Synthesis and Optimization of Steam System Networks. 2. Multiple

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Ind. Eng. Chem. Res. 2010, 49, 9154–9164

Synthesis and Optimization of Steam System Networks. 2. Multiple Steam Levels Tim Price† and Thokozani Majozi*,†,‡ Department of Chemical Engineering, UniVersity of Pretoria, South Africa, and Modelling and Digital Science, CSIR, Pretoria, South Africa

The use of steam in heat exchanger networks (HENs) can be reduced by the application of heat integration with the intention of debottlenecking the steam boiler and indirectly reducing the water requirement [Coetzee and Majozi. Ind. Eng. Chem. Res. 2008, 47, 4405-4413]. By reducing the steam flow rate, the return condensate temperature to the boiler is compromised which adversely affects the boiler operation. A means of maintaining efficient boiler operation is to reheat the return flow to the boiler. Steam systems typically employ turbines of which the exhaust is frequently used as a heating utility in the background process. Since turbines operate at various steam levels, a means for incorporating these steam levels into the HEN optimization framework is necessary. Consequently this paper concerns the restructuring of all steam system heat exchangers using conceptual and mathematical analysis to create a series HEN with the aim of reducing the overall steam flow rate, while maintaining the boiler efficiency. In an example problem, it was found that the steam flow rate could be reduced by up to 26.3% while still maintaining the boiler efficiency. 1. Introduction Pinch analysis has found numerous applications in a wide range of process integration areas, most specifically mass and heat integration. Heat integration has the ultimate goal of reducing external utilities by maximizing process to process heat exchange, but it can also be used in the optimal placement of utilities.2 Cooling water systems comprising a cooling tower and associated heat exchanger network (HEN) have been examined in detail and optimized by Kim and Smith.3 Majozi and Moodley4 optimized and designed cooling water systems incorporating multiple cooling towers using mixed integer linear programming. They considered the cooling system in a holistic fashion and improved the cooling tower operation by reducing the cooling water flow rate to the HEN. A consequence of this is increased return temperature which is beneficial to cooling tower operation. More recently, Panjeshahi et al.5 presented a combined approach of designing optimal cooling water systems using mathematical modeling where both the HEN and cooling tower operational characteristics were considered. This work shows the importance of incorporating entire systems into optimization and design, as opposed to the optimization of isolated areas. The work on steam network synthesis by Coetzee and Majozi1 encompasses a graphical targeting technique on a T/H diagram. The minimum steam flow rate is found by creating a pinch point between the system and the utility curves. Using this minimum steam flow rate, an appropriate HEN is then designed using a linear programming (LP) model. The whole process can also be done simultaneously using a mixed integer linear programming (MILP) model. By considering multiple steam pressure levels, the graphical targeting technique becomes obsolete as it cannot cater for the extra dimension of various steam pressure levels. Thus, the MILP model derived by Coetzee and Majozi1 is altered to accommodate multiple steam pressure levels and is used to target a minimum flow rate as well as design the network. * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +27 86 633 5729. † University of Pretoria. ‡ Modelling and Digital Science, CSIR.

However, the effects of minimizing the steam flow rate on the entire steam system have not been considered. The efficient operation on the steam boiler is dependent on the condensate return flow rate and temperature. Reducing the steam flow rate reduces both of these operation parameters and as such affects the steam boiler. The advantages of reducing the steam flow rate in steam systems include decreased water consumption in retrofit operations and a smaller boiler for the grassroot design of plants. This paper consists of an explanation on how the boiler efficiency is calculated, followed by the methodology and mathematical model. A case study demonstrating the effectiveness of the formulation is then presented, followed by the conclusions. 2. Problem Statement The steam flow rate to a HEN can be reduced using pinch analysis. This has been done successfully by Coetzee and Majozi,1 albeit without consideration of boiler efficiency. The steam system also includes steam turbines with fixed steam flow rates. The saturated steam from the turbines can be used to provide energy to the background process; however, this energy exists at various pressure levels that must be considered in the problem. The problem addressed in this investigation can be formally stated as follows. Given. a steam boiler with known efficiency a set of heat exchangers linked to the boiler with limiting temperatures and fixed duties turbines operating at multiple pressure levels with fixed power ratings which deliver saturated condensate to the set of background processes determine the minimum steam flow rate and corresponding HEN that can be achieved by integrating the steam turbine exhaust into the steam system while maintaining the boiler efficiency. 3. System Description Figure 1 shows the steam system in the context of this investigation. The steam boiler produces high pressure steam, shown as stream 1. This steam then proceeds to a let down valve, shown as stream 2, or is taken to a high pressure turbine,

10.1021/ie1008579  2010 American Chemical Society Published on Web 08/20/2010

Ind. Eng. Chem. Res., Vol. 49, No. 19, 2010

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Figure 1. Typical steam system layout.

Figure 2. Steam system layout embracing multiple steam levels.

shown as stream 3. Stream 1 is superheated high pressure steam, whereas the heating utility for processes is typically saturated steam. This is because heat exchanger networks are often arranged in parallel where each heat exchanger receives saturated steam and the latent energy is used to heat the appropriate process stream. Thus the let down valve provides a means to reduce the pressure of stream 2 such that it becomes saturated. Also, the latent energy of steam tends to decrease as temperature and pressure increase. As such the temperature and pressure of the saturated steam used as a heating utility should, if at all possible, be as close to the required heating utility temperature so as to take as much advantage of the latent heat as possible. Thus the steam leaving the let down valve is at a lower pressure than stream 2. This is referred to as the process pressure from here onward. When the steam leaves the processes, it is either saturated or subcooled condensate. This condensate, represented by stream 4, is then collected in a condensate tank before passing through a condenser en route to a pump to prevent cavitation. Before entering the boiler, the condensate passes through a preheater.

Some make up water is also added to the boiler after some polishing, but this has been omitted from the diagram for the sake of simplicity. By incorporating the various streams used in the turbine section of the steam system, the three processes seen in Figure 1 can be combined into one. The steam flow to the turbines is fixed since the turbines have a fixed power rating, thus the only area for steam reduction is the high pressure steam that proceeds through the let down valve. This can be seen in Figure 2. Figure 2 also shows the various steam levels for the steam system. PHP represents high Pressure, PMP, medium pressure, PLP, low pressure, and PPP, the specified process pressure. 4. Boiler Efficiency Constraint 1 relates the variation of efficiency ηb to the effects of changing steam load, capacity and operating conditions as would be encountered in a realistic situation.6 In its simplest form, it represents the ratio of the energy content of the steam to the energy content of the fuel.

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ηb )

(cP∆Tsat

q(F/FU) + q)[(1 + b)(F/FU) + a]

(1)

In constraint 1, q is the heat load of the steam (i.e., the latent and superheated sensible heat), F is the steam load raised by the boiler which consists of the condensate return of all the steam using processes in the system, be they HENs or turbines. FU is the maximum capacity of the boiler. The parameters a and b are taken from a study by British Gas in work done by Pattison and Sharma.7 These parameters relate the heat loss percentage to the load percentage and as such are used to define an expression for the heat losses of the boiler.6 Constraint 1 is not a strict definition of efficiency as it does not include the heat added to subcooled liquid entering the boiler, which would normally occur in the numerator. The constraint is in this form since the strict definition of efficiency using the heat losses term from Pattison and Sharma7 would only relate efficiency to the load percentage of the boiler, as the cP∆Tsat term would cancel in constraint 1. This adaptation is used as a comparative tool for the various HEN arrangements. Reducing the steam flow rate to a set of heat exchangers with fixed duties has the effect of lowering the return temperature to the boiler. This is because the method for reducing steam flow rate utilizes the sensible heat of the condensed steam. This affects the ∆Tsat variable in constraint 1. The mass flow rate to the boiler is also reduced, affecting the variable F. These effects must be considered, since they could adversely change the operation of the steam boiler. It may also be possible to improve the efficiency as defined in constraint 1 by changing these variables in a favorable way.

Constraints 2-7 are the mass balance constraints for the heat exchangers and the HEN. Figure 3 shows the superstructure used to derive the mass balance constraints. In Figure 3, i and j refer to different heat exchangers within a given set of heat exchangers I. The various steam levels are included by adding the index pp, mp, or lp to the various flow rates that deal with steam or saturated condensate. The steam levels pp, mp, or lp all belong to the set of steam levels L. Constraint 2 is the mass balance of the inlet to the HEN.

∑

FS )

The inlet to any heat exchanger Fin,i is made up of saturated steam SSi,l and recycled hot liquid FRRj,i from any other heat exchanger j, as seen in constraint 3. The condensate can be either saturated or subcooled as long as it is meets the limiting temperature of the heating utility stream for heat exchanger i. Fin,i )

∑ SS

i,l

+

l∈L

∑ FRR

∀i ∈ I

j,i

(3)

j∈I

Similarly, the outlet of each heat exchanger, Fout,i, consists of condensate recycled to any other heat exchanger j, as well as any return to the boiler FRi. Thus constraint 4 constitutes the outlet heat exchanger mass balance. Fout,i ) FRi +

∑ FRR

i,j

∀i ∈ I

(4)

j∈I

Constraint 5 is the return flow mass balance, where FB is the total return to the boiler, made up of the return flows of all the heat exchangers.

5. Methodology The methodology consists of two parts. First, the minimum steam flow rate is determined and an associated HEN is found. This is done with no consideration for boiler efficiency so as to have a flow rate target to work toward. The second part of the methodology deals with constraints concerning the boiler efficiency. An attempt is made to reconstruct the HEN so as to maintain the initial boiler efficiency. If the initial boiler efficiency cannot be maintained using the minimum steam flow rate, a compromise in either the minimum flow rate or the boiler efficiency will have to be made. 5.1. Steam Reduction and Initial Network Design. The first part of the objective is to reduce the steam flow rate to the HEN. Coetzee and Majozi1 showed that a graphical targeting technique can be used to find a minimum steam flow rate to the HEN. A simple LP model can then be used to create the network associated with the reduced flow rate. A mathematical model can also be used to simultaneously target for a minimum flow rate as well as design the network. However, this particular model exhibits a mixed integer linear programming (MILP) structure instead of an LP structure. This particular model forms the basis of the work presented in this paper. The inclusion of additional steam levels as well as consideration of the boiler efficiency changes the formulation by Coetzee and Majozi.1 The properties of steam at various levels (i.e., high pressure (HP), process pressure (PP), medium pressure (MP), and low pressure (LP)) make it necessary to specify how much steam from a particular level is being used to provide heat to a heat exchanger. The HP steam is not used directly in the process as it is let down so that the latent energy can be used as a source of heat. This let down results in the PP steam.

(2)

SSi,l

i∈I,l∈L

FB )

∑ FR

i

(5)

i∈I

Finally, constraints 6 and 7 are the overall mass balances around each heat exchanger i and the total HEN respectively Fin,i ) Fout,i

∀i ∈ I

FS ) FB

(6) (7)

The constraints above only cater for mass balances and not energy balances. A heat exchanger can be supplied with heat from two sources, namely saturated steam from the boiler and turbine exhaust or recycled/reused condensate. Binary variables are introduced to specify that a particular heat exchanger i can either be supplied with energy in the form of latent heat or sensible heat. The binary variable xi is associated with recycled/ reused condensate. Since a process stream can be heated by steam at more than one pressure level, the binary variable associated with saturated steam, yi,l, has an index relating to the steam pressure level as well as if steam at more than one pressure level is used to heat a stream more than one heat exchanger will be required since the multiple pressure levels are incompatible in the same heat exchanger. Constraint 8 is used to restrict a process stream to at most a single steam pressure level. Constraint 9 is used to allow a stream to be heated by at most m extra steam pressure levels. Either one of these constraints can be implemented in the model however constraint 9 is more flexible and may lead to a greater reduction in the steam flow rate.

∑y

i,l

l∈L

e1

∀i ∈ I

(8)


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Figure 3. HEN superstructure.

∑y

i,l

e1+m

∀i ∈ I

(9)

l∈L

Apart from the limitations for the steam pressure levels, a heat exchanger can receive either steam or condensate, since a mixture is to be avoided. The binary variables xi and yi,l are used to control whether a heat exchanger receives saturated steam or saturated condensate to heat the process stream. It is possible for a process stream to be heated by both however this will require an additional heat exchanger and cause what is known as a heat exchanger split. To implement these restrictions, upper limits for both saturated steam and recycled/ reused condensate needed to supply heat to each heat exchanger i must be known. Constraints 10 and 11 show these limits. SSUi,l ) FRRUi )

Qi λl

∀i ∈ I ∀l ∈ L Qi

cP(TLin,i

- TLout,i)

∀i ∈ I

(10)

(11)

In constraints 10 and 11, Qi is the heat duty for each heat exchanger i. λl is the latent heat of evaporation of the saturated steam at steam level l. In constraint 11, cP is the specific heat L L and Touti are the limiting capacity of the condensate while Tin,i temperatures for steam, as calculated from the limiting process stream data and the global ∆Tmin for the HEN. These constraints are used along with the binary variables to restrict the flow of steam or condensate to heat exchanger i. Constraint 12 limits the steam while constraint 13 limits the condensate. SSi,l e SSUi,lyi,l

∑ FRR

j,i

∀i ∈ I ∀l ∈ L

e FRRUi xi

∀i ∈ I

(12) (13)

and there can be at most n + m streams heated by steam and condensate. yi,l + xi ) 1 +

+

∑x

i,l

∑y

i,l

i∈I

(14)

g |i|

(15)

e |i| + n + m

(16)

∑x

i

i∈I

i∈I

i

i∈I

In constraint 16, n is the number of allowable stream splits, meaning that n streams can be heated by latent heat from saturated steam and sensible heat from saturated and subcooled condensate. In constraints 9 and 16, m is the number of extra steam pressure levels that can provide latent heat to a stream. Constraint 17 shows the energy gained from saturated steam, and constraint 18 shows that from condensate. Each steam level has its own latent heat of evaporation, thus the steam flow rates differ for a particular stream at different pressure levels. QSi,l ) SSi,lλl QLi )

∑

(cpSLj,i,lTsat,l) +

j∈I,l∈L

∀i ∈ I ∀l ∈ L

∑ (c L

p j,iTout,j)

(17)

-

j∈I

(cpFout,iTout,i)

∀i ∈ I

(18)

In constraint 18, SLj,i,l and Lj,i are the saturated condensate and subcooled condensate flow rates, respectively. Since the saturated condensate flow rates can be from various pressure levels which have different saturation temperatures, these must all be included by summing over each steam level as well as each source of saturated condensate. Constraint 19 states that the heat supplied to any heat exchanger i is made up of the sum of the latent and sensible heat.

j∈i

In the case where only steam or only condensate is to be used, Constraint 14 is implemented. This will have to be used with constraint 8. Constraints 15 and 16 are then used if some process streams can be heated with both steam and condensate. If no process streams may be heated by multiple steam levels then constraints 15 and 16 should be used with constraint 8. Then there can be n streams heated by steam and condensate. If multiple steam levels are allowed for the same process stream then constraints 15 and 16 should be used with constraint 9

∑y

∀i ∈ I

Qi )

∑Q

S i,l

+ QLi

∀i ∈ I

(19)

l∈L

The variable FRRj,i in the mass balance constraints is made up of the sum of the various saturated condensate flows at different steam levels, SLj,i,l and the recycled/reused condensate flow Lj,i found in the energy constraints. This relationship is shown in constraint 20. Constraints 21 and 22 are restrictions for the amount of saturated condensate and subcooled condensate that can be transferred from heat exchanger j to i.

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∑ SL

FRRj,i )

j,i,l

+ Lj,i

∀i, j ∈ I

(20)

l∈L

∑ SL

i,j,l

∀i ∈ I ∀l ∈ L

e SSi,l

(21)

j∈I

∑L

i,j

e

j∈I

∑ FRR

∀i ∈ I

j,i

(22)

j∈I

Another constraint to consider is the restriction that no subcooled liquid can be recycled locally. Constraint 23 deals with this. Li,j ) 0

∀i, j ∈ I i ) j

(23)

The steam flow rates to the turbines are fixed, thus the steam at the pressure level of the turbine condensate has a fixed flow rate. This flow rate can be seen as an upper bound for the saturated flow variables at that level. Constraints 24 and 25 show this upper bound for the medium and low pressure steam levels.

∑ SS

i,mp

mp ∈ L

e SSUi,mp

(24)

i∈I

∑ SS

i,lp

e

lp ∈ L

SSUi,lp

subcooled condensate. The saturated condensate can come from various steam levels and thus these are included. Thus, two terms, FRSi,l and FRLi are used to represent FRi in constraint 29, respectively. Using these variables, the total return temperature to the boiler, Tboil, can then be calculated with constraint 30. Then the efficiency can be calculated using these variables in constraint 1, shown in constraint 31. In constraint 29, Tproc is the process outlet temperature. The unused MP and LP steam can therefore be used in other parts of the plant. In constraint 30, Qpreheat is the heat added by the preheater. Finally, the boiler efficiency is calculated by

Tproc )

∑

FRSi,lTsat,l +

i∈I,l∈L

(cP(Tsat

L i out,i

i∈I

FS Tboil ) Tproc +

ηb )

∑ FRL T

Qpreheat FScP

(29)

(30)

q(FS/FU) - Tboil) + q)[(1 + b)(FS/FU) + a]

(31)

(25)

i∈I

U is the set steam flow rate through the In constraint 24, SSi,mp high pressure turbine, leaving at medium pressure. In constraint U is the steam flow rate through the medium pressure 25, SSi,lp turbine, leaving at low pressure. When using steam at various levels, it is important to ensure that steam with a low saturation temperature is not used as the hot utility for a heat exchanger that requires a high inlet temperature. Constraints 26 and 27 ensure the steam temperature is above the limiting utility inlet temperature.

SSi,l ) 0 SLj,i,l ) 0

∀i ∈ I

Tsat,l e TLin,i

∀i, j ∈ I

Tsat,l e TLin,i

(26) (27)

Finally the objective function for this set of constraints is to minimize the overall high pressure steam flow rate. This is because the steam flow rates from the turbines, which occur at every other steam level, are constant. Thus, the objective function is written as min Z ) SSi,pp

(28)

Constraints 2-28 constitute the basic steam reduction and network design model for multiple steam pressure levels. Constraint 18 is the only nonlinear constraint in the formulation. This can be linearized by utilizing the lower bound outlet temperatures for the individual heat exchangers, which are constant. This has been proven to provide an optimal solution by Savelski and Bagajewicz8 in their work on wastewater minimization which is also applicable to energy optimization. Thus, an MILP model can be formulated and then solved to yield the preliminary steam flow rate reduction and HEN design. 5.2. Boiler Efficiency Considerations and Altered HEN. To calculate the boiler efficiency as defined in constraint 1, several variables are required. The outlet temperature of the process that has undergone heat integration must be known and can be calculated by constraint 29. The return flow rate to the boiler is represented by the term FRi. This can take two forms however: the first being saturated condensate and the other

5.2.1. Maintaining Boiler Efficiency Using Sensible Heat. Since the steam flow rate reduction causes a decrease in the return boiler temperature, a means of reheating the boiler feed to its original return temperature must be found. The preheater may not be able to heat the boiler return water to a temperature high enough to retain the original boiler efficiency as calculated by constraint 1. Thus it is suggested that the boiler return condensate be heated in heat exchangers which utilize the sensible heat of the superheated steam from the boiler and from the high pressure turbine exhaust. In most instances, this energy is lost during the pressure let down. Thus using this method, the energy can be reclaimed and used to maintain the boiler efficiency. It will not be possible to utilize all of the sensible heat for this purpose. The reason for this is that any condensation of the steam will compromise the energy supplied to the HEN. Since it is expected that the HEN outlet temperature is well below the saturation temperature, it may also be possible to eliminate the condenser from the steam system as there is little risk of cavitation from pumping water. The stack preheater can then be used to preheat any make up water for the process, or to further heat the boiler feedwater. Figure 4 shows a simple diagram of this proposed alteration to the steam system. The new boiler return water temperature can be calculated using constraint 32. Tboil ) Tproc + (PP)(hsup,hp - hsat,hp)θ + MP(hsup,mp - hsat,mp)φ FScP

(32)

In constraint 32, hsup,hp and hsup,mp are the enthalpies of the superheated HP and MP steam leaving the boiler and high pressure turbine, respectively, and hsat,hp and hsat,mp are the enthalpies of the saturated HP and MP steam at the boiler and high pressure turbine outlet conditions, respectively. θ and Φ are the fractions of the sensible energy that can be used safely without the risk of condensation for the high and medium pressure steam, respectively. These fractions are arbitrarily chosen with a large enough safety factor that is deemed appropriate so as to not compromise the saturated steam by


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Figure 4. Proposed steam system to maintain boiler efficiency.

condensation. MP and LP are equivalent to SSi,mp and SSi,lp as used before but are easier to recognize in the context of Figure 4. Constraints 29-32 can be used to create a second part to the mathematical model formulated before. Using the minimum steam flow rate FS from the original formulation, an attempt to maintain the boiler efficiency can be made. Two cases can be considered, each using the same basic constraints but focusing on two objectives. Case 1: Maintain Boiler Efficiency with a Slight Compromise in Minimum Flow Rate. First, the primary objective is to maintain boiler efficiency. This can mean that the minimum steam flow rate may not be reached, if there is not enough sensible heat available to preheat the boiler feed stream. The minimum steam flow rate is first determined using the first part of the methodology. The steam flow rate is then relaxed by adding a slack variable and the constraints relating to boiler efficiency are included. The boiler efficiency is then fixed while the slack variable is minimized, thus the total steam flow rate, i.e. the sum of FS and the slack variable, can be found with constant boiler efficiency. A feasible HEN must also be found with this formulation. Constraints 33-35 show the new forms of constraints 29-31, with constraint 34 replacing constraint 30 as before. Constraint 36 is the objective function.

Tproc )

∑

FRSi,mTsat,m +

i∈I,m∈M

∑ FRS T

FS + slack

i out,i

i∈I +

(33)

Tboil ) Tproc + (FS + slack+)(hsup,hp - hsat,hp)θ + MP(hsup,mp - hsat,mp)φ (FS + slack+)cP

(34) ηb )

(cP(Tsat

q((FS + slack+)/FU) - Tboil) + q)[(1 + b)((FS + slack+)/FU) + a] (35) min Z ) slack+

(36)

This formulation includes several nonlinear terms. One means to deal with this situation is to implement Quesada and

Grossmann9 type relaxation linearizations. The solution to the relaxed problem is then found and used as a starting point for the exact, nonlinear model. In the case where the relaxed solution and the exact solution coincide, then a globally optimum minimum flow rate is found. Case 2: Maintain Minimum Flow Rate with Slight Compromise in Boiler Efficiency. Second, the objective could be to achieve the minimum steam flow rate with only a slight compromise in boiler efficiency. The efficiency term is relaxed by adding a slack variable to the efficiency constraint. The minimum steam flow rate is set as a parameter while the relaxed efficiency is increased to its original value by minimizing the slack variable. If the slack variable is zero, then the boiler efficiency can be maintained with the minimum steam flow rate. If it is not then the boiler efficiency will be compromised by minimizing the steam flow rate. Constraint 37 is the final form of the efficiency constraint, and constraint 38 is the objective function for case 2. ηb - slack- )

(cP(Tsat

q(FS/FU) - Tboil) + q)[(1 + b)(FS/FU) + a] (37)

min Z ) slack-

(38)

A further extension to the formulation can be made in the case where the boiler efficiency is maintained. If this occurs, then it is possible for the boiler efficiency to be increased by utilizing as much of the sensible heat from the superheated steam as possible. If this is the case, constraints 39 and 40 can be used to maximize the boiler efficiency. ηb + slack+ )

(cP(Tsat

q(FS/FU) - Tboil) + q)[(1 + b)(FS/FU) + a] (39)

max Z ) slack+

(40)

Either of these two cases can be explored, depending on the circumstances of the steam system being examined. For example, arid regions may have a shortage of water and thus minimizing the steam flow rate could be the primary objective,

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Figure 5. Additional HEN heat exchanger in multiple pressure level system.

or where water is in abundance, maintaining or maximizing the boiler efficiency can be used as the objective. 5.2.2. Maintaining Boiler Efficiency Using a Dedicated Preheater. The boiler efficiency can also be maintained by using a dedicated preheater in the HEN to ensure the return stream to the boiler is at a sufficient temperature. Figure 5 shows the proposed steam system. The condensate tank and condenser have been omitted. The set of heat exchangers i has to be extended by 1 to accommodate the preheater. A i* will be used to denote the extra heat exchanger in the formulation. The inlet to the i* heat exchanger, stream 5, is at temperature Tproc. The outlet, stream 6, is at temperature Tboil. The utility limiting temperatures for the i* heat exchanger are thus Tproc and Tboil in addition to the global ∆Tmin. Constraints 41 and 42 represent the limiting temperatures for the i* heat exchanger, and constraints 43-45 are the steam duty, condensate duty, and total duty, respectively. TLin,i* ) Tpump + ∆Tmin

∀i* ∈ I

(41)

TLout,i* ) Tboil + ∆Tmin

∀i* ∈ I

(42)

∀i* ∈ I ∀l ∈ L

(43)

QSi*,l ) SSi*,lλl QLi* )

∑

j∈I,l∈L

(cPSLj,i*,lTsat,l) +

∑ (c L

P j,i*Tout,j)

j∈I

(cPFout,i*(Tboil + ∆Tmin)) Qi* )

∑Q

S i*,l

+ QLi*

-

∀i* ∈ I

∀i* ∈ I

(44) (45)

l∈L

The boiler efficiency is to be maintained and as such is fixed for this formulation, thus case 1 from section 5.2.1 will be implemented for the efficiency constraints. The calculation of Tboil in constraint 34 is changed since sensible heat is no longer used to preheat the boiler feed. Constraint 46 now shows the calculation of Tboil. Tboil ) Tproc +

Qi* (FS + slack+)cP

(46)

Constraints 33-36 with constraint 46 replacing constraint 34 are then used to complete the formulation. As before, the linearization technique of Quesada and Grossmann9 is used to deal with the continuous nonlinear elements while the Glover10 transformation is used for the products of binary and continuous variables. This improvement can be easily combined with that of the sensible heat preheaters as many of the elements are the same. Figure 6 shows an example of this steam system, with both types of preheaters. 6. Case Study The case study presented by Coetzee and Majozi1 is used here to show how boiler efficiency is affected by a reduction in steam flow rate and how the formulations above can be used to maintain the original efficiency, or how the reduced flow rate affects the efficiency. Table 1 contains the hot utility stream data for the case study. The pressure level in brackets shows which steam pressure level provides heat to the respective heat exchanger. This is based on the limiting temperature data for the stream as well as the steam saturated temperature. The hot utility streams shown in Table 1 include the streams that were originally heated using saturated condensate from the turbine exhaust, i.e. the situation shown in Figure 2 is represented. Table 2 contains information about the steam levels, and Table 3 contains information about the turbine component of the steam system. All of the mathematical programming was completed in GAMS. The solvers utilized were Cplex for the linear models and Dicopt (which utilizes Cplex as the MIP solver and Conopt as the NLP solver) for the nonlinear models. Using constraint 31, the original boiler efficiency was calculated as 63.4%. This is for a parallel HEN and the system saturated steam information as given in Table 2. 6.1. Steam Reduction and Maintained Boiler Efficiency. The number of heat exchanger splits was varied until a minimum flow rate was found. This minimum corresponded to two heat exchanger splits. Steam at different pressure levels cannot be combined and used as a source of heat in a single heat exchanger. In the event of a single stream requiring heat from steam at different pressure levels, more than one heat exchanger will be required. The number of these possible heat exchangers


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Figure 6. Combination of different preheater concepts. Table 1. Hot Stream Data stream number

TLT (°C)

TSL (°C)

duty (kW)

1 2 3 4 5 6 7 total

35 35 219 89 217 54 54

55 55 225 195 217 80 80

135 320 3620 12980 1980 635 330 20000

Table 2. Steam System Data11 description

T (°C)

P (kPa)

Tsat (°C)

high pressure (HP) from boiler medium pressure (MP) system pressure for case study intermediate pressure (IP) low pressure (LP) deaerator outlet feed pump outlet to the boiler

399 327 225 209 221 113 116

4238 1 480 2550 377 164 164 6310

254 197 225 141 113 113 277

Table 3. Relevant Data for Turbine Portion of Steam System condensate leaving the HP turbine, not proceeding to the LP turbine (MP in Figure 4) condensate leaving the MP turbine (LP in Figure 4)

2.0 kg/s 0.23 kg/s

was also varied and it was found that one stream needed two heat exchangers each for the latent energy heating. After steam reduction, the steam flow rate was reduced from 10.73 to 7.91 kg/s, a 26.3% reduction. Figure 7 shows the network layout using the minimum steam flow rate. This however has the effect of reducing the process outlet temperature to 63.0 °C. This reduction in return temperature and flow rate corresponds to a new boiler efficiency of 58.19%, an 8.2% reduction. From this figure, it can be seen that the process stream associated with heat exchanger 4 requires heating from both saturated steam at two pressure levels as well as saturated condensate. Thus the HEN requires two extra heat exchangers. A heat exchanger split with respect to two saturated steam pressure levels is indicated by the main heat exchanger number being accompanied by a ^ symbol. All of the heat exchangers that are associated with any form of split, be it from different pressure levels or from a difference in steam and condensate are indicated by the main heat exchanger number accompanied by a subscript. Heat exchanger 4 in Figure 7 is made up of 41^,

42^, and 43 representing three separate physical heat exchangers. 41^ and 42^ represent the heat exchangers for the two separate steam pressure levels, and 43 represents the additional heat exchanger to accommodate steam condensate. 6.1.1. Maintained Efficiency Using Sensible Heat. The loss of boiler efficiency can be remedied by reheating the boiler feed. The streams that receive more than one steam stream represent more than one heat exchanger. Since 1 such stream exists, the total number of heat exchangers is 10. Using the first premise of maintained boiler efficiency, it was found that the boiler efficiency could be sustained without compromising the steam flow rate. As in the formulation, part of the superheated energy from the HP and MP steam is used to preheat the boiler feed. This was successfully accomplished using 60.1% of the available sensible heat from the HP steam and 80% from the MP steam. To avoid the risk of condensation the maximum allowable fraction of sensible heat (θ and Φ) that can be used was set to 80%. With the reduction in steam flow rate, the return temperature had to be increased to 119.0 °C. Given that the steam flow rate was not changed, the HEN as seen in Figure 7 was also not changed. Since the boiler efficiency could be maintained for the minimum steam flow rate, it follows that the second premise of minimum steam flow rate with minimally decreased boiler efficiency would also yield the same answer. As expected, the minimum steam flow rate required 60.1% of the available sensible heat from the HP steam and 80% from the MP steam once again. Both formulations gave the same result since there was enough sensible energy to heat the boiler feed to the point where the original efficiency could be maintained. If the amount of sensible energy was reduced, or the maximum allowable sensible energy was reduced, the models may show how they compromise either efficiency or minimum flow rate to maintain the other, depending on the objective function. For this purpose, the amount of sensible heat available was reduced to 30% of the original available amount for both the HP and MP sensible heat sources. To maintain boiler efficiency using this reduced amount of sensible heat, it was found that the minimum steam flow rate was indeed compromised. The new steam flow rate for the

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Figure 7. Minimum flow rate network layout.

Figure 8. New network layout to maintain boiler efficiency for reduced flow rate.

relaxed solution was found to be 8.22 kg/s. The exact solution then resulted in a flow rate of 8.33 kg/s, which still yielded a 22.4% reduction from the original parallel HEN. This new flow rate did require a new HEN to satisfy the duties of the various process streams. This network was designed by the model and can be seen in Figure 8 below. Since 2 streams require heating

from steam at different pressure levels, the total number of heat exchangers is 10. In the figure, it can be seen that the process streams associated with heat exchangers 4 and 7 required an extra heat exchanger to accommodate the multiple steam levels needed to heat these streams. As seen in Figure 8, some saturated liquid is returned


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Figure 9. HEN arrangement showing no use of additional heat exchanger. Table 4. Comparison of Model Numerical Results model steam flow rate reduction additional preheater using sensible energy additional preheater 30% sensible heat (case 1) additional preheater 30% sensible heat (case 2) additional preheater incorporated into HEN

flow rate reduction (%)

return temperature reduction (°C)

boiler efficiency reduction (%)

26.3 26.3

8.3 0

22.4

53.1 -2.9 (return temperature had to be increased) -2.3

26.3

19.3

3.2

23.7

-2.2

0

to the boiler from heat exchangers 4 and 7. This is as a result of it being much warmer than the subcooled condensate. This saturated liquid, even in small amounts, raises the process outlet temperature to 92.1 °C. For the change in return flow rate, the return temperature to the boiler is raised to 118.4 °C. This demonstrates how the boiler efficiency can indeed be maintained, even with low amounts of sensible heat, but still showing a fairly large saving in steam flow rate. Using the second objective to utilize the minimum steam flow rate, the available sensible heat was once again reduced to 30% of the original amount. It was found that the minimum flow rate could be maintained but a small decrease in boiler efficiency was noted. The new boiler efficiency was calculated as 61.40%, a decrease of 3.2% from the original boiler efficiency. The process outlet temperature increased to 70.6 °C, while the boiler return temperature dropped to 96.8 °C. The network required for this flow rate is the same as that of Figure 7, since there is no change in flow rate. 6.1.2. Maintained Efficiency Using Dedicated Preheater. The dedicated preheater shown in section 5.2.2 was implemented into the steam system. The minimum steam flow rate was found to be 6.48 kg/s. This is 14.1% higher than the minimum steam flow rate, but still a fairly considerable 23.7% reduction from

0

comments boiler efficiency not considered 80% of MP, 60.1% HP sensible heat required to preheat return steam reduction compromised to maintain boiler efficiency boiler efficiency compromised to achieve minimum steam flow rate sensible heat not required while saving still made

the original flow rate. Figure 9 shows the HEN arrangement for this solution. The additional heat exchanger is distinguished with a * symbol. The process outlet temperature for this network configuration was calculated as 106.0 °C. This is fairly high and may lead to a risk of pump cavitation. The duty of the dedicated preheater was calculated to be 409.8 kW. A lower limit duty of 800 kW was set for the dedicated preheater in an attempt to force the system to a lower process outlet temperature. The same minimum flow rate of 6.48 kg/s was calculated, as well as a preheater duty of 1033.0 kW. This higher duty resulted in a process outlet temperature of 89.4 °C which greatly reduced the risk of cavitation. The drawback of this is the additional heat exchanger area that is required in the HEN which will increase the capital cost. Table 4 summarizes the numerical results from the various models. By utilizing the sensible heat that would normally be lost during steam letdown, the boiler efficiency of the system can be maintained while still reducing the steam flow rate. In the event that sensible heat cannot be used, considering a holistic optimization framework allows steam flow rate savings to be made while still maintain the boiler efficiency by including a dedicated boiler return preheater.

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7. Conclusions The following conclusions can be made about reducing steam flow rate while maintaining boiler efficiency: • Reducing steam flow rate affects the boiler efficiency by lowering boiler return temperature and flow rate. • Preheating the return flow to a slightly higher temperature will maintain boiler efficiency for a reduced steam flow rate. • By incorporating the background processes of steam turbines into the HEN and minimizing the flow rate, some of the excess exhaust from the turbines can be used in the HEN. • The system let down valve can be replaced by two heat exchangers that will utilize the sensible heat of the high pressure steam and the exhaust from the high pressure turbine to preheat the boiler feedwater. • In the event of there not being enough sensible heat to maintain the boiler efficiency with the minimum steam flow rate, a compromise in either the boiler efficiency or the minimum steam flow rate must be made. Nomenclature Sets I ) {i or j|i or j ) 1, 2, ..., I} is the set of heat exchangers L ) {l|l ) hp, mp, or lp} is the set of various saturated steam pressure levels Parameters a ) regression parameter b ) regression parameter cp ) heat capacity, kJ/kg · K FU ) maximum boiler capacity, kg hsup,hp ) enthalpy of superheated HP steam, kJ/kg hsup,mp ) enthalpy of superheated MP steam, kJ/kg hsat,hp ) enthalpy of saturated HP steam, kJ/kg hsat,mp ) enthalpy of saturated MP steam, kJ/kg q ) latent and superheated sensible heat of HP steam, kJ/kg L Tin,i ) limiting utility inlet temperature for heat exchanger i, (°C) L Tout,i ) limiting utility outlet temperature for heat exchanger i, (°C) λl ) latent heat of steam, kJ/kg θ ) fraction of HP sensible heat used for preheating Φ ) fraction of MP sensible heat used for preheating Binary Variables

{ {

1 0 1 yi,1 ) 0 Continuous xi )

if heat exchanger i receives heat from condensate otherwise if heat exchanger i receives heat from steam level 1 otherwise Variables

F ) mass flow to the boiler, kg

FB ) total return flow to the boiler, kg/s Fin,i ) total flow rate entering heat exchanger i, kg/s Fout,i ) total flow rate leaving heat exchanger i, kg/s FRi ) condensate returning to the boiler from heat exchanger i, kg/s FRRj,i ) reused/recycled condensate from heat exchanger j to heat exchanger i, kg/s FS ) total saturated steam flow rate from the boiler, kg/s Lj,i ) subcooled condensate reuse from heat exchanger j to heat exchanger i, kg/s ηb ) boiler efficiency Qi ) duty of heat exchanger i, kW slack ) slack variable used in objective functions SLj,I,l ) saturated condensate reuse/recycle from heat exchanger j to heat exchanger i at pressure level l, kg/s SSi,l ) saturated steam flow rate to heat exchanger i at pressure level l, kg/s Tboil ) temperature of boiler return flow, °C Tproc ) outlet temperature from HEN, °C ∆Tsat, l ) temperature difference between boiler return and saturated HP steam, °C

Literature Cited (1) Coetzee, W. A.; Majozi, T. Steam System Network Design Using Process Integration. Ind. Eng. Chem. Res. 2008, 47, 4405–4413. (2) Linnhoff, B.; Hindmarsh, E. The Pinch method for heat exchanger network. Chem. Eng. Sci. 1983, 38, 745–763. (3) Kim, J. K.; Smith, R. Cooling water system design. Chem. Eng. Sci. 2001, 56, 3641–3658. (4) Majozi, T.; Moodley, A. Simultaneous targeting and design for cooling water systems with multiple cooling water supplies: An Automated Approach. Comput. Chem. Eng. 2008, 32, 540–551. (5) Panjeshahi, M. H.; Ataei, A.; Gharaie, M.; Parand, R. Optimal design of cooling water systems for energy and water conservation. Chem. Eng. Res. Des. 2008, 87, 200–209. (6) Shang, Z.; Kokossis, A. A transshipment model for the optimization of steam levels of total site utility system for multiperiod operation. Comput. Chem. Eng. 2004, 28, 1673–1688. (7) Pattison, J. R.; Sharma, V. Selection of boiler plant and oVerall system efficiency; Studies in energy efficiency in buildings; British Gas, 1980. (8) Savelski, M. J.; Bagajewicz, M. J. On the optimality of water utilisation systems in process plants with single contaminants. Chem. Eng. Sci. 2000, 55, 5035–5048. (9) Quesada, I.; Grossmann, I. E. Global optimization of bilinear process networks with multi component flows. Comput. Chem. Eng. 1995, 19, 1219– 1242. (10) Glover, F. Improved linear integer programming formulation of nonlinear problems. Manage. Sci. 1975, 455–460. (11) Harrel, G. Steam System SurVey Guide; ORNL/TM-2001/263, Oak Ridge National Laboratory: Oak Ridge, TN, 1996.

ReceiVed for reView April 11, 2010 ReVised manuscript receiVed July 25, 2010 Accepted August 3, 2010 IE1008579

Synthesis and Optimization of Steam System Networks. 2. Multiple

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