (MJ) SHi ) total sensible heat of hot stream i (MJ) tp ) elapse time of period p (min or h) TCj(in) ) inlet temperature of cold stream j TCj(out) ) outlet temperature of cold stream j THi(in) ) inlet temperature of hot stream i (hot) Zip ∈ {0, 1} ) 1 denotes the existence of hot stream i at period p (cold) Zjp ∈ {0, 1} ) 1 denotes existence of hot stream i at period p F(HTM) ) density of HTM (kg/L) Γ ) large positive upper bound ∆Tmin ) minimum difference temperature of match (i, HTM) and match (HTM, j) Continuous Variables Fai ) heat capacity flow rate of HTM matches with hot stream i Frj ) heat capacity flow rate of HTM matches with cold stream j qai ) heat flow rate of hot stream i removed by HTM qcui ) heat flow rate of hot stream i removed by cold utility qhuj ) heat flow rate of cold stream j heated by hot utility qrj ) heat flow rate of cold stream j heated by HTM Qkp ) initial quality of tank k at time period p (kg) tcj ) outlet temperature of match (HTM, j) in series unit thi ) outlet temperature of match (i, HTM) in series unit Tk ) temperature of tank k Vk ) volume of tank k Binary Variables zai ∈ {0, 1} ) 1 denotes existence of heat exahanger i of match (i, HTM)

zaki ∈ {0, 1} ) 1 denotes existence of stream coming from tank k to cool hot stream i zaik(H2T) ∈ {0, 1} ) 1 denotes existence of hot stream i flowing into tank k zcui ∈ {0, 1} ) 1 denotes existence of cooler i zhuj ∈ {0, 1} ) 1 denotes existence of heater j zqkp ∈ {0, 1} ) 1 denotes existing quality in tank k at period p zrj ∈ {0, 1} ) 1 denotes existence of heat exahanger j of match (HTM, j) zr(T2C) ∈ {0, 1} ) 1 denotes existence of stream coming from tank kj k to heat cold stream j zrjk(C2T) ∈ {0, 1} ) 1 denotes existence of cold stream j flowing into tank k ztk ∈ {0, 1} ) 1 denotes existence of storage tank k (T2H)

Literature Cited (1) de Boer, R.; Semeding, S. F.; Bach, P. W. Heat storage systems for use in an industrial batch plant (Result of) A case study. The Tenth International Conference on Thermal Energy Storage, Ecostock; New Jersey, May 31–June 2, 2006. (2) Kemp, I. C.; Macdonald, E. K. Application of pinch technology to separation, reaction and batch processes. Inst. Chem. Eng. Symp. Ser. 1988, 109, 239. (3) Kemp, I. C.; Deakin, A. W. The cascade analysis for energy and process integration of batch processes (part 1: calculation of energy targets; part 2: network design and process scheduling; part 3: a case study). Chem. Eng. Res. Des. 1989, 67, 495. (4) Zhao, X. G.; O’ Neill, B. K.; Roach, J. R.; Wood, R. M. Heat integration for batch processes (part 1: process scheduling based on cascade analysis; part 2: heat exchanger network design). Chem. Eng. Res. Des. 1998, 76, 685. (5) Vaklieva-Bantcheva, N.; Ivanov, B. B.; Shan, N.; Pantelides, C. C. Heat exchanger network design for multipurpose batch plants. Comput. Chem. Eng. 1996, 20, 989. (6) Corominas, J.; Espuna, A.; Puigjaner, L. Method to incorporate energy integration considerations in multiproduct batch processes. Comput. Chem. Eng. 1994, 18, 1043. (7) Lee, B.; Reklaitis, G. V. Optimal scheduling of cyclic batch processes for heat integrationsI. basic formulation. Comput. Chem. Eng. 1995, 19, 883. (8) Lee, B.; Reklaitis, G. V. Optimal scheduling of cyclic batch processes for heat integrationsII. extended problems. Comput. Chem. Eng. 1995, 19, 907. (9) Kemp, I. C. Pinch analysis and process integration; Elsevier: Amsterdam, 2007. (10) Sadr-Kazemi, N.; Polley, G. T. Design of energy storage systems for batch process plants. Trans. Inst. Chem. Eng. 1996, 74, 584. (11) Krummenacher, P.; Favrat, D. Indirect and mixed direct-indirect heat integration of batch processes based on Pinch Analysis. Int. J. Appl. Thermodyn. 2001, 4, 135. (12) Krummenacher, P. Contribution to the heat integration of batch processes (with or without heat storage). Ph.D. These, Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland, 2001; no. 2480. (13) Papageorgiou, L. G.; Shah, N.; Pantelides, C. C. Optimal scheduling of heat-integrated multipurpose plants. Ind. Eng. Chem. Res. 1994, 33, 3168. (14) Georgiadis, M. C.; Papageorgiou, L. G. Optimal scheduling of heatintegrated multipurpose plants under fouling conditions. Appl. Thermal Eng. 2001, 21, 1675. (15) Brooke, A.; Kendrick, D.; Meeraus, A.; Raman, R.; Rosenthal, R. E. GAMS: A User’s Guide; Scientific Press: Redwood City, 2003. (16) Gremouti, I. D. Integration of batch processes for energy savings and debottlenecking. MSc. Thesis, Department of Chemical Engineering, University of Manchester Institute of Science and Technology (UMIST), U.K., 1991.

ReceiVed for reView August 06, 2007 ReVised manuscript receiVed February 1, 2008 Accepted February 26, 2008 IE0710667

Design and Optimization of Indirect Energy Storage Systems for Batch

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