Design of Indirect Heat Recovery Systems with Variable-Temperature

Ind. Eng. Chem. Res. 2009, 48, 4375–4387

4375

Design of Indirect Heat Recovery Systems with Variable-Temperature Storage for Batch Plants Cheng-Liang Chen* and Ying-Jyuan Ciou Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, ROC

This article aims to propose an iterative method for designing an indirect heat recovery system including the associated variable-temperature storage in a batch plant. A recirculated heat transfer medium (HTM) is applied to absorb surplus heat load from hot process streams and temporarily reserved in storage tanks, where the accumulated energy is then released to later cold process streams by exchanging heat with HTM. This work is a direct extension of one recent article where the restraint of constant temperature for HTM in each storage tank is relaxed to magnify the heat recovery potential. The design problem is formulated as a mixed-integer nonlinear program (MINLP) based on proposed superstructures. A novel iterative solution strategy for setting the variable temperatures of HTM in storage is provided by linking the network optimization tool (a GAMS program) and the streams temperature simulation software (a MATLAB program). One numerical example is explored to demonstrate the applicability of the proposed indirect heat exchanger network design method for batch processes. 1. Introduction The major ways to implement heat integration in batch process plants can be classified into direct and indirect heat interchange methods. An indirect heat recovery system in a batch plant applies a heat transfer medium (HTM) to absorb surplus heat from hot process streams and then to release accumulated energy to later cold process streams, where the recirculated HTM is temporarily reserved in storage tanks to relax the limitation of heat interchange between noncoexistent process streams. The indirect heat exchange method is much less schedule-sensitive than the direct one. A well-designed indirect heat recovery system for a batch plant can thus reduce the utility consumption with minimum sacrifice of operation flexibility. Recently, de Boer et al.1 evaluated the technical and economical feasibility of different types of heat storages for industrial processes. On the basis of the novel progress in thermal storage technologies, the HTM can play a significant role in transferring cooling or heating loads of process streams in industries. Methods for designing an indirect heat exchange system were proposed by several authors. Sadr-Kazemi and Polley2 proposed the design of an indirect heat storage system with minimum utility by pinch analysis approach. Later, Krummenacher and Favrat3 followed Sadr-Kazemi’s approach and discussed the problem of the minimal number of heat storage tanks. At the same time, Krummenacher4 also proposed a heat exchange network (HEN) with closed and open storage tanks by applying a genetic algorithm optimization method in problem solving. Papageorgiou et al.5 proposed a mathematical programming framework to calculate the variation of mass and energy holdups of heat transfer media over time under a known operating policy. Georgiadis and Papageorgiou6further extended the original method to considering a fouling problem during heat integration in multipurpose batch plants. However, these mathematical programming studies are based on an important assumption that the operating policy for each heat integrated operation is fixed and known a priori. Recently, Chen and Ciou7 proposed a novel mixedinteger nonlinear program (MINLP) formulation for determining the utility consumption of an indirect heat recovery system, where the HEN and the associated thermal storage policy for recirculated hot/cold HTMs are determined simultaneously. * To whom correspondence should be addressed. Tel.: 886-233663039. Fax: 886-2-23623040. E-mail: [email protected].

One impractical restraint for the recent study7 is the fixed temperature of each HTM storage tank. The temperatures of input/output streams for each HTM storage tank thus must be confined to their storage temperatures. Such a fixed-temperature/ variable-mass (FTVM) storage constraint may result in enlarged storage capacities with degraded heat recovery potential. This extended article aims at applying variable-temperature/variablemass (VTVM) storage for maximizing the energy recovery potential with reduced tank sizes. Furthermore, the annual cost for the VTVM storage system is also targeted for providing more practical design. An iterative method linking the network optimization tool (a GAMS program) and the numerical HTM temperature simulation tool (one MATLAB program)8 is prepared for estimating the variable HTM temperatures in storages. The rest of this paper is organized as follows. The problem statement of indirect heat integration in batch plants is introduced in section 2. Next, the concept of an indirect heat recovery network which allows variable storage temperatures is elucidated, and superstructures for subsequent modeling are proposed. In section 4, the MINLP design model is formulated for considering possible network configurations with VTVM storage tanks. An iterative solution strategy connecting the GAMS tool for network design and the MATLAB simulation for variable temperature in storages is discussed in section 5. A numerical example is thereafter presented in section 6 to demonstrate the proposed design procedure. Finally, a conclusion is made for summarizing the work. 2. Problem Statement The problem of indirect HEN design with VTVM storage policy in batch plants addressed in this paper can be stated as follows. Given 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 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 HTM under various operating temperatures. Also given are input and output temperatures of hot and cold process streams; latent and sensible heats or the heat capacity flow rate of each stream;

10.1021/ie8013633 CCC: $40.75  2009 American Chemical Society Published on Web 04/09/2009

4376 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009

3. Concept and Superstructures for Indirect Thermal Storage Systems

Figure 1. Conceptual structure of an indirect heat exchanger network with two-hot/one-cold process streams and one recirculated heat transfer medium.7

the existence of streams at different operating periods; minimum temperature approach for transferring heat between two streams; properties of heat transfer medium such as density and heat capacity; costs of HTM and utilities; and cost data of heat exchangers. The objective then is to determine the heat integration strategy in batch plants with the associated VTVM storage policy for periodically dependent inlet flows, based on the targets of the external utilities and annual cost of the VTVM storage system. To find the solution of the proposed indirect heat recovery system in batch plants, the following are also provided: the number of thermal storage reservoirs and their variable nonlinear temperature profiles, 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 variable outlet temperature profiles of hot and cold process streams of each match in series-type heat exchange units, and the sizes of storage tanks and additional heat exchangers. Then, the overall HEN with VTVM storage is constructed on the basis of this information.

The concept of an indirect heat recovery system with variable storage temperatures is almost the same as that of fixed reservoiring temperature case recently studied by Chen and Ciou.7 As illustrated in Figure 1, the conceptual structure of an indirect thermal recovery system consists of absorption and rejection parts. In the absorption segment, the recirculated cold heat transfer medium (HTM) absorbs surplus energy from hot process streams and the resulting hot HTM is temporarily reserved in a hot tank. In the rejection segment, the accumulated thermo-inertia is released to subsequent cold process streams and the incurring cold HTM is sent back to a cold reservoir. On the basis of the operational conditions, the heat exchangers in the indirect HEN consists of the series-type units (S) and the parallel-type units (P), where the latter also includes the multistream heat exchangers (M) and the jacket-type heat exchangers with coil inside (C). As shown in Figure 1, the series/ parallel-types of units are classified according to the configuration of matching process streams with the recirculated HTM and external utilities. In a series-type unit, a process stream exchanges energy with the HTM first and is then followed by a supplementary external utility. Whereas in a parallel unit, the process stream exchanges energy with the HTM and the complementary hot/cold utilities simultaneously. The options assigned in advance for matching series/parallel heat exchange units are thus based on the configuration of matching process streams with the recirculated HTM and the external utilities. Thermal storage is a key to relax the time limitation of timedependent operations in batch plants and to integrate the heat of noncoexistent process streams. A common and typical thermal storage, fixed temperature/Variable-mass (FTVM) heat storage, operates according to the temperature change of the material and is used to integrate the heat of process streams.7 However, FTVM storage might require a large volume in tank size because of the tight restriction of fixed temperature on the recirculated HTM in each tank. Should the tight constraint of fixed storage temperature be relaxed, it is reasonable to expect that indirect heat integration with a Variable-temperature/Variable-mass (VTVM) heat storage policy can reduce significantly the external

Figure 2. Configuration for 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) during time period p.

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4377

utility consumption; in addition, the size of VTVM storages will also be smaller than that of FTVM course. To model the design problem of indirect heat recovery system with VTVM thermal storages, superstructures for series/paralleltypes of heat exchange units and thermal storage reservoirs are proposed in Figure 2, where the overall cyclic operating time is divided into several operating periods p ∈ TP according to the existence of hot/cold process streams. The Nomenclature gives definitions for all relevant indices and sets, parameters, and variables for subsequent modeling. In Figure 2a, a hot (out) ) stream with the supply and target temperatures (TH(in) i , THi is treated by HTM and cold utility in series. The HTM removes (ave) that the heat from the hot stream with a heat flow rate qaip causes the outlet temperature of hot stream to decrease from (ave) (T2H) to thip . The two binary variables, zak′i and za(H2T) , are TH(in) i ik used to define that which storage provides or receives the HTM and, furthermore, decides the supply and target temperatures of HTM (T′k and Tk). If the two binary variables with unity (T2H) ) za(H2T) ) 1, then the cold HTM with average values, zak′i ik (ave) flows from cold tank k′ to hot reservoir k in temperature Tk′p period p via a heat exchange unit to cool hot process stream i. Similarly, in Figure 2c, the case in which binary variables with ) zr(C2T) ) 1, indicates that the HTM comes unity values, zr(T2C) kj jk′ from hot tank k and flows into cold reservoir k′ via a heat exchange unit to heat cold stream j. The superstructures of parallel heat exchanger units for hot and cold streams are shown in Figure 2b and d. As shown Figure 2b, a hot stream is cooled down by HTM and cold utility simultaneously. The HTM removes the heat from the hot stream (ave) , and the surplus is taken away by the with heat flow rate qaip (ave) . In addition, Figure 2d is cold utility with heat flow rate qcuip provided for cold stream j. The binary variables in the superstructures of series/parallel heat exchanger units give the information of the direction of HTM. Therefore, the superstructure of storage tank k is proffered in Figure 2e, where the quantity at time point p and temperature of the tank k is defined by Qkp and Tkp. Each flow is marked with its temperature and heat capacity flow rate. One of the main difficulties in designing the indirect heat recovery system is the handling of possible nonlinear temperature profile of HTM in VTVM storages. It is not straightforward to find the remaining energy in storage tanks over each operating period, because the HTM temperature profiles cannot be predicted in advance. Instead of directly estimating the nonlinear temperature profiles over each period, the HTM temperatures in tank k at the start and end of period p, Tkp and Tk,p+1, and the average temperature of HTM over period p, T(ave) kp , are simultaneously adopted as design variables to model the design problem of indirect heat integrated network. A simple relation between T(ave) kp , Tkp, and Tk,p+1 will be established, and the estimation of the average HTM temperatures will be updated iteratively with the determined heat recovery network, which will be discussed in more detail in the following sections. Likewise, some average properties are also used in later formulation during period p, such as the average heat flow rates (ave) (ave) and qrjp ) and the average intermediate outlet temper(qaip and tc(ave) atures of process streams (th(ave) ip jp ), as depicted in Figure 2. Note that the flow rates of HTM (Fai and Frj) as well as the output temperatures of HTM (Tai and Taj) are assumed to be time-independent to simplify the formulation. 4. Superstructures Modeling and MINLP Formulation On the basis of the definitions and assumptions on proposed superstructures, the design of indirect HEN with VTVM storage in a batch process can be formulated as follows.

4.1. Overall Heat Balance on the Recirculated Heat Transfer Medium. By way of the thermal storage in an indirect heat recovery system, the total energy absorbed by the HTM from hot process streams in the absorption part is equal to that released to later cold process streams in the rejection segment, as follows.

∑ ∑

(hot) qa(ave) ) ip tpZip

i∈HP p∈TP

∑ ∑

(cold) qr(ave) jp tpZjp

(1)

j∈CP p∈TP

(cold) Here, tp is the elapsed time of operating period p; Z(hot) ip and Zjp denote predefined existence of hot i and cold j process streams; (ave) (ave) and qrjp represent the average heat absorption rate of qaip cold HTM from hot stream i and the average heat rejection rate of hot HTM to cold stream j at time period p, respectively. Take eq 1 as an example; it reveals the advantage to avoid the (ave) numerical integration of energy by using mean values (qaip and qr(ave) ). Similarly, the mean values are also used commonly jp in other heat balance equations in the next subsection. 4.2. Heat Balance around Series- and Parallel-Type Heat Exchangers. Equation 2 summarizes the heat balance around series-type heat exchangers. In a series-type heat exchange unit, the total heat including stream i’s latent heat LHi and sensible heat SHi is removed by the recirculated HTM (ave) ) and the cold utility (with an (with an average rate of qaip (ave) average rate of qcuip ) in series at time period p. Noted that the latent heat is divided by duration time to transfer the unit from megajoules to megajoules per hour. In addition, the sensible heat is corresponding to the multiplication of its heat capacity flow rate (FHi) and the temperature difference. Also noted that when a hot stream i does not exist at a certain time (hot) (ave) ) 0), its average heat flow rates (qaip and period p (Zip (ave) qcuip ) will become zero and the outlet temperature of stream (ave) ) will be equivalent to the input i matching HTM (thip temperature of stream i (TH(in) i ). The formulation of cold stream j in series-type heat exchange units can be described similarly.

[

LHi

∑

tpZ(hot) ip

p∈TP

qa(ave) ip

LHizai

∑

tpZ(hot) ip

+

qcu(ave) ip

LHi(1 - zai)

∑

tpZ(hot) ip

tpZ(cold) jp

p∈TP

Z(cold) (cold) jp

tpZjp

p∈TP

LCj(1 - zrj)

∑

p∈TP

+

(TC(out) j

-

TC(in) j )FCj

qr(ave) + qhu(ave) jp jp

LCjzrj

[

+ (th(ave) - TH(out) )FHi Z(hot) ) qcu(ave) ip i ip ip ∀i ∈ HPS, p ∈ TP

LCj

∑

] ]

∀i ∈ HPS, p ∈ TP

p∈TP

∑


(ave) Z(hot) + (TH(in) - th(ave) ip i ip )FHi ) qaip

p∈TP

[ [

]

+ (TH(in) - TH(out) )FHi Z(hot) ) i i ip

tpZ(cold) jp

+

(tc(ave) jp

-

Z(cold) ) jp

∀j ∈ CPS, p ∈ TP

TC(in) j )FCj

]

) qr(ave) jp

∀j ∈ CPS, p ∈ TP (cold) + (TC(out) - tc(ave) ) qhu(ave) j jp )FCj Zjp jp


(2)


In parallel-type heat exchange units, the recirculated HTM and the cold utility remove the total heat including stream i’s latent heat LHi and sensible heat SHi, simultaneously. LHi + SHi

∑

Z(hot) (hot) ip tpZip

) qa(ave) + qcu(ave) ip ip

p∈TP

LCj + SCj

∑

∀i ∈ HPP, p ∈ TP

Z(cold) (cold) jp

tpZjp

p∈TP

(3)

) qr(ave) + qhu(ave) jp jp ∀j ∈ CPP, p ∈ TP

4.3. Heat Balance on Recirculated HTM around Units. (T2H) and zr(T2C) , provide information about Binary variables, zak′i kj the source of recirculated HTM. Therefore, the heat removed or absorbed by the HTM can be calculated according to the differences of average temperatures between connecting tanks, as shown in eqs 4 and 5. Note that the output temperatures of recirculated HTM, Tai and Trj, are assumed constants and do (H2T) and zr(C2T) ). not relate to the path of discharging HTM (zaik′ jk (hot) (Tai - T(ave) - qa(ave) e qj(1 - za(T2H) ) k'p )FaiZip ip k'i ∀i ∈ HP, k ∈ ST, p ∈ TP (hot) - qa(ave) g -qj(1 - za(T2H) ) (Tai - T(ave) k'p )FaiZip ip k'i ∀i ∈ HP, k ∈ ST, p ∈ TP

(4)

(T(ave) - Trj)FrjZ(cold) - qr(ave) e qj(1 - zr(T2C) ) kp jp jp kj ∀j ∈ CP, k ∈ ST, p ∈ TP - Trj)FrjZ(cold) - qr(ave) g -qj(1 - zr(T2C) ) (T(ave) kp jp jp kj ∀j ∈ CP, k ∈ ST, p ∈ TP

(5)

As mentioned previously, the average temperature of HTM in tank k at period p, T(ave) kp , is the mean value of its variable temperature profile. However, it is not easy to have an accurate temperature profile in advance in a VTVM storage system. Hence, a parameter, Rkp, is directly used to provide a simple estimate of the average temperature of HTM in tank k from the start/end temperatures in period p. An iterative strategy for adaptation of these Rkp values will be discussed in more detail in the next section. Tkp + Tk,p+1 Rkp ∀k ∈ ST, p ∈ TP2 TkNP + Tk1 RkNP ∀k ∈ ST ) 2

) T(ave) kp (ave) TkN P

(6)

4.4. Heat Balance around Series Type Units at a Time Point. In series-type heat exchangers, the outlet temperatures of streams matching HTM are time-varying. Therefore, in addition to the average outlet temperatures during period p (th(ave) ip (ave) and tcjp ), the outlet temperatures at time point p (thpip and tcpjp) should also be determined. Notably, the temperatures at the end of period p are limited by the supply and target temperatures of process streams. e thpip e TH(in) TH(out) ∀i ∈ HPS, p ∈ TP i i TC(in) j

e tcpjp e

TC(out) j


(cold) and Zpjp , are used to define the existence of process streams at time point p. For instance, if a hot process stream i exists (hot) during time period p (Zip ) 1), it should subsist on the bounding of period p (Zp(hot) ) 1 and Zp(hot) ip i,p+1 ) 1). Furthermore, (hot) (cold) because of the condition of cyclic operation, Zpi1 and Zpj1 can be used to define the existence of process streams at the start time in the first period as well as the ending time in the final period NP. Notice that at most one of all paths supplying (T2H) HTM is allowable (∑∀k∈ST Zak′i e 1; ∑∀k∈ST Zrkj(T2C) e 1). Hence, the supply temperature of HTM (Tkp) can be decided by binary variables and the heat balance of HTM in series-type heat exchanger is formulated at time point p as follows.

[

(

(TH(in) - thpip)FHi - Tai i

Zp(hot) ip zai ) 0

[

(tcpjp -

TC(in) j )FCj

(∑

∀k∈ST

) ]

za(T2H) Tk'p Fai × k'i


zr(T2C) Tkp kj

) ]

Frj × - Tr(out) j

)0 ∀j ∈ CPS, p ∈ TP

(8)

Notably, the average intermediate temperatures of process (ave) (ave) streams (thip and tcjp ) and the temperatures at the start/end of period p (thpip and tcpjp) are closely related to the temperatures of HTM in storages T(ave) kp , Tkp, and Tk,p+1. The heat balances of (ave) the average temperatures during time period p (T(ave) kp , thip , and (ave) tc(ave) ) and the temperatures at time point p (T , th , and tc(ave) jp kp ip jp ) are formulated in eqs 2, 4, 5, and 8. By observing these heat balances, the time-invariant variables (Fai, Frj, Tai, and Trj) imply that these average temperatures are the same as that of their start/end temperatures. On the basis of the relation, if the average temperature of storage k is correct (T(ave) kp ), the average outlet temperature of streams matching HTM form the storage is also precise. Accordingly, the ratio, Rkp, in eq 6 is a crucial key in affecting the precision of those average temperatures (ave) (ave) (T(ave) and tcjp ). kp , thip 4.5. Calculation of Approach Temperatures. A sufficiently large temperature driving force at each time point is needed to guarantee feasible heat transfer between HTM and the hot/cold process streams. The HTM operating temperature is nonlinear and monotonous, the driving force during the whole period p can be assumed by the large enough driving force on two extremities of period p. The input temperature of HTM is (T2H) (C2T) associated with the paths (zak′i and zrjk′ ), similar to that provided in the former subsection. 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):

[ (∑ [( ∑ thpip -

)] ]

(hot) (T2H) zak'i Tk'p + ∆Tmin Zpip g0

zr(T2C) Tkp kj

∀k∈ST

If a specific stream exists during time period p, not only the heat balance of its average outlet temperatures (see former subsections) but also that of the outlet temperatures on the bounds of period p should be considered. Two parameters, Zp(hot) ip

-

Zp(cold) zrj jp

∀k'∈ST

(7)

∑

∀k∈ST

)

- ∆Tmin -

∀i ∈ HPS, p ∈ TP j (1 - zrj) g 0 +T

(cold) tcpjp Zpjp

For parallel-type units (M and C):

∀i ∈ CPS, p ∈ TP (9)


[ (∑ [( ∑ TH(out) i

∀k'∈ST

∀k∈ST

)] ]

(hot) (T2H) zak'i Tk'p - ∆Tmin Zpip g0

Tk,p+1Qk,p+1Cp(HTM) )

∀i ∈ HPP, p ∈ TP (out) (cold) j (1 - zrj) g 0 zr(T2C) T ∆T TC Zp +T kj kp min j jp

)

Tai e (TH(in) - ∆Tmin)zai ∀i ∈ HPS ∪ HPM i Trj e

(TC(in) j

- ∆Tmin)zrj

∀j ∈ CPS ∪ CPM

(11)

)

- ∆Tmin)zrj Trj e (TC(out) j

(12)

∀j ∈ CPC

-

(ave) zrkj(T2C)Tkp )Frjtp

+

∀k ∈ ST, p ∈ TP-

∀j∈HP

∀j∈CP

] ]

(hot) (ave) ZiN (zaik(H2T)Tai - zaki(T2H)TkN )FaitNP + P P (cold) ZjN (zrjk(C2T)Trj P

-

(ave) zrkj(T2C)TkN )FrjtNP P

+

TkNPQkNPCp(HTM) ∀k ∈ ST (14)

4.7. Logical Constraints. Logical constraints proposed by Chen and Ciou7 are used to define the existence of units and tanks. Furthermore, the restrictions can simplify the complexity of network and are classified into six categories as follows. (1) Logical constraints about temperature order of storage tanks: k

k

∑

za(T2H) e li

∑

zr(T2C) lj

∑ za

(H2T) il

∀i ∈ HP, k ∈ ST

l)1 k

l)1

Tai e (TH(out) - ∆Tmin)zai ∀i ∈ HPC i

[∑ [∑

l)1 k

For a jacket exchanger with a coil (C):

(cold) Zjp (zrjk(C2T)Trj

TkpQkpCp(HTM) Tk1Qk1Cp(HTM)

For series-type units and multistream exchangers (S and M):

] ]

(hot) (ave) Zip (zaik(H2T)Tai - zaki(T2H)Tkp )Faitp +

∀i∈HP

∀j∈CP

∀j ∈ CPP, p ∈ TP (10)

The outlet temperature of HTM (Tai and Trj) is assumed to be time-invariant and obeys the following constraints. In seriestype units and multistream exchangers, the outlet temperature of HTM should be less than the inlet temperature of the matching stream minus the minimum approach temperature when the match occurs (zai ) 1, zrj ) 1). Because of the closest approaching temperature, the outlet temperature of HTM is less than the outlet temperature of the matching stream minus the minimum approach temperature in jacket exchanger with a coil.

[∑ [∑

g

∑

(15) ∀j ∈ CP, k ∈ ST

zr(C2T) jl

l)1

j ztk g Tkp g Tk+1,p + DTztk ∀k ∈ STTk+1,p + T (16) j ztk T _ztk e Tkp e T ∀k ∈ ST (2) Logical constraints on heat exchange units:

4.6. Remaining Mass and Energy in Energy Reservoirs. The remaining mass and accumulated energy of HTM in each tank k at the end of each period p are relevant to determining suitable size and temperature of the tank. The remaining mass of storage k at the start of period p + 1, Qk,p+1, equals the tank’s initial mass at period p, Qkp, plus the input mass and minus the outlet mass. For cyclic operation, the initial mass will be equivalent to that at the end of the whole cycle.

Qk,p+1

)

[∑ [∑ [∑ [∑

∀i∈HP

∀j∈CP

Qk1

)

(zr(C2T) jk

Fai

] ] ] ]

(hot) Zip tp + Cp(HTM) Frj (cold) - zr(T2C) ) (HTM) Zjp tp + Qkp kj Cp

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

∀k ∈ ST, p ∈ TPFai (hot) (za(H2T) - za(T2H) ) (HTM) ZiN t + ik ki P NP Cp ∀i∈HP Frj (cold) (zr(C2T) - zr(T2C) ) (HTM) ZjN tNP + QkNP jk kj P Cp ∀j∈CP ∀k ∈ ST (13)

q_zaiZ(hot) e qa(ave) e qjzaiZ(hot) ip ip ip (hot) (ave) q_zcuiZip e qcuip e qjzcuiZ(hot) ip

e qr(ave) e qjzrjZ(cold) q_zrjZ(cold) ∀j ∈ CP, p ∈ TP jp jp jp (ave) (cold) (cold) q_zhujZjp e qhujp e qjzhujZjp ∀j ∈ CP, p ∈ TP (17) (3) Logical constraints on heat capacity flow rate of HTM: j zai ∀i ∈ HP F _ zai e Fai e F j F _ zrj e Frj e Fzrj ∀j ∈ CP

(18)

(4) Logical constraints on inlet/outlet streams of storage tanks:

∑ ∑

∀k∈ST ∀k∈ST

∑ ∑

∀i∈HP ∀i∈HP

It is assumed that the possible heat loss in the storage system can be negligible. The accumulated energy of storage k at the start of period p + 1, Tk,p+1Qk,p+1Cp(HTM), thus equals the tank’s initial energy at period p, TkpQkpCp(HTM), plus the input energy and minus the outlet energy during period p. Notice that, for tank k at period p, the outlet energy is calculated by using the average temperature of HTM T(ave) kp .

∀i ∈ HP, p ∈ TP ∀i ∈ HP, p ∈ TP

za(T2H) ) zai ) ki zr(T2C) ) zrj ) kj za(H2T) + ik za(T2H) + ki

∑ ∑

∀j∈CP ∀j∈CP

∑ ∑

∀k∈ST ∀k∈ST

za(H2T) ik ∀i ∈ HP zr(C2T) jk

(19) ∀j ∈ CP

zr(C2T) g ztk ∀k ∈ ST jk zr(T2C) g ztk ∀k ∈ ST kj

za(T2H) + za(H2T) e ztk ∀i ∈ HP, k ∈ ST ki ik + zr(C2T) e ztk zr(T2C) kj jk

∀j ∈ HP, k ∈ ST

(20)

(21)

(5) Logical constraints on remaining mass of storage tanks: j ztk ∀k ∈ ST, p ∈ TP Q _ ztk e Qkp e Q

(22)


ztk g ztk+1 ∀k ∈ ST-

(23)

.

(6) Restraints on number of storage tanks:

∑

2e

ztk e MNT

(24) (ex) qjp )

k∈ST

4.8. Sizing Equation of Heat Exchangers. The equipment of a VTVM storage system used to integrate the heat of noncoexistent process streams involves the additional heat transfer equipment of process streams matching the HTM and the storage tanks for recirculated HTM. Therein, the heat transfer equipment can be categorized into several types, such as an additional heat exchanger of HTM in series-type, a multistream exchanger, and a coil-in-jacket exchanger, and its size is calculated by the following equations. The heat transfer area of stream i matching with the HTM at (ex) ), time point p (aip) associates with its heat transfer rate (qip the overall heat transfer coefficient (Ui), and the log mean temperature difference (LMTDip). In a series unit, the heat (ex) ) is transfer rate of stream i matching HTM at time point p (qip the capacity flow rate of stream i (FHi) multiplied by its temperature difference (TH(in) i - thpip). In multistream exchanger, the heat of process stream is treated simultaneously by the external utility and the HTM so the heat transfer rate in the exchanger is the entire processed heat load divided by the existing time. In the coil of a jacket exchanger, the heat transfer rate of a specific process stream exchanged by HTM can be calculated by the temperature difference and the heat capacity of HTM.

{

∑

∀k'∈ST

∀i ∈ HPS, p ∈ TP ∀i ∈ HPM, p ∈ TP

(T2H) (hot) zak'i Tk'p)Zpip zai ∀i ∈ HPC, p ∈ TP

(25)

The log mean temperature difference (LMTDip) can be (out) and dthip denote the approximated9 as follows, where dth(in) i input and output temperature differences of the match between the hot stream i and the HTM, respectively. For the coil-injacket exchanger, the smallest temperature difference is chosen to obtain a large enough area of heat transfer equipment. LMTDip ≈ . . dthi(in)

)

. (out) dthip )

[

(out) dthi(in)dthip

{

(out) dthi(in) + dthip (out) Zpip zai + σ 2

]

∀i ∈ HPS ∪ HPM

{

∑ za - ∑ za

(T2H) Tk'p k'i

∀k'∈ST

THi(out)

(T2H) Tk'p k'i

∀k'∈ST

Frj(

. . )

. (out) dtcjp

∀j ∈ CPM, p ∈ TP

)

∑

∀k∈ST

(cold) zrj ∀j ∈ CPC, p ∈ TP zrkj(T2C)Tkp - Trj)Zpjp

(27)

[

(out) dtcj(in)dtcjp

{

]

(out) dtcj(in) + dtcjp (cold) zrj + σ Zpjp 2

1/3

∀j ∈ CP, p ∈ TP Trj Trj -

{

∑ ∑

∀k∈ST

∀k∈ST

TCj(in) TCj(out)

∀j ∈ CPS ∪ CPM ∀j ∈ CPC

zrkj(T2C)Tkp - tcpjp ∀j ∈ CPS, p ∈ TP zrkj(T2C)Tkp-TCj(out) ∀j ∈ CPM ∪ CPC, p ∈ TP (28)

4.9. Costs of Variable Temperature Storage System. The VTVM storage system mentioned in prior subsections involves the additional heat transfer equipment of the process streams matching HTM and storage tanks for the recirculated HTM. The annual cost of heat transfer equipment consists of fixed cost (CFi/j) and area cost (CAi/j). (29)

Here, τ(ex) is the payback time for annualizing heat transfer equipments. The annual cost of storage tanks can be approximated as the following,10 where is the Marshall and Swift (MS) index and Fm is material correction factor. Furthermore, a cylinder tank is used and its height is twice its diameter. T(tank) is the payback time for annualizing storage tanks. AC(tank) ) k

MS ( 280 )(957.8820D

Hk0.82(2.18 + Fm))/T(tank)

1.066

k

Hk ) 2Dk ∀k ∈ ST Qkp π Vk ) Dk2Hk g ∀k ∈ ST, p ∈ TP 4 F

∀k ∈ ST

(30)

THi(out) - Tai ∀i ∈ HPC thpip -

∑

∀p∈TP

LMTDjp ≈

dtcj(in)


(cold) zrj FCj(tcpjp - TCj(in))Zpjp LCj + SCj (cold) Zpjp zrj (cold) tpZjp

1/3

∀i ∈ HP, p ∈ TP THi(in) - Tai

{

∀j ∈ CP, p ∈ TP

(ex) j g (CFj + CAjaCB AC(ex) ∀j ∈ CP, p ∈ TP j jp )/τ

∑

Fai(Tai -

(ex) qjp UjLMTDjp

(ex) i AC(ex) g (CFi + CAiaCB ∀i ∈ HP, p ∈ TP i ip )/τ

(ex) qip aip ) ∀i ∈ HP, p ∈ TP UiLMTDip . (hot) FHi(THi(in) - thpip)Zpip zai LHi + SHi (hot) Zpip zai (hot) (ex) tpZip qip ) ∀p∈TP

ajp )

∀i ∈ HPS, p ∈ TP ∀i ∈ HPM ∪ HPC, p ∈ TP

The HTM in storage tanks constitutes one investment cost, with τ(HTM) as the annualizing factor, to absorb and release heat of process streams repeatedly. Notably, at any time, the total amount of HTM is the same due to cyclic operation as shown in eq 13. Thus, the amount of HTM is equivalent to the sum of the quantity of all tanks at any time point, say the initial time (p ) 1). Q(HTM)

)

AC(HTM)

)

(26)

The following constraints concern the area (ajp) for cold (ex) stream j, heat transfer rate (qjp ), and log mean temperature difference (LMTDjp), which are similar to those restraints for hot stream i.

∑

∀k∈ST (HTM)

C

Qk1

(31)

Q(HTM) /τ(HTM)

The additional annual investment cost (AAIC) for building a VTVM storage system is defined as follows, including the additional exchangers, recirculating HTM, and storage tanks.


AAIC )

∑

∑

AC(ex) + i

∀i∈HP

∀j∈CP

AC(ex) + j

∑

∀k∈ST

AC(tank) + k AC(HTM) (32)

The next step is to verify the operating cost and the annual cost of utility (ACU) comprising the costs of cold and hot utilities. In batch process, the utility is used intermittently. (hot) tp)/BT) is multiplied to describe the Therefore, a fraction ((Zip proportion of processing time of a specific stream to the whole cyclic process time, where BT is the operating time of one batch and the units of utility cost are dollar hours per megajoule year. ACU ) Ccu

∑ ∑

∀i∈HP ∀p∈TP

Z(hot) ip tp qcu(ave) + ip BT

Chu

∑ ∑

∀j∈CP ∀p∈TP

Z(cold) tp jp qhu(ave) (33) jp BT

4.10. Design Objectives and MINLP Formulations. The indirect heat recovery design problem for batch processes can be formulated as the following mixed-integer nonlinear program (P1). The objective is to minimize the total utility (UVT), where xVTU and ΩVTU denote the set of all design variables and the feasible searching space defined by all constraints eqs 1-24. P1: min

xVTU∈ΩVTU

UVT )

∑ ∑

(ave) Z(hot) + ip tpqcuip

∀i∈HP ∀p∈TP Z(cold) tpqhu(ave) jp jp ∀j∈CP ∀p∈TP

∑ ∑

ΩVTU ) {xVTU |set of constraints, eqs 1-24} Another reasonable objective is to minimize the additional annual cost for operating a VTVM storage system (ACVT) including the annual equipment cost of the VTVM storage system (AAIC) and the annual cost of utility (ACU). The design problem can be formulated as the following mixed-integer nonlinear program (P2), where xVTC and ΩVTC denote the set of all design variables and the feasible searching space defined by all constraints eqs 1-33 P2: min

xVTC∈ΩVTC

ACVT ) AAIC + ACU

ΩVTC ) {xVTC |set of constraints, eqs 1-33}

This design objective is used to compare the benefit between the optimized design with a VTVM storage system and the original design where the process streams are served by hot/ cold utilities only. Accordingly, the annual operating cost of the original design (ACU(ori)) is also obtained as the following. ACU(ori) ) Ccu

∑ ∑

∀i∈HP ∀p∈TP

LHi + SHi + BT Chu

Figure 3. Flowsheet of iterative update of Rkp values by linking a GAMS program for network design and a MATLAB program for variable temperature simulation.

∑ ∑

∀j∈CP ∀p∈TP

LCj + SCj (34) BT

For solving the MINLP formulations with the design of the VTVM storage system by using the average temperature of the storage during each period in batch process plants, the general algebraic modeling system (GAMS)11 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. Note that the given parameters, Rkp, are resolved

Figure 4. Network from GAMS solution for the actual temperature simulation of HTM.

by a MATLAB program for temperature simulation and its updated strategy is further explained in next section. 5. Solution Strategy for Iterative Parameters, Rkp Precise estimation for key parameters, Rkp, will seriously affect the adequacy of the heat exchanger network (HEN) design and the variable temperatures of HTM in the storages. An iterative method linking the GAMS program for HEN design and a subsequent MATLAB program for variable temperature simulation8 is prepared to update the estimation of the values of Rkp, and its flow diagram is sketched in Figure 3. The iterative updating procedure consists of four steps as follows. Step 1: Determine the Optimal Heat Exchanger Network (HEN). Given ratio parameters, Rkp, the MINLP design problems P1 or P2 are directly solved to determine the indirect heat recovery network with associated storage policy, where the GAMS is used as the solution tool. Note that, during the iterative procedure, the binary variables of HEN are fixed for convergence. The other variables are calculated afresh after each iteration, i.e. the heat capacity flow rates of HTM, heat flow rates of exchangers, temperatures of HTM in storages, and so on. Step 2: Simulate the Actual Temperatures. The resulting indirect heat recovery network from step 1 includes the flowing (H2T) , zai′k , zr(T2C) , paths of recirculated HTM (the values of za(T2H) ki kj (C2T) and zrj′k ), their heat capacity flow rates (Fai and Frj), and the input temperatures of HTM (Tai and Trj), as shown in Figure 4. These temporary results can be used to simulate the actual HTM quantity (Q′kt) and the actual HTM temperature (T ′kt) in tank k during period p, i.e., for t ∈ [tp0, tp0 + tp]. The actual remaining HTM and temperature (Q′kt and T ′kt) at the initial time of each period p (t ) t0p) are provided by GAMS design solution


(Qkp and Tkp). The differential equations for modeling the thermal quantity of HTM in tank k can be formulated as follows. dQ′kt dt

)

[ [∑

∑

] ]

Fai

(za(H2T) - za(T2H) ) + ik ki (HTM) Cp ∀i∈HP Frj Z(cold) (zr(C2T) - zr(T2C) ) jp jk kj (HTM) Cp ∀j∈CP ∀k ∈ ST, p ∈ TP, t ∈ [t0p, t0p + tp] Z(hot) ip

I.C. Q′kt ) Qkp,

at t ) t0p, ∀k ∈ ST, p ∈ TP (35)

[∑

d(T kt′ Qkt′ Cp(HTM)) ) dt

∀i∈HP

[∑

(cold) Zjp Frj(zrjk(C2T)Trj

-

′

zrkj(T2C)T kt)

∀j∈CP

I.C. T kt′ ) Tkp,

] ]

(hot) Zip Fai(zaik(H2T)Tai - zaki(T2H)T kt′ ) +

∀k ∈ ST, p ∈ TP, t ∈ [t0p, t0p + tp]

at t ) t0p, ∀k ∈ ST, p ∈ TP (36)

Integrating the above equations and rearranging the resulting solutions, the actual temperature of HTM in storage k, T ′kt, is a natural log function as the following. (Akp - Bkp + Ckp - Dkp)t + Qkp 1 ) ln (Akp - Bkp + Ckp - Dkp) (A - B + C - D )t0 + Q kp kp kp kp p kp . AkpTai + CkpTrj - (Akp + Ckp)T kt′ -1 ln (Akp + Ckp) AkpTai + CkpTrj - (Akp + Ckp)Tkp ∀k ∈ ST, p ∈ TP, t ∈ [t0p, t0p + tp] (37) Akp )

∑

∀i∈HP

Ckp )

∑

∀j∈CP

(hot) Zip

Fai (HTM)

Cp

(cold) Zjp

zaik(H2T)

Frj Cp(HTM)

Bkp )

∑

(hot) Zip

∀i∈HP

zrjk(C2T) Dkp )

∑

Fai

Cp(HTM) Frj (cold)

zaki(T2H)

zrkj(T2C) Cp(HTM) ∀k ∈ ST, p ∈ TP

[ [

(

- th′it)FHi - Tai -

(tc′jt - TC(in) j )FCj -

∑

∀k∈ST

(∑

∀k∈ST

estimate at the end of each period p, the discrepancy in remaining energy for each tank can be used as a stopping criteria for the iterative update of Rkp values. The actual average temperature of HTM in tank k over period p (T(ave) kp ) is computed, which is equivalent to the integrated value of the actual HTM temperature profile form t0p to t0p + tp divided by the elapsed time tp. T ′kp(ave) )

(T2H) zak'i T k′t ′

) ]

(hot) Fai Zip zai

)0

∀i ∈ HPS, p ∈ TP, t ∈ [tp0, tp0 + tp] (cold) zr(T2C) T kt′ - Tr(out) Frj Zjp zrj ) 0 kj j

) ]

∀j ∈ CPS, p ∈ TP, t ∈ [tp0, tp0 + tp] (38)

Step 3: Calculate the Discrepancy in Remaining Energy for All Tanks. The actual temperature of HTM in tank k at the end of period p, T ′k,tpo+tp, can be directly calculated from eq 37. However, T ′k,tpo+tp might be inconsistent to the terminal temperature Tk,p+1 from the GAMS design program in step 1, if the Rkp values are inadequate as shown in Figure 5. In order to reduce the discrepancy in temperature

(∫

tp0+tp

tp0

)

T ′kt dt /tp ∀k ∈ ST, p ∈ TP

(39)

The actual remaining energy under MATLAB simulation in tank k over period p (Ekp ′ ) is the integral of the actual average HTM temperature multiplied by the actual remaining quantity and the heat capacity of HTM from tp0 to tp0 + tp. The actual average temperature and the heat capacity of HTM are both constant during period p so that the integral value of the actual quantity (Q′kt) from tp0 to tp0 + tp can be easily found by the trapezoidal integration method. E′kp

Zjp

∀j∈CP

Note that the constant capacity flow rates and outlet temperatures of the HTM (Fai, Frj, Tai, and Trj) result in a simple solution for the actual temperatures; otherwise, the numerical method would be needed to find an approximated solution and the iterative solution procedure will be quite tedious. For series-type heat exchangers, the actual outlet temperatures of process streams matching HTM (thit′ , tcjt′ ) can be calculated by heat balance from the actual temperature of HTM. Note that and the mean values of the actual outlet temperatures are th(ave) it determined by the GAMS design program. tc(ave) jt (TH(in) i

Figure 5. Actual temperature profile of HTM and the update of the Rkp value.

)

∫

tp0+tp

tp0

Cp(HTM)T ′kp(ave)Q′kt dt (Qk,p+1 + Qkp)tp 2 (Q + QkNP)tNP k1 (ave)

) Cp(HTM)T ′kp(ave) EkN ′ P ) Cp(HTM)T ′ kNP

2

∀k ∈ ST, p ∈ TP∀k ∈ ST (40)

Similarly, the remaining energy based on the GAMS design in tank k at period p (Ekp) can also be determined. Ekp

EkNP

)

∫

)

(ave) Cp(HTM)Tkp

)

(ave) Cp(HTM)TkN P

t0p+tp

t0p

(ave) Cp(HTM)Tkp Qkt′ dt

(Qk,p+1 + Qkp)tp 2 (Qk1 + QkNP)tNP 2

∀k ∈ ST, p ∈ TP∀k ∈ ST (41)

The difference of the actual remaining energy from the MATLAB simulator and the remaining energy from the GAMS ′ - Ekp) refers the adequacy of Rkp values. To designer (Ekp guarantee the actual temperatures of HTM in storages at the end of period p to achieve the values from the GAMS design, the external utility is directly used to make-up/remove the deficit/surplus energy. Then, the summed up value of the ′ - Ekp|) is the total external absolute energy difference (|Ekp energy (edk) to heat up or cool down the HTM in tank k. In addition, the error in remaining energy of tank k at period p (ek) is the percentage of the absolute value of the energy difference divided by the actual remaining energy and is used to measure the solution accuracy.


∑

edk ) ek )

∀p∈TP

∑

∀p∈TP

|E′kp - Ekp |

Table 1. Stream Dataa 7

∀k ∈ ST

|E′kp - Ekp | × 100% ∀k ∈ ST E′kp

(42)

Step 4: Update the Iterative Parameters, Rkp. To eliminate the error of terminal temperatures in period p between the solution of the GAMS design tool and the MATLAB temperature simulation, the ratio parameter Rkp is updated as follows. Rkp )

T ′kp(ave) (Tk,p+1 + Tkp)/2

RkNP )

T kN ′(ave) P (Tk1 + TkNP)/2

∀k ∈ ST, p ∈ TP(43) ∀k ∈ ST

In Figure 5, the old parameter, R(old) kp , is substituted into eq 6 to obtain the average temperature of HTM (T(ave) kp ) and, furthermore, to design the heat recovery system. However, the error in remaining energy is bigger than the permissible error (ε). Therefore, eq 43 is used to determine a new parameter, R(new) kp , for the actual average temperature of HTM (T ′kp(ave)) and, finally, the parameter causes that the temperature at the end of period p in the MATLAB simulation will trace to that of the GAMS design. In sum, if the Rkp values are accurate enough, the average temperature of HTM (T (ave) kp ) will be equivalent to the actual ′(ave)), by collocating eqs 6 and average temperature of HTM (T kp 43. 6. Numerical Example One existing method for the design of indirect heat integration system was studied recently by Chen and Ciou,7 where fixed temperature for storage as well as input/output streams for each storage tank was assumed and the external utilities were minimized for heat interchange network design. Consider the same problem in the article of Chen and Ciou7 with two-hot/ two-cold streams, as listed in Table 1, where all streams are processed in series-type heat exchange units. Furthermore, the original design applies external utilities (510 and 470 MJ for cold and hot utilities, the sum of sensible heat (SHi and SCj)) for heating/cooling process streams as given in Table 1. The investment costs are also shown in Table 2. There are four time periods which are defined according to the discontinuous existence of process streams during period p (an existing stream (hot) (cold) or Zjp ) and at time is designated with unity value for Zip (hot) (cold) point p (Zpip and Zpjp ), as shown in Table 3. In the following two cases with different tank sizes, the example is used to design the indirect HEN with a VTVM storage policy. The design of using the FTVM storage policy studied by Chen and Ciou7 is also compared. Furthermore, the solution of minimizing the annual cost for operating the VTVM storage system (ACVT) is investigated in the last case. 6.1. Case 1: Minimizing External Utilities with Two Small Storage Tanks. Chen and Ciou7 investigated the same heat recovery system with fixed storage temperatures in reservoirs for minimizing the external utility expenditure. The results show the amount of heat recovery and utility usage are 285 and 410 MJ/cycle (42% of original design), respectively, and the size of both storage tanks are 298 L. In case 1, the two tanks with the same volumes (298 L) are used to design the heat recovery system with variable storage temperatures for minimizing utilities (P1). The resulting HEN is depicted in Figure 6a. The hot stream H1 is used for further elucidation. During the existing time of

°C

h

MJ/(h °C)

MJ

hot stream

THi(in)

THi(out)

t(s)

t(f)

LHi

SHi

FHi

type

H1 H2

170 150

60 30

0.3 0.0

0.9 0.5

0 0

330 180

5 3

S S

°C

h

MJ/(h °C)

MJ

cold stream

TCj(in)

TCj(out)

t(s)

t(f)

LCj

SCj

FCj

type

C1 C2

20 80

135 140

0.3 0.5

0.5 1.0

0 0

230 240

10 8

S S

HTM glycerol 18-290 °C; C(HTM) ) $5/kg, Cp(HTM) ) 0.0024 MJ/(kg °C), F(HTM) ) 1.2578 kg/L utility Chu ) $60 h/(MJ y), Ccu ) $18 h/(MJ y) a

∆Tmin ) 10 °C; DT ) 10 °C, ε ) 1%.

Table 2. Data for Investment Cost 75000 + 500a0.6 ($) 0.85 (MJ/(h m2 °C)) 3.67 1231.4 1 (h) 5 (y)

heat exchanger overall heat transfer coefficient material correction factor Marshall and Swift index (MS) operating time of one batch payback time τ(ex), τ(tank), τ(HTM)

Table 3. Existence of Streams for Illustrative Example period

1

2

tp(h)

0.3

0.2

3

4

1

2

0.4

0.1

0.3

0.2

(hot) Zip

H1 H2

0 1

1 1

0 0

1 0

4

0.4

0.1

(hot) Zpip

1 0

0 0

0 1

1 1

(cold) Zjp

C1 C2

3

1 1

1 0

1 1

0 1

(cold) Zpjp

0 1

0 1

0 1

1 0

hot stream H1, there are one heat exchanger and one cooler for reducing the temperature of H1. The cold HTM coming from tank 2 absorbs heat of H1 with average heat rates of 509 and 407 MJ/h in the second and third periods. The total heat load of H1 absorbed by HTM in one batch operation is the sum of the average heat rate multiplied by the duration time (509 MJ/h × 0.2 h + 407 MJ/h × 0.4 h ) 265 MJ). The remaining heat load of H1 is treated by cold utility with 41 and 143 MJ/h average heat rates in the second and third periods. The energy transferring causes the average outlet temperatures of H1 matching HTM to be 68 and 89 °C during the second and the third periods, and the outlet temperatures of HTM matching H1 keep a constant value (160 °C) at the same time (from 0.3 to 0.9 h). Notably, the figures of the second row shown in the box of each tank in Figure 6a indicate the average temperatures of HTM during periods 1-4, and those of the third row in parentheses display the temperatures at time points 1-5. The amount of heat recovery is 351 MJ/cycle which is supplied from parts of hot streams (265 MJ from H1 and 86 MJ from H2) and then totally released to cold streams (220 MJ to C1 and 131 MJ to C2). The indirect heat integration network with two VTVM reservoirs can significantly reduce the external utility to 278 MJ/cycle (28.4% of original design). Comparing with the FTVM storage system studied by Chen and Ciou,7 the VTVM storage system can save 132 MJ/cycle with the same storage tanks. The actual outlet temperatures of streams matching HTM are shown in Figure 6b, which are simulated by a MATLAB program based on the network designed by a GAMS program as shown in Figure 6a. Each figure in Figure 6b from top to bottom indicates the actual outlet temperatures (solid line) of


Figure 6. Design results for case 1. (a) Final heat interchange network with two VTVM storage tanks. (b) Actual outlet temperature profile of streams matching HTM. (c and d) Remaining quantities and temperature profile for the thermal reservoirs and the recirculated HTM rates to corresponding streams.

H1, H2, C1, and C2, respectively. The HTM temperature (dotted line) extremely influences the outlet temperature of the corresponding stream, and its profile is also drawn in Figure 6b. Note that for hot streams, the temperature of HTM is lower than that of the corresponding hot stream; for cold streams, the HTM temperature is higher than that of corresponding cold stream. The black points are the outlet temperatures of streams matching HTM from GAMS solution. Take the hot stream H1 for example, from 0.3 to 0.9 h; the outlet temperature of hot stream H1 matching HTM decreases from 107 to 60 °C and then increases to 106 °C. Meanwhile, the temperature of HTM absorbing energy of H1 drops from 88 to 35 °C and then raises to 87 °C. After several iterations, the actual outlet temperature of H1 can match well with the temperature points calculated by GAMS design program at each time point. Figure 6c and d also show the time-dependent remaining quantities (the upper part) and the temperature profile (the middle part) in the two reservoirs and the flow rates of recirculated HTM which exchanges heat with corresponding streams (the lower part). For example, there are two input flow rates of HTM from H1 (H1T) and H2 (H2T) and two output flow rates from hot tank 1 to process streams C1 (TC1) and C2 (TC2). The required volume of both storage tanks is 298 L that is the same as that of FTVM storage system. The solution (338 equations with 94 continuous and 24 binary variables for GAMS) is obtained in 20 iterative times with 643 s

of CPU time, and the energy errors (energy differences) in tank 1 and 2 are 0.38% (0.46 MJ) and 0.34% (0.21 MJ), respectively. 6.2. Case 2: Minimizing External Utility with Two Large Storage Tanks. In this case, the reservoirs are assumed large enough to reach the minimal utility usage by two VTVM storage tanks. The resulting network is shown in Figure 7. The cold HTM in tank 2 is charged with 5.0 and 3.0 MJ/(h °C) heat capacity flow rates to absorb heat of hot streams H1 and H2 and then flows to hot tank 1. Next, tank 1 supplies HTM to heat up C1 and C2 with 9.5 and 5.2 MJ/(h °C) heat capacity flow rates. The recirculated HTM recovers 399 MJ/cycle heat load which is supplied heat load from hot streams (493 MJ/h × 0.2 h + 486 MJ/h × 0.4 h ) 293 MJ from H1 and 195 MJ/h × 0.3 h + 236 MJ/h × 0.2 h ) 106 MJ from H2), and then, the absorbed energy is released to the cold streams (1128 MJ/h × 0.2 h ) 226 MJ to C1 and 343 MJ/h × 0.4 h + 361 MJ/h × 0.1 h ) 173 MJ to C2). In addition, the utility consumption is further reduced to 182 MJ/cycle (18.6% of original design). It is noted that the required volumes of both storage tanks are 451 and 1060 L in case 2 as shown in Figure 7. Meanwhile the amount of heat recovery of cold stream C2 raises from 131 MJ in case 1 to 173 MJ in case 2. The heat capacity flow rate of HTM for heating up C2 increases from 4.2 MJ/(h °C) in case 1 to 5.2 MJ/(h °C) in case 2. Therefore, the size of tank 2 is almost three times as large as that of case 1 to balance the


Figure 7. Design results for case 2. (a) Final heat interchange network with two VTVM storage. (b) Actual outlet temperature profile of streams matching HTM. (c and d) Remaining quantities and temperature profile for the thermal reservoirs and the recirculated HTM rates to corresponding streams.

high temperature (90 °C) HTM matching C2 and charging into tank 2 during period 5. The solution (338 equations with 94 continuous and 24 binary variables for GAMS) is obtained in three iterations with 186 s of CPU time, and the energy errors (energy differences) in tank 1 and 2 are 0.59% (0.86 MJ) and 0.04% (0.07 MJ), respectively. 6.3. Analysis of Annual Cost of a VTVM Storage System, ACVT. The result of minimizing the annual cost for operating a VTVM storage system (P2) is fortuitously the same as that of case 1. By comparing with case 2 (minimizing utility) and case 1 (minimizing the annual cost ACVT), it can be found that less external utility is required with the expense of enlarged storage tanks. The raised amount of HTM used for integrating heat of process streams results in reduced utility consumption and even the annual cost for operating a VTVM storage system. Table 4 shows the annual costs of additional equipment (AAIC) and operating utility (ACU), and the annual cost for operating a VTVM storage system (ACVT) when various quantities of HTM (Q(HTM)) are applied. Therein, the additional annual equipment cost (AAIC) consists of the sum of four additional (HTM) ), and the sum of two exchangers (∑AC (ex) i/j ), HTM (AC ). The original design has a utility cost storage tanks (∑AC (tank) k ($37,380/y) before integration by HTM, which can be calculated by eq 34 from the data in Table 1. In case 1, the cost of ACVT is $25,629/y which consists of $15,633/y for additional annual

Table 4. Detailed Costs under Different Quantities of Recirculated HTM kg (HTM)

Q

0 188 438 656 1252 1526 1850 2304

$/y (ex) ∑ACi/j

0 8221 9073 9236 9773 10003 10003 10003

(HTM)

AC

0 188 438 656 1252 1526 1850 2304

∑ACk(tank) 0 3663 6122 7335 9906 10769 11850 13146

AAIC

ACU

ACVT

0 37380 37380 12072 15209 27281 15633 9996 25629 17227 9060 26287 20931 7356 28287 22298 6275 28573 23703 6275 29978 25453 6275 31728

annotation ACU(ori) case 1 case 2

investment cost and $9,996/y for annual operating utility cost. Comparing with the original design, although AAIC is spent for building up a VTVM storage system, the cost for ACU saves $27,384/y ($37,380-9,996/y), which is even larger than AAIC. Therefore, the VTVM storage system can save $11,751/y ($37,380-25,629/y) in total. The data in Table 4 indicates that all terms in equipment costs increase with the increased quantity of HTM recirculated in the HEN. It is worth mentioning that variations of the payback factors of different equipments will affect the slope of AAIC and, furthermore, the optimal solution. Fortunately, the increased amount of the recirculated HTM can also enhance the heat integration of process streams and, meanwhile, reduce the


Figure 8. Cost analysis for illustrative example.

supplementary external utilities. Notice that applying more HTM than that of case 2, the best solution with minimal utility consumption, cannot further reduce utility consumption, yet meanwhile the cost of storage tanks is increased. The effects of recirculated amount of HTM on costs are shown in Figure 8. Comparing case 2 (minimizing utility) with case 1 (minimizing ACVT), the utility cost of case 1 rises to $3,721/y, but the equipment cost saves $6,665/y. That is because while the amount of HTM is greater than 500 kg in case 1, the curve of utility cost rises gradually but the equipment cost rises substantially. On the basis of analysis in Figure 8, a small amount of HTM can reduce utility consumption significantly. Consequently, in case 1, 438 kg of HTM in two small tanks recovers the heat of process streams and, furthermore, causes the cost for operating the VTVM storage system to be $25,629/y (68.6% of AC(ori)). 7. Conclusion The design of an indirect heat recovery system with a variable temperature storage policy in batch plants is studied in this article. A recirculated heat transfer medium (HTM) is used to absorb surplus heat load from hot process streams and the accumulated energy is then released to later cold process streams. Therefore, the limitation of time-dependent process streams is relaxed. Superstructures modified from the work of Chen and Ciou7 are presented for modeling the time-dependent heat exchange operations. The design problem is formulated as a mixed-integer nonlinear program (MINLP) to minimize the consumption of external utility and total annual cost (TAC) for operating a VTVM storage system. In order to catch nonlinear temperature profile exactly in an indirect VTVM storage system, an iterative update method is prepared by linking a GAMS program for network design and a MATLAB program for storage temperature simulation. A numerical example is used to demonstrate the applicability of the proposed method for indirect heat integration in batch plants. 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. Nomenclature AbbreViations FTVM ) fixed-temperature/variable-mass HEN ) heat exchanger network

HTM ) heat transfer medium VTVM ) variable-temperature/variable-mass MINLP ) mixed-integer nonlinear program Indices i ) index for hot process streams j ) index for cold process streams k ) index for storage tanks, 1, ..., NS p ) index for time periods, 1, ..., NP, and time points, 1, ..., NP + 1 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 in a multistream exchanger} HPC ) {i|i is a hot process stream in 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 in a series unit} CPP ) {j|j is a cold process stream in a parallel unit} ) CPM ∪ CPC CPM ) {j|j is a cold process stream in a multistream exchanger} CPC ) {j|j is a cold process stream in a jacket exchanger with a coil} ST ) {k|k is a storage tank in 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 ACU(ori) ) annual cost of utility for original design BT ) operating time of one batch C(HTM) ) cost per kilogram of HTM ($/kg) Ccu ) cost of cold utility (($ h)/(MJ y)) Chu ) cost of hot utility (($ h)/(MJ y)) Cp(HTM) ) heat capacity of HTM CAi/j ) area cost of heat exchanger of stream i/j matching HTM CBi/j ) exponential coefficient for heat exchanger cost of stream i/j matching HTM CFi/j ) fixed cost of heat exchanger of stream i/j matching HTM DT ) minimum difference temperature of two neighboring tanks j ,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 Fm ) material correction factor qj,q_ ) upper/lower bound of heat flow rate LCj ) total latent heat of cold stream j LHi ) total latent heat of hot stream i MNT ) maximum number of tanks j ,Q ) upper/lower bound of quality Q Rkp ) ratio of actual average temperature during period p to average value of temperatures at time points p and p + 1 for tank k (for GAMS) SCj ) total sensible heat of cold stream j SHi ) total sensible heat of hot stream i tp ) elapse time of period p (h) tp0 ) initial time of period p (h) Tj ,T _ ) upper/lower bound of temperature TCj(in) ) inlet temperature of cold stream j TCj(out) ) outlet temperature of cold stream j THi(in) ) inlet temperature of hot stream i THi(out) ) outlet temperature of hot stream i Ui/j ) overall heat transfer coefficient of stream i/j (hot) Zip ) ∈{0,1}, ) 1 denotes the existence of hot stream i during period p

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4387 ) ∈{0,1}, ) 1 denotes the existence of cold stream j during period p (hot) Zpip ) ∈{0,1}, 1 denotes the existence of hot stream i at time point p (cold) Zpjp ) ∈{0,1}, ) 1 denotes the existence of cold stream j at time point p ε ) permissible error in remaining energy F(HTM) ) density of HTM (kg/L) σ ) a small positive bound τ* ) payback time (y) ∆Tmin ) minimum temperature difference of match (i, HTM) and match (HTM, j) MS ) Marshall and Swift index Continuous Variables ai/j,p ) heat exchanger area of stream i/j matching HTM at time point p AAIC ) annual additional investment cost for a VTVM storage system AC(HTM) ) annual cost of HTM AC(ex) i/j ) annual cost of heat exchanger of stream i/j matching HTM (tank) ACi/j ) annual cost of tank k ACVT ) the annual cost objective for indirect batch heat exchanger network (BHEN) with a VTVM storage system ACU ) annual cost of utility dtcj(in) ) inlet temperature difference between cold stream j and HTM dtcj(out) ) outlet temperature difference between cold stream j and HTM dth(in) i ) inlet temperature difference between hot stream i and HTM dthi(out) ) outlet temperature difference between hot stream i and HTM Dk ) diameter of tank k ek ) error in remaining energy of tank k (by MATLAB) Ekp ) remaining energy of tank k at period p (by MATLAB) E′kp ) actual remaining energy of tank k at period p (by MATLAB) edk ) energy difference of tank k (by MATLAB) Fai ) heat capacity flow rate of HTM matching hot stream i Frj ) heat capacity flow rate of HTM matching cold stream j Hk ) height of tank k LMTDi/j,p ) log mean temperature difference of HTM and the matching stream i/j at time point p (ex) qi/j,p ) heat flow rate of stream i/j matching HTM at time point p (ave) qaip ) average heat flow rate removed by HTM from hot stream i at time period p (MJ/h) (ave) qcuip ) average heat flow rate removed by cold utility from hot stream i at time period p (MJ/h) qr(ave) ) average heat flow rate supplied by hot utility to cold stream jp j at time period p (MJ/h) (ave) qrjp ) average heat flow rate supplied by HTM to cold stream j at time period p (MJ/h) Qkp ) initial quality of tank k at time period p (kg) Q′kt ) initial quality of tank k at absolute time t (kg) (by MATLAB) Q(HTM) ) entire quantity of HTM for all tanks Rkp ) ratio of actual average temperature during period p to average value of temperatures at time points p and p + 1 for tank k (for MATLAB) tc(ave) ) average outlet temperature of cold stream j matching HTM jp in series unit at time period p tcjt′ ) actual average outlet temperature of cold stream j matching HTM in series unit at absolute time t (by MATLAB) tcpjp ) initial outlet temperature of cold stream j matching HTM in series unit at time period p (ave) thip ) average outlet temperature of hot stream i matching HTM in series unit at time period p (cold) Zjp

thit′ ) actual average outlet temperature of hot stream i matching HTM in series unit at absolute time t (by MATLAB) thpip ) initial outlet temperature of hot stream i matching HTM in series unit at time period p Tkp ) initial temperature of HTM in tank k at time period p T kt′ ) actual temperature of HTM in tank k at absolute time t (by MATLAB) (ave) Tkp ) average temperature of tank k at time period p (ave) Tkp ) actual average temperature of tank k at time period p (by MATLAB) Tai ) output temperature of HTM matching hot stream i Trj ) output temperature of HTM matching cold stream j UVT ) the utility objective for indirect BHEN with a VTVM storage system Vk ) volume of tank k Binary Variables zai∈{0,1}, ) 1 denotes existence of heat exchanger i of the match between hot stream i and HTM zaki(T2H)∈{0,1}, ) 1 denotes existence of HTM coming from tank k to cool hot stream i zaik(H2T)∈{0,1}, ) 1 denotes existence of HTM matching hot stream i and flowing into tank k zcui∈{0,1}, ) 1 denotes existence of cooler of the match between hot stream i and cold utility zhuj∈{0,1}, ) 1 denotes existence of heater of the match between cold stream j and hot utility zrj∈{0,1}, ) 1 denotes existence of heat exchanger j of the match between cold stream j and HTM zrkj(T2C)∈{0,1}, ) 1 denotes existence of HTM coming from tank k to heat cold stream j zr(C2T) ∈{0,1}, ) 1 denotes existence of HTM matching cold stream jk j and flowing into tank k ztk∈{0,1}, ) 1 denotes existence of storage tank k

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ReceiVed for reView September 10, 2008 ReVised manuscript receiVed January 18, 2009 Accepted February 13, 2009 IE8013633

Design of Indirect Heat Recovery Systems with Variable-Temperature

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