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Steam system network synthesis with hot liquid reuse: II. Incorporating shaft work and optimum steam levels Sheldon G. Beangstrom a , Thokozani Majozi a,b,∗ a b

Department of Chemical Engineering, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Johannesburg 2000, South Africa

a r t i c l e

a b s t r a c t

i n f o

Article history: Received 28 April 2015 Received in revised form 14 October 2015 Accepted 20 October 2015 Available online 2 November 2015 Keywords: Heat integration Steam network synthesis Steam system optimization Steam ﬂowrate minimization

In this ﬁrst of a series of two papers, the effects of varying steam levels on the total steam ﬂowrate are analyzed mathematically for the traditional parallel conﬁguration as well as for the case of hot liquid reuse. It is demonstrated that in the case of parallel heat exchangers utilizing only latent heat, a minimum total steam ﬂowrate is obtained by optimally selecting steam levels, but that in the case of hot liquid reuse, introducing multiple steam levels increases the minimum total steam ﬂowrate attainable under those conditions. The ﬂowrate attained utilizing hot liquid reuse, however, remains lower than when only utilizing latent heat. It is concluded that the lowest steam ﬂowrate is attained using hot liquid reuse and only a single level of steam, but that the presence of additional steam levels resulting from turbines requires a more holistic approach to the synthesis of steam networks. © 2015 Elsevier Ltd. All rights reserved.

Synopsis In the ﬁrst paper of this series, it is shown that the lowest total steam ﬂowrate is attained with only a single steam level and hot liquid reuse, but that the need for shaft work necessitates additional steam levels. Existing models for steam ﬂowrate minimization fail to take into account the effects the steam turbines and the temperatures of intermediate steam levels have on the minimum total steam ﬂowrate, keeping the ﬂowrate and operating temperatures of the ﬁxed. An MINLP formulation is developed with a holistic approach to the steam system. Turbine ﬂowrates and steam level temperatures are treated as variables, allowing the model to target the minimum total steam ﬂowrate, whilst also synthesizing the heat exchanger network, placing the intermediate steam levels, and sizing the turbines. Application of this technique to a case study yielded a 28.6% reduction in total steam ﬂowrate, compared to using only latent heat in a traditional parallel conﬁguration. 1. Introduction Pinch analysis has become a commonplace tool for optimizing chemical processing plants. The introduction of personal

∗ Corresponding author at: School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Johannesburg, 2000, South Africa. Tel.: +27 11 717 7567; fax: +27 86 565 7517. E-mail address: [email protected] (T. Majozi). http://dx.doi.org/10.1016/j.compchemeng.2015.10.016 0098-1354/© 2015 Elsevier Ltd. All rights reserved.

computers capable of solving large problems rapidly using mathematical programming has further boosted the popularity of pinch analysis. Although originally a method of minimizing the energy utility requirements of a plant, heat integration has been applied in related ﬁelds, such as cooling water system design (Kim and Smith, 2001), steam system network synthesis (Coetzee and Majozi, 2008), and wastewater minimization (Wang and Smith, 1994). The success of pinch analysis lies in the ability to determine a target before committing to a detailed design. Pinch analysis can also bring about meaningful energy savings in retroﬁt designs, while requiring very little capital expenditure (Zhang et al., 2013). Wang and Smith (1994) developed a graphical method of minimizing waste water by allowing spent water to be reused in process with less stringent requirements. The method is based on a limiting concentration proﬁle, which represents the concentration requirements of the processes on a concentration/mass-load plot. Waste water may be represented on the same plot and minimized by reducing the ﬂowrate until a pinch is seen. Having obtained a target for the minimum waste water ﬂowrate, a mass exchanger network involving reuse can be synthesized to meet this target. Kim and Smith (2001) used a similar method to reduce the ﬂowrate of cooling water in a heat exchanger network in order to optimize the effectiveness of cooling towers. More recently, Gololo and Majozi (2011) created a mathematical model, which combines multiple cooling towers with the heat exchanger network. This is necessary in order to gain a holistic view of the cooling tower network, since cooling towers have an effect on the heat exchanger network and vice versa.

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Nomenclature Sets: I J L

Set of all heat exchangers, Set of all heat exchangers (alias), Set of all steam levels.

Parameters: cp Heat capacity of water [kJ/kg·◦ C], U Upper limit on reuse in i [kg/s], FRRi Qi Duty required by stream i [kW], U SSi,l Upper limit on steam of level l in i [kg/s], Tiin,L

Tiout,L

Limiting the utility inlet temperature for i [◦ C], Limiting the utility outlet temperature for i [◦ C].

Continuous variables: Al Regression parameter for level l [MW], Bl Regression parameter for level l [-], Inlet ﬂow to i [kg/s], Fiin Fiout Outlet ﬂow from i [kg/s], Liquid reuse to i from j of level l [kg/s], FRRi,j,l FRi Boiler return from i [kg/s], Total steam supply of level l [kg/s], FSl is

H l

HLi,j,l m qin QiSS QiHL SLi,j,l SSi,l Tlsat TR TS TUSl Wls

Speciﬁc isentropic enthalpy change between levels [MWh/t], Hot liquid to i from j of level l [kg/s], Additional heat exchanger splits permitted [-], Heat load of steam [MWh/t], Heat supplied to i from saturated steam [kW], Heat supplied to i from liquid [kW], Saturated liquid to i from j of level l, Saturated steam supplied to i from level l [kg/s], Saturation temperature of level l [◦ C], Total return to boiler [kg/s], Total steam supply [kg/s], Steam supplied to turbine of level l [kg/s], Shaft work produced by turbine l [MW].

Binary variables: Binary variable for use of liquid in i [−], xi yi,l Binary variable for use of steam in i of level l [−]. Greek: l

Latent heat of level l [kJ/kg].

On the other end of the spectrum, Coetzee and Majozi (2008) developed methods of minimizing the ﬂowrate of steam required by a heat exchanger network by reusing the hot liquid condensate produced by certain heat exchangers. A graphical procedure was developed in which the minimum ﬂowrate is targeted much like in the method of Kim and Smith (2001). A mathematical programming model is then used to synthesize the heat exchanger network. Coetzee and Majozi (2008) also present a more detailed mathematical programming model that simultaneously performs the targeting and synthesis tasks. Continuing on this work, Price and Majozi (2010) considered the effects of reduced ﬂowrate on boiler efﬁciency, the presence of multiple steam levels and also pressure drop in the heat exchanger network. The use of mathematical programming to synthesis optimum designs in the chemical industry has also become a popular tool. Grossmann and Santibanez (1980) demonstrated the power of binary variables in MILP formulations for process synthesis. Papoulias and Grossmann (1983) went on to use binary variables

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with superstructures to synthesize and optimize whole processes. The mathematical models by Coetzee and Majozi (2008) and Price and Majozi (2010) both use binary variable to indicate the existence of streams, while synthesizing the heat exchanger network. In the ﬁrst paper of this series it is shown that when minimizing the steam ﬂowrate in a heat exchanger network using hot liquid reuse, the best option is to only use a single high pressure steam level as considered by Coetzee and Majozi (2008). The need for producing shaft work in a turbine does, however, leads to the creation of additional steam levels that would be wasted if not used in the heat exchanger network. The model by Price and Majozi (2010) optimizes the use of multiple steam levels, but does not consider the turbines or production of these steam levels. Rather, the model assumes that multiple levels of steam are available at ﬁxed levels and ﬂowrates. Considering the results of the ﬁrst paper in this series, it is clear that a model is required that includes the interactions of the turbines and steam levels with the heat exchanger network. El-Halwagi et al. (2009) consider the case of cogeneration, while utilizing waste combustibles as a heating source for both utility steam and generation steam. This method, however, only considers the use of latent heat in the heat exchangers, without looking at the potential of utilizing the sensible heat in the condensate. Mavromatis and Kokossis (1998) developed a turbine hardware model to estimate shaft work production, while performing heat integration. It is used to select a set of steam levels from a number of steam levels for maximum shaft work recovery, while performing total site heat integration. Shang and Kokossis (2004) used the turbine hardware model as well as a boiler model to develop a transhipment model to optimize the steam levels for total site heat integration, where the efﬁciencies of these units are incorporated into the design. At present, there is no known model that considers the interaction of the turbine and steam levels with the ﬂowrate of steam through the heat exchanger network, as discussed in the ﬁrst paper of this series. In this paper, a model is developed that takes into account these interactions, while simultaneously targeting the minimum steam ﬂowrate and synthesizing the steam system. A case study is used to demonstrate the model and to compare it to previous results.

2. Problem statement The ﬂowrate of steam to a heat exchanger network (HEN) can be minimized using pinch analysis by reusing hot liquid. This has been successfully demonstrated (Coetzee and Majozi, 2008; Price and Majozi, 2010), although without consideration for shaft work targets or the effects of level temperatures on the total steam ﬂowrate. The problem to be addressed here can be formally stated as follows: Given: • • • • •

a set of cold process streams, the ﬁxed duties required by each cold process stream, the supply and target temperatures for each cold process stream, the minimum temperature difference for heat exchange, a shaft work target. Determine:

• the minimum total steam ﬂowrate, • the intermediate steam levels required to satisfy the energy demands of the heat exchangers and shaft work requirements, • the network of heat exchangers that will fulﬁl this target.

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Fig. 1. General steam system layout.

3. System description Although steam systems vary from plant to plant, they all follow common design practices, which can be generalized as in Fig. 1. High pressure superheated steam is raised in a boiler, from which it is distributed to all the steam using operations. Since conventional design practice only utilizes latent heat in heat exchangers, the portion of steam sent to provide high pressure utility heating passes through a let-down valve to bring it to saturation conditions. This allows the steam to condense isothermally, which allows for simpler heat exchangers since only condensation of steam need be considered. Heat exchangers are also generally laid out in a parallel conﬁguration, for which only using latent heat is ideal. Depending on the plant in question, another portion of high pressure superheated steam is sent to a steam turbine. This might be due to the need for shaft work, or for deliberately creating another steam level, while gaining the full potential of the higher steam level. As shown in the ﬁrst paper of this series, when the process stream to be heated does not call for a high temperature and only latent heat is used, then it is beneﬁcial to use a lower steam level for its higher latent heat value. Thus, a portion of this intermediate steam is brought to saturation conditions through a let-down valve and passed to the HEN, while the remainder may be passed to a second turbine to create another steam level. Steam from each level is assigned to different sets of process streams based on the temperature requirements of each stream. After steam from various steam levels has condensed inside the heat exchangers, the condensate is collected for return to the boiler. Occasionally, there will be a buffer tank to reduce any ﬂuctuations in ﬂowrate caused by the control system. Since the condensate is at saturation point, when using certain types of pump (e.g., centrifugal) it must ﬁrst be cooled before pumping it to the boiler to prevent cavitation in the pump. The condensate then passes through an economizer where it is preheated with waste heat before entering the boiler. The economizer and the boiler are shown as two separate entities in Fig. 1 for the sake of demonstration. In reality, the economizer is integrated into the boiler. Shang and Kokossis (2004) found that in order to maintain boiler efﬁciency, as the steam ﬂowrate was reduced the boiler return temperature must increase. From a thermodynamic perspective though, for a ﬁxed heating duty, as the utility ﬂowrate decreases, so too does the return temperature. To counteract this, Price and Majozi (2010) proposed a modiﬁed system (Fig. 2). Here, the

sensible superheat from the high pressure steam level is used to heat the boiler return to the requisite temperature for maintained efﬁciency. This also brings the high pressure steam level closer to saturation and lessens the need for a let-down valve. Price and Majozi (2010) also propose that the process streams be treated as a whole, to allow better use of the multiple steam levels. Here, one process stream might be heated by two steam levels. Having all the process streams together also allows condensate from one level to be used to heat a process stream that would otherwise be conﬁned to the set of streams belonging to a different steam level. 4. Heat exchanger network layout The distinctive ﬁngerprint of process integration on the HEN of a steam system is the presence of both serial and parallel connections. The need for this parallel-serial hybrid stems from the various sources of heat. As depicted in Fig. 3a, reuse of condensate for heating results in two distinct regimes within the HEN, viz. latent heat and sensible heat regimes. Whilst the latent heat regime entails only a parallel conﬁguration of heat exchangers, the sensible heat regime involves a complex combination of parallel and series connections. Some heat exchangers may even need to be split between latent and sensible heat. The distinction between the simple parallel layout of latent heat using heat exchangers, and the more detailed layout of the sensible heat using heat exchangers is seen in Fig. 3b. Split heat exchangers occur whenever a particular process stream requires heat from two or more different heat sources. At this point, it is worth emphasizing the distinction between serial and parallel heating since this will feature prominently when using multiple steam levels. Consider the case of a single cold process stream being heated using two levels of saturated steam. Fig. 4 shows this process, along with the limiting temperature proﬁle that arises due to the minimum temperature difference for heat exchange Tmin . In the ﬁrst instance (Fig. 4a), the lower steam level may have sufﬁcient energy to supply the entire process, but is limited by its temperature. In the second instance (Fig. 4b), both steam levels are hot enough to successfully heat the cold process stream, but the lower steam level may have insufﬁcient energy, necessitating an addition steam level. In Fig. 4a, since the medium pressure steam is not hot enough to heat the cold stream to its target temperature Ttar , it can only raise

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Fig. 2. Steam system for reduced ﬂowrate, while maintaining boiler efﬁciency.

Fig. 3. The division between latent and sensible heat using processes (a) on a T–H diagram, (b) in the HEN.

Fig. 4. T–H diagram and heat exchanger layout for (a) one steam level colder than Ttar , (b) both steam levels hotter than Ttar .

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Table 1 Turbine regression constants (Mavromatis and Kokossis, 1998).

General case Ws < 1.2 MW Ws < 1.2 MW

procedure, one can see that such simpliﬁcations are acceptable and provide an easy to use model.

a1 [MW]

a2 [MW/K]

a3 [−]

a3 [−]

−0.538 −0.131 −0.928

0.00,364 0.00,117 0.00,623

1.112 0.989 1.12

0.00,052 0.00,152 0.00,047

6. Method

the temperature of the cold stream to Tint . The remainder of the duty required is provided by high level steam. In Fig. 4b, since both steam levels are hot enough to bring the cold stream to its target temperature, but the total heating duty is shared between them, the cold stream can be split in two and heated to Ttar in parallel heat exchangers before being recombined. It is also possible to use serial heat exchangers in this case, but heat exchangers in series will cause a higher total pressure drop than heat exchangers in parallel. 5. Shaft work targeting In order to develop a model with a holistic approach to both the HEN and the power block containing the turbines, one needs a method of estimating the performance of a turbine. Turbines, however, are intricate pieces of machinery, requiring very detailed simulation to achieve an accurate value of the power output. In process integration, the model describing the turbines needs to be simple enough for an optimization routine to handle efﬁciently, while still retaining enough accuracy to ensure that the resulting design is within the optimum range. Mavromatis and Kokossis (1998) derive the following model for estimating the shaft work of a turbine operating at full load: 1 Ws = B

is

H M − A

(1)

where Ws is the shaft work in MW, H is is the speciﬁc isentropic enthalpy change of the steam in MWh/t, M is the steam ﬂowrate in t/h, and A and B are calculated using Eqs. 2 and 3. sat A = a1 + a2 Tin

(2)

sat b1 + b2 Tin

(3)

B=

The regression constants in Eqs. 2 and 3 are given in Table 1. The ﬁrst set of values is a general set, while the second two sets of values give better accuracy if the expected shaft work output is roughly known. This turbine model is claimed to be accurate to 3% of the actual value, if using the latter parameters in Table 1. To calculate the speciﬁc isentropic enthalpy change of steam, Singh (1997) gives the following correlation: is

H =

T sat 1854 − 1931 × qin

(4)

where qin is the speciﬁc heat load of the steam entering the turbine in MWh/t. This correlation is claimed to have an accuracy of between 5 and 10%, depending on operating conditions. The heat load of steam comprises the latent heat and sensible superheat. If it is assumed that the latent heat is dominating, then the following simpliﬁcation of Eq. 4 is obtained: is

H =

T sat sat 391.8 − 2.215 × Tin

The primary objective of this model is to minimize the total steam ﬂowrate coming from the boiler, while simultaneously determining the intermediate steam levels required, the steam ﬂowrates need by the turbines in order to meet shaft work demands, and synthesizing the HEN. Fig. 5 shows the superstructure used in this model. In this instance, there are three steam levels and two turbines, but the model is fully customizable as the superstructure and the model have the ﬂexibility to allow for additional steam levels and turbines. In the ﬁrst paper of this series, it was shown that when utilizing hot liquid reuse, only one high level of steam should be used. The demand for shaft work, however, leads to the creation of additional steam levels along with their associated penalties on the total steam ﬂowrate. The distinguishing feature of this model lies in leaving the temperatures and ﬂowrates of the lower steam levels as variables, and including the shaft work targets with the model constraints whilst minimizing the total steam ﬂowrate. In Fig. 5, i and j are heat exchangers. The full declaration of variables can be found in the nomenclature. The temperatures under consideration here are the limiting inlet and outlet temperatures of the hot utility, taking into account the minimum temperature difference for heat transfer. These are used to calculate upper limits on the ﬂowrate of liquid and steam using Eqs. 6 and 7, respectively.

FRRiU =

cp

Qi Tiin,L

− Tiout,L

∀i ∈ I

(6)

⎧ 0 if Tlsat ≤ Tiout,L ⎪ ⎪ ⎪ sat ⎪ out,L ⎪ ⎪ Q ⎨ Tl − Ti i if Tiout,L < Tlsat < Tiin,L U l in,L out,L SSi,l = ∀i ∈ I Ti − Ti ⎪ ⎪ ⎪ ⎪ ⎪ Qi ⎪ ⎩ if T sat ≥ T in,L l

l

(7)

i

The initial constraints in the model concern mass balances over various parts of the HEN. Eq. 8 states that the total supply of steam to the steam system is comprised of the ﬂowrates of the individual steam levels, while Eq. 9 states the total return to the boiler is comprised of the return from the individual heat exchangers. Since mass is conserved, these two ﬂowrates must be equal (Eq. 10). Eq. 11 stipulates that each steam level is made up of the ﬂowrate of steam supplied at that level to each of the heat exchangers. TS =

FSl

(8)

l∈L

TR =

(9)

FRi

i∈I

TS = TR FSl =

(10) SSi,l

∀l ∈ L

(11)

i∈I

(5)

using the relation for latent heat discussed in the ﬁrst paper of this series, and repeated in this paper as Eq. 22. The substitution of latent heat for the heat load of steam within the context of hot utilities is justiﬁed, since in many cases the latent heat does dominate the heat load. Noting that Eq. 4 is not perfectly accurate, and accepting the fact that this model is only to be used in a targeting

Performing mass balances across individual heat exchangers, Eq. 12 represents the inlet stream to the heat exchanger, while Eq. 13 represents the outlet stream. Eq. 14 ensures that mass is conserved over each heat exchanger, while Eq. 15 states that each reuse stream is made up of saturated liquid and/or hot liquid. Fiin =

l

SSi,l +

j,l ∈ J,L

FRRi,j,l

∀i ∈ I

(12)

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207

Fig. 5. Superstructure multiple steam level ﬂowrate minimization.

Fiout = FRi +

∀i ∈ I

FRRj,i,l

(13)

j,l ∈ J,L

∀i ∈ I

Fiin = Fiout

(14)

∀i, j ∈ I, l ∈ L

FRRi,j,l = SLi,j,l + HLi,j,l

(15)

To prevent a heat exchanger from recycling hot liquid back to itself, Eq. 16 is implemented. A similar constraint can be implemented to prevent reuse between particular heat exchangers for reasons of safety, risk of contamination, or excessive distance between units. FRRi,j,l = 0 ∀i, j ∈ I i = j

(16)

Eq. 17 states that that the total amount of saturated liquid being supplied to others by a particular heat exchanger cannot exceed the amount of saturated steam supplied to that heat exchanger, since it is the condensation of this steam that makes saturated liquid available. Likewise, Eq. 18 states that a heat exchanger cannot send more hot liquid for reuse than it is supplied with:

∀i ∈ I, l ∈ L

SLi,j,l ≤ SSi,l

j∈J

HLi,j,l ≤

j∈J

SLj,i,l +

j∈J

(17)

HLj,i,l

∀i ∈ I, l ∈ L

(18)

QiSS =

SSi,l l

∀i ∈ I

cp SLj,i,l Tlsat +

(19)

j,l ∈ J,L

cp HLj,i,l Tjout,L − cp Fiout Tiout,L ∀i ∈ I

j,l ∈ J,L

(20)

QiSS

+ QiHL

= Qi

∀i ∈ I

∀l ∈ L

(22)

In order to control which heat exchangers are using saturated steam or hot liquid, the binary variables yi, j and xi are introduced to represent the use of steam and liquid, respectively. Eqs. 23 and 24 prevent any ﬂowrate from exceeding its upper bound whilst using the binary variables to determine the existence of streams. U yi,l SSi,l ≤ SSi,l

∀i ∈ I, l ∈ L

FRRi,j,l ≤ FRRiU xi

(23)

∀i ∈ I

(24)

j,l ∈ J,L

The use of binary variables also gives control over how many heat exchangers must be split between sensible and latent heat. In order to prohibit any heat exchanger from being split, then the following constraint applies: xi +

j∈J

l∈L

QiHL =

l = 2726 − 4.13Tlsat

yi,l ≤ 1 ∀i ∈ I

(25)

l∈L

Constraints must be used to ensure that the required duty is supplied to each heat exchanger. Eqs. 19 and 20 calculate, respectively the amount of latent and sensible heat supplied to each heat exchanger, while Eq. 21 ensures that the latent and sensible heat combined fulﬁls the heat demand of the heat exchanger. In Eq. 20, the outlet temperatures of the heat exchangers have been ﬁxed at their limiting outlet temperatures as this ensures a minimum ﬂowrate (Coetzee and Majozi, 2008).

required to calculate the latent heat of steam required in Eq. 19. Eq. 22 shows the correlation between the latent heat and saturation temperature used by Mavromatis and Kokossis (1998). It is valid between the temperatures of 100 and 300 ◦ C, and is accurate to within 2% of the true value.

(21)

Since in this model the saturation temperatures of the intermediate steam levels are not ﬁxed, but treated as variable, a method is

To allow up to m heat exchangers to be split, then the following to constraints should be applied in the place of Eq. 25. This will be more capitally intensive, but will provide a lower ﬂowrate.

xi +

i∈i

i∈I

yi,l ≥ i

(26)

i,l ∈ I,L

xi +

yi,l ≤ i + m

(27)

i,l ∈ I,L

Turning to the power block, a mass balance can be performed over each turbine. Note that if there are L steam levels, then there are only L–1 intervals over, which the mass balance in Eq. 28 can be applied. TUSl+1 + FSl+1 ≤ TUSl

∀ l ∈ L l < L

(28)

Eq. 29 gives the shaft work produced by each turbine. Eq. 30 ensures that together, the turbines produce the required total amount of shaft work. If a particular piece of equipment requires a dedicated turbine, then an individual shaft work target can be set for that turbine in addition to Eq. 30. If more than one

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Fig. 6. Resulting steam system.

turbine is required between two speciﬁc steam levels, Eq. 29 must be modiﬁed to reﬂect each individual turbine. Wls =

1 Bl

is

3.6H l TUSl − Al

∀l ∈ L l < L

s Wls = Wdemand

(29) (30)

l<|L|

Again, since the saturation temperatures of the steam levels are not ﬁxed, certain parameters in Eq. 29 cannot be predetermined. Therefore, the following constraints must be added to the model: the coefﬁcients for Eqs. 32 and 33 can be found in Table 1. is

H l =

sat Tlsat − Tl+1

391.8 + 2.215Tlsat

Al = a1 + a2 Tlsat

∀l ∈ L l < L

∀l ∈ L l < L

(31) (32)

∀l ∈ L l < L

(33)Bl = b1 + b2 Tlsat Finally, the objective to minimize the total steam ﬂowrate supplied by the boiler, is expressed in Eq. 34. Thus, along with the above constraints constitute a MINLP model, which includes the power block in the design of the steam system. minimise (TS)

(34)

7. Case study In order to demonstrate this model and its beneﬁts, an illustrative example is given. The problem is taken from a South African petrochemical plant, and is used by Coetzee and Majozi (2008), and Price and Majozi (2010). Table 2 gives the data relating to the seven cold streams. High pressure steam is available at a saturation temperature of 225 ◦ C and a constant heat capacity of 4.3 kJ/kg ◦ C is used. A Tmin of 10 ◦ C was applied globally. The turbine model was used to calculate the amount of shaft work produced under the conditions given in Price and Majozi (2010). The result, 110 kW, is set as the shaft work target in the new model.The model was solved in GAMS v22.0 on an Intel® CoreTM i3-2100 3.1 GHz processor and 2 GB of RAM, using Dicopt with Conopt as the NLP solver, and Cplex as the MIP solver. Since the Dicopt algorithm requires a feasible starting point, the temperatures of the steam levels were ﬁxed at levels recommended by

Table 2 Cold stream and limiting utility data (Coetzee and Majozi, 2008). Stream

Tsup (◦ C)

Ttar (◦ C)

Tin, L (◦ C)

Tout, L (◦ C)

Duty (kW)

1 2 3 4 5 6 7

25 25 209 79 207 44 44

45 45 215 185 207 70 70

55 55 225 195 217 80 80

35 35 219 89 217 54 54

135 320 3620 12,980 1980 635 330

Harrel (1996) and the resulting MILP problem solved. The solution to this MILP was used as the starting point for the MINLP problem with variable temperatures. A CPU time of 0.01 s was required for the MILP and 0.52 s for the MINLP, giving a total of 0.53 s. Three major iterations were required between the NLP and MIP solvers. The objective value, being the minimum total steam ﬂowrate supplied by the boiler, was found to be 7.8066 kg/s.Using the levels and turbine steam ﬂowrates given by Price and Majozi (2010) as a base case, it was calculated that a design utilizing only latent heat in a traditional parallel conﬁguration would require 10.94 kg/s of high pressure steam, 8.71 kg/s supplied directly to the process, and 2.23 kg/s supplied to the turbines. Using the new formulation, a total steam ﬂowrate of 7.81 kg/s of high pressure steam was achieved, 4.81 kg/s supplied directly to the process, and 2.92 kg/s supplied to the turbine. This represents a 28.6% reduction in the total steam ﬂowrate requirement of the steam system. Fig. 6 shows the resulting steam system with the relevant ﬂowrates in kg/s. It can be seen that the network in Fig. 6 superﬁcially resembles the result in Coetzee and Majozi (2008) with the exception of a turbine and an additional steam level, and that the ﬂowrate achieved, 7.81 kg/s, is slightly higher than the ﬂowrate of 7.67 kg/s achieved by Coetzee and Majozi (2008). This is to be expected, since as was shown in the ﬁrst paper of this series, when utilizing hot liquid reuse, the best option is to only use a single steam level. Coetzee and Majozi (2008); however, do not take into account the production of shaft work, which would require additional steam. Although the model in this case was set up to consider three steam levels and two turbines, the model chose to eliminate one steam level and one turbine. It should be noted

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that the ﬂowrate and temperatures of the supply to the turbine should not be considered as the ﬁnal design, but as the starting point to a detailed design involving rigorous simulations. 8. Conclusion A novel MINLP model has been developed to include the power block within the steam system. The model aims to minimize the total steam ﬂowrate supplied by the boiler and simultaneously synthesize the resulting steam system, while fulﬁlling shaft work targets and the heat demands of cold streams. In an illustrative example, the new model produced a 28.6% reduction in total steam ﬂowrate compared to the convention design method. This proves that the total steam ﬂowrate can be reduced by holistically optimizing the steam levels and turbine ﬂowrates with the steam system. Furthermore, the results conﬁrm the observations made in the ﬁrst paper of this series that introducing additional steam levels increases the minimum steam ﬂowrate when utilizing hot liquid reuse.

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