On Synthesis and Optimization of Steam System Networks. 1

Ind. Eng. Chem. Res. 2010, 49, 9143–9153

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On Synthesis and Optimization of Steam System Networks. 1. Sustained Boiler Efficiency Thokozani Majozi*,†,‡ and Tim Price† Department of Chemical Engineering, UniVersity of Pretoria, South Africa, and Modelling and Digital Science, CSIR, Pretoria, South Africa

The traditional steam system comprises a steam boiler and the associated heat exchanger network (HEN). Most research published in literature tends to address both the elements of the steam system as separate entities instead of analyzing, synthesizing, and optimizing the overall system in a holistic manner. True optimality of the steam system can only be achieved if the analysis is conducted within an integrated framework. Process integration has proven to be a powerful tool in similar situations. This paper presents a process integration technique for network synthesis using conceptual and mathematical analysis without compromising boiler efficiency. It was found that the steam flow rate to the HEN could be reduced while maintaining boiler efficiency by utilizing sensible heat from the high pressure steam leaving the boiler. In the event of too little sensible energy being available, a compromise in either minimum steam flow rate or boiler efficiency must be made. A dedicated preheater can also be added to the HEN so as to guarantee the boiler efficiency is maintained; however, this will compromise the minimum steam flow rate. It was found that the flow rate could be reduced by 29.6% while still maintaining the boiler efficiency for an example problem. 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 exchange1 but can also be used in the optimal placement of utilities.1 Cooling water systems comprising a cooling tower and associated HEN has been examined in detail and optimized by Kim and Smith.2 Coetzee and Majozi3 optimized and designed cooling water systems incorporating multiple cooling towers using mathematical programming. Grossmann and Santibanez4 optimized steam systems using a MILP approach. Heat exchanger networks are, however, not considered in the scope. 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. Pillai and Bandyopadhyay6 show how the two favored forms of process integration methodologies, mathematical optimization and the conceptual approach of pinch analysis, can be combined to create a powerful process integration algorithm. They apply their algorithm to the problem of resource allocation, specifically those resources common to chemical plants. The work on steam network synthesis by Coetzee and Majozi3 encompasses a graphical targeting technique on a T/H diagram. In the context of this work the steam system comprises a HEN and a steam boiler. In this method a HEN is represented by a composite curve. A hot utility curve is constructed and then appropriately shifted such that a pinch point is found. This utility curve then corresponds to the minimum steam flow rate. Using this minimum steam flow rate, an appropriate HEN is then designed accordingly. This network design is done using * Corresponding author. E-mail: [email protected]. Fax: +27 86 633 5729. † University of Pretoria. ‡ CSIR.

mathematical programming, where an LP model is used. The whole process, including the targeting, can also be done simultaneously using an MILP model. The MILP model contains binary variables and as such was found to take more CPU processing time. Both methods did, however, result in the same minimum flow rate. The effects of minimizing the steam flow rate on the entire steam system have, however, not been considered. The efficient operation of 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 a review of the graphical targeting technique of Coetzee and Majozi,3 followed by an explanation on how the boiler efficiency is calculated. The methodology and mathematical model are presented next, followed by a case study and conclusions. 2. Problem Statement The steam flow rate to HENs can be reduced using pinch analysis. This has been done successfully by Coetzee and Majozi,3 albeit without any consideration of the boiler efficiency. 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, and turbines and background heat exchangers also linked to the boiler, determine the minimum steam flow rate and corresponding HEN while maintaining boiler efficiency. In the event that the minimum steam flow rate cannot be achieved without compromising the boiler efficiency, two situations arise that are catered for in this paper. First the boiler efficiency is maintained while the minimum steam flow rate is compromised slightly or second

10.1021/ie1007008  2010 American Chemical Society Published on Web 08/20/2010

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

Figure 1. Focus area for Coetzee and Majozi.3

Figure 4. Hot utility supply curve.

Figure 2. Representation of hot and cold streams in a heat exchanger.

Saturated steam provides heat to a HEN in two ways. First, the latent energy is used, shown in constraint 1. Once steam has condensed, it is still at its saturation temperature and thus an opportunity exists to use the sensible heat from the hot condensate, shown in constraint 2. Constraints 1 and 2 can be combined and solved for the mass flow rate, as shown in constraint 3. ˙ latent ) m Q ˙λ

(1)

˙ sensible ) m Q ˙ cp(Tin - Tout)

(2)

m ˙ )

Figure 3. T/H diagram showing the composite curves.

the minimum steam flow rate is achieved at the expense of the boiler efficiency. 3. Graphical Targeting Technique The following section elaborates on the graphical targeting technique developed by Coetzee and Majozi.3 Their work focuses purely on streams that appear on the “hot” side of the process integration process. This is shown in Figure 1 below. Linnhoff and Hindmarsh1 have optimized process to process heat exchange, while Kim and Smith2 explored process integration in cooling systems while heating systems have not been explored in as much detail. Heat exchangers are designed with a ∆Tmin to ensure that there is always a temperature driving force to transfer heat from hot utilities to cold process streams. Figure 2 shows a representation of the hot utility and cold process streams in a counter current heat exchanger. The ∆Tmin should not be violated and so Th,in and Th,out in Figure 2 become the lower bound for all utilities that can supply heat to that heat exchanger. By combining many heat exchanger temperature boundaries, a composite curve for the hot utility stream for an entire HEN can be constructed. Figure 3 shows such a composite curve on a T/H diagram. This composite curve serves as the lower bound for the hot utility system and should not be violated.

˙ Q λ + cp(Tsat - Tout)

(3)

Phase change, which is associated with latent heat, is generally considered to be associated with constant temperature. This is represented by a horizontal line on the T/H diagram. Sensible energy, which is associated with a change in temperature, will thus be represented by a slanted line on the T/H diagram. Figure 4 shows the two kinds of energy along with a hot utility composite curve. By altering the mass flow rate of the steam, the two lines can be shifted. According to constraint 1, the length of the latent energy line is affected by the mass flow rate. Similarly, in constraint 2, changing the mass flow rate alters the slope of the sensible energy line. The latent and sensible energy lines combine to form the hot utility supply curve. The dashed utility curve in Figure 4 shows the effect of increasing the mass flow rate. The hot utility supply curve can never cross the lower bound represented by the utility composite curve. Thus a minimum flow rate will exist at a point where the composite curve and the utility curves touch, which is known as the pinch point. The targeting procedure begins by identifying all the possible pinch points on the composite curve. Once all the points have been found, each is used to find the minimum steam flow rate associated with that point. Then the hot utility supply curve is constructed to test whether the composite curve is violated. If this is the case, another pinch point is chosen and the procedure is repeated3. The pinch points are chosen in order of decreasing temperature difference between the point and the saturation temperature of the steam. This is done since the maximum temperature difference invariably corresponds to minimum mass flow rate, according to constraint 3. Figure 4 shows a pinch point that


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Figure 5. Steam system used as in a typical plant. Table 1. Steam System Data7 description

T (°C)

P (kPa)

Tsat (°C)

high pressure (HP) from boiler medium pressure (MP) process pressure (an example) intermediate pressure (IP) low pressure (LP) deaerator outlet feed pump outlet to the boiler

399 327 225 209 221 113 116

4238 1480 2550 377 164 164 6310

254 197 225 141 113 113 277

does not violate the composite curve and thus represents a feasible minimum steam flow rate. 4. System Description The steam system in the context of this investigation is shown in Figure 5. The steam boiler produces superheated 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, shown as stream 3. Table 1 shows typical steam pressure levels, temperatures, and saturation temperatures that will be used for this chapter.

Figure 6. Tendency of latent heat to decrease as P and T increase.

Stream 1 is a 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. Figure 6 (taken from steam tables) shows this tendency. 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 as stream 4 is at a lower pressure than stream 2. This is referred to as the process pressure from here onward. Stream 3 passes through the turbine where energy is recovered in the form of shaft work Ws, which is typically used to drive other process equipment. The exhaust from the high pressure turbine is generally at medium pressure. This steam then either passes through a medium pressure turbine as stream 5 where

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Figure 7. Sensitivity of constraint 4 to changes in F.

shaft work W′s is recovered or passes through a let down valve and proceeds to a process as a heating utility (stream 6). The purpose of this let down valve is the same as that mentioned above. The exhaust of the medium pressure turbine is at low pressure and proceeds as a further heating utility to background processes as stream 7. A let down valve is indicated in stream 7; however, as seen in Table 1, the low pressure steam is extremely close to saturation and as such this valve may not be necessary. The bypass streams around process 2 and process 3 have been included for completeness. The outlet from the hot utility using processes is typically saturated condensate. These streams then combine to form stream 8 and proceed to a condensate tank. The condensate then passes through a condenser to ensure the condensate is far away from the saturation temperature so as to prevent the risk of cavitation during pumping. The risk of cavitation is mostly associated with centrifugal pumps that are the most prevalent in boiler systems. Makeup water is also typically added in this region although this has been omitted for simplicity. After being pumped, the return stream to the boiler passes through a preheater or economizer to heat the boiler feed, shown as stream 9. 5. Boiler Efficiency Constraint 4 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.8 In its simplest form it represents the ratio of the energy content of the steam to the energy content of the fuel. ηb )

(cP∆Tsat

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

(4)

In constraint 4, q is the heat load of the steam (i.e., the latent and 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.9 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. Constraint 4 is not a strict definition of efficiency as it does not include the heat added to subcooled liquid entering the boiler. The constraint is in this form since the strict definition of efficiency using the heat losses term from Pattison and Sharma9 would only relate efficiency to the load percentage of the boiler. This adaptation is used as a comparative tool for the various HEN arrangements presented in this paper.

Table 2. Typical Values Used To Calculate Boiler Efficiency As Seen in Figure 7 parameter

value

q (sum of the latent and superheated energy) FU (maximum steam load of boiler) cP (specific heat capacity of boiler feedwater) a (regression parameter) b (regression parameter) Tsat (saturated steam temperature at boiler pressure) Tret (initial return temperature for 100% ηb calculation) F (initial return flow rate for 100% ηb calculation)

2110 [(kJ/kg) · K] 20.19 (kg/s) 4.3 (kJ/kg) 0.0126 0.2156 253.20 (°C) 116.10 (°C) 18.17 (kg/s)

The effect of reducing the steam flow rate in the steam system will reduce F in constraint 4. One means of maintaining the efficiency ηb is to increase the temperature of the stream returning to the boiler, which effectively reduces the value of ∆Tsat. Figure 7 shows the percentage increase in return temperature necessary to retain the boiler efficiency for a specified decrease in the return mass flow rate. Typical values of the other parameters were used to create Figure 7, and these are shown in Table 2. In Figure 7 it can be seen that for a fairly substantial decrease in steam flow rate a relatively small increase in the return temperature is required to maintain the efficiency defined by constraint 4. The ∆Tsat value is calculated by subtracting Tret from Tsat. It must be noted that the effect of the processes that utilize steam from the turbine exhaust is also considered when the efficiency is calculated as defined by constraint 4. The steam flow rate to the turbines is not tampered with and as such it remains constant in the model presented in this report. 6. Methodology The first part of the objective is to reduce the steam flow rate to the HEN. 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. 6.1. Steam Reduction and Initial Network Design. Coetzee and Majozi3 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 report, so the constraints from that model are shown below.


FRRUi )

Figure 8. HEN superstructure (Coetzee and Majozi3).

Constraints 5-10 are the mass balance constraints for the heat exchangers and the HEN. Figure 8 shows the superstructure used to derive the mass balance constraints. In Figure 8, i and j refer to different heat exchangers within a given set of heat exchangers I. Constraint 5 is the mass balance at the inlet to the HEN. FS )

∑ SS

(5)

i

i∈I

The inlet to any heat exchanger, Fin,i, is made up of saturated steam SSi and recycled hot liquid FRRj,i from any other heat exchanger j, as seen in constraint 6. The condensate can be either saturated or subcooled as long as it meets the limiting temperature of the heating utility stream for heat exchanger i. Fin,i ) SSi +

∑ FRR

i,j

∀i ∈ I

(6)

j∈I

Similarly, the outlet of each heat exchanger, Fout,i, consists of water recycled to any other heat exchanger j, as well as any return to the boiler FRi. Thus constraint 7 constitutes the outlet heat exchanger mass balance. Fout,i ) FRi +

∑ FRR

i,j

∀i ∈ I

(7)

∑

∑ FRR

j,i

yi + xi ) 1

(8)

∑y

The constraints above only cater for mass balances and not energy balances. A heat exchanger can be supplied with energy from two sources, namely, saturated steam directly from the boiler or recycled condensate from another heat exchanger. Thus binary variables are introduced to specify that a particular heat exchanger i can be supplied with energy in the form of either latent heat or sensible heat. The binary variable yi denotes saturated steam and xi denotes recycled/reused condensate. To implement this, the upper limits for both saturated steam and recycled/reused condensate needed to supply energy to each heat exchanger i must be known. Constraints 11 and 12 show these limits. SSUi )

Qi λ

∀i ∈ I

∑y

(11)

i

i∈I

(9) (10)

+

i∈I

Finally, constraints 9 and 10 are the overall mass balances around each heat exchanger i and the total HEN respectively.

FS ) FB

(13)

∀i ∈ I

e FRRUi xi

i

∀i ∈ I

(12)

(14)

j∈i

i∈I

Fin,i ) Fout,i

∀i ∈ I

SSi e SSUi yi

Constraint 8 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. FRi

∀i ∈ I

- TLout,i)

In constraint 11 and constraint 12, Qi is the heat duty for each heat exchanger and λ is the latent heat of evaporation of the saturated steam. In constraint 12, cP is the specific heat L L and Tout,i are the lower capacity of the condensate while Tin,i limiting temperature values for the utility, as calculated from the limiting process stream data and the global ∆Tmin for the HEN. These upper limits are shown in constraint 13 and constraint 14 for the saturated steam and the condensate, respectively. They will form part of the binary variable restrictions that follow. They ensure that if the appropriate binary variable allows steam or condensate to pass through a heat exchanger, the flow rate is less than the upper limit defined in constraints 11 and 12. If the binary variable is set to zero, then the flow rate through that particular heat exchanger will also be zero. These upper limits along with the applicable binary variable restrictions can be used to control which kind of heat is supplied to each heat exchanger i. There are two forms of binary variable restrictions. The first, constraint 15, states that a heat exchanger can be supplied with energy from either latent or sensible heat only. Constraints 16 and 17 state that at most n heat exchangers can be split, meaning that a combination of latent and sensible energy is used to heat the process stream. This occurs in two separate heat exchangers since saturated steam and condensate cannot enter a single heat exchanger together. This has the effect of increasing the capital cost of the heat exchanger network but can possibly result in further reduction in the steam flow rate. Thus using constraints 13-15 for no heat exchanger splits or constraints 13, 14, 16, and 17 for n heat exchanger splits the type of heat that each heat exchanger receives can be controlled.

j∈I

FB )

Qi cP(TLin,i

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∀i ∈ I

(15)

g |i|

(16)

e |i| + n

(17)

∑x

i

i∈I

+

∑x

i

i∈I

Constraint 18 shows the energy gained from saturated steam and constraint 19 from condensate. QSi ) SSiλ QLi )

∑ (c SL P

j∈i

j,iTsat)

+

∀i ∈ I

∑ (c L

P j,iTout,j)

(18)

-

j∈I

(cPFout,iTout,i)

∀i ∈ I

(19)

In constraint 19, SLj,i and Lj,i are the saturated liquid and subcooled liquid flow rates respectively. Constraint 20 states that the heat supplied to any heat exchanger i is made up of the sum of the latent and sensible heat. Qi ) QSi + QLi

∀i ∈ I

(20)

The variable FRRi,j in the mass balance constraints is made up of the SLi,j and Li,j variables found in the energy constraints.

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This relationship is shown in constraint 21. Constraints 22 and 23 are restrictions for the amount of saturated liquid and subcooled liquid that can be transferred from heat exchanger i to j. FRRi,j ) SLi,j + Li,j

∑

∀i,j ∈ I

∀i ∈ I

i,j

e

j∈I

∑ FRR

(22)

∀i ∈ I

j,i

(23)

j∈I

Another constraint to consider is the restriction that no subcooled liquid can be recycled locally. This may seem obvious from a design sense but it is possible mathematically. Constraint 24 deals with this. Li,j ) 0

∀i, j ∈ I

i)j

(24)

Finally, the objective function for this set of constraints is to minimize the overall steam flow rate. min Z ) FS

(25)

Constraints 5-25 constitute the basic steam reduction and network design model as found in Coetzee and Majozi.3 Constraint 19 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 Bagajewicz10 in their work on wastewater minimization, which is also applicable to energy optimization. This can only be done if the mass flow rate to the heat exchanger is allowed to vary. Thus a MILP model can be formulated and then solved to yield the preliminary steam flow rate reduction and HEN design. 6.2. Boiler Efficiency Considerations and Altered HEN. To calculate the boiler efficiency as defined in constraint 4 several variables are required. The outlet temperature of the process that has undergone heat integration must be known and can be calculated by constraint 26. 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 subcooled condensate. Thus two terms, FRSi and FRLi, are used to represent FRi in constraint 26, respectively. The total mass flow to the boiler, including the stream from the turbines, must be considered by constraint 27. Using these variables, the total return temperature to the boiler, Tboil, can then be calculated with constraint 28. Then the efficiency can be calculated using these variables in constraint 4, shown in constraint 29. In constraint 26, Tproc is the process outlet temperature. In constraint 27, Tturb and Fturb are the turbine outlet temperature and mass flow rate, respectively. In constraint 28, Qpreheat is energy added by the preheater. Finally, the boiler efficiency is calculated by constraint 29.

∑ FRS T

i sat

Tproc )

+

i∈I

∑ FRL T

L i out,i

i∈I

FS

(27)

Qpreheat (FS + Fturb)cP

(28)

Tboil ) Tpump +

ηb ) SLi,j e SSi

(TprocFS) + (TturbFturb) (FS + Fturb)

(21)

j∈I

∑L

Tpump )

(26)

q((FS + Fturb)/FU) (cP(Tsat - Tboil) + q)[(1 + b)((FS + Fturb)/FU) + a] (29)

6.2.1. Maintaining Boiler Efficiency Using Sensible Heat. Since the steam flow rate reduction causes a decrease in the temperature of the boiler return stream, a means of reheating this stream to its original temperature must be found. The stack economizer may not be able to heat the boiler return water to a temperature high enough to retain the original boiler temperature as calculated by constraint 4. Thus it is suggested that the boiler return condensate be heated in a heat exchanger that utilizes the sensible heat of the superheated steam from the boiler. In most instances this energy is lost during the pressure let down. Thus with 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 subcooled condensate. The preheater can then be used to preheat any make up water for the process, or to further heat the boiler feedwater. Figure 9 shows a simple diagram of this proposed alteration to the steam system. The new boiler return temperature can be calculated using constraint 30: Tboil ) Tpump +

FS(hsup - hsat)θ (FS + Fturb)cP

(30)

In constraint 30, hsup is the enthalpy of the superheated steam leaving the boiler, hsat is the enthalpy of saturated steam at the boiler outlet conditions, and θ is the fraction of this energy that can be used safely without the risk of condensation. Constraints 26-30 can be used to create a second part to the mathematical model formulated before. With the minimum steam flow rate FS from the original formulation by Coetzee and Majozi,3 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: Maintaining Boiler Efficiency with a Slight Compromise in Minimum Flow Rate. First the primary objective can be to maintain boiler efficiency. This may mean that the minimum steam flow rate may not be attained if there is not enough sensible heat available to preheat the boiler feed stream. The method of Coetzee and Majozi3 is first used to find the minimum steam flow rate. This minimum flow rate is then relaxed using a slack variable in all the constraints relating to boiler efficiency. The boiler efficiency is then fixed while the deviation from the minimum steam flow rate, represented by the slack variable, is minimized. A feasible HEN corresponding to the new flow rate must also be found with the formulation. Constraints 31-34 show the new forms of constraints 26-29 with constraint 33 replacing constraint 28 as before. Constraint 35 is the objective function.


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Figure 9. Proposed steam system to maintain boiler efficiency.

∑ FRS T

i sat

Tproc )

Tpump )

i∈I

∑ FRS T

ηb - slack- )

i out,i

i∈I

FS + slack

+

(Tproc(FS + slack+)) + (TturbFturb) (FS + slack+ + Fturb)

Tboil ) Tpump + ηb )

+

(FS + slack+)(hsup - hsat)θ (FS + slack+ + Fturb)cP

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

(32)

(33)

q((FS + slack+ + Fturb)/FU)

(cP(Tsat - Tboil) + q)[(1 + b)((FS + slack+ + Fturb)/FU) + a] (34) min Z ) slack+

q((FS + Fturb)/FU)

(31)

(35)

This formulation includes several nonlinear terms. One means to deal with this situation is to implement Quesada and Grossmann11 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: Maintaining Minimum Flow Rate with a Slight Compromise in Boiler Efficiency. Second, the minimum flow rate must be achieved with the smallest possible decrease in boiler efficiency. The boiler efficiency is relaxed by adding a slack variable to the efficiency constraint (the slack variable is added to the right-hand side of the constraint, thus subtracted from the left-hand side) while the minimum flow rate is fixed. This slack variable is once again minimized, reducing the deviation from the original boiler efficiency. If the slack variable is zero, then the boiler efficiency can be maintained. If it is not, then the boiler efficiency will be compromised by minimizing the steam flow rate. Constraint 36 is the final form of the efficiency constraint, and constraint 37 is the objective function for case 2.

(36) (37)

A further extension to this formulation can be made in the case where the slack variable is found to be zero, and therefore, the boiler efficiency can be 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 38 and 39 can be used to maximize the boiler efficiency. ηb + slack+ ) q((TS + Fturb)/FU) (cP(Tsat - Tboil) + q)[(1 + b)((FS + Fturb)/FU) + a] max Z ) slack+

(38)

(39)

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, or where water is in abundance maintaining or maximizing the boiler efficiency can be used as the objective. 6.2.2. 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. This preheater will operate as all others in the HEN; however, the duty and limiting temperatures will be unknown until the optimization has taken place. Figure 10 shows the steam system set up with this addition. The original preheater, or economizer, is hereafter assumed to become part of the boiler itself. Thus the stream leaving the dedicated preheater becomes the boiler return stream. The additional heat exchanger expands the set of i heat exchangers by 1. Constraints 5-25 will occur as normal, with the addition of the following three constraints that will ensure the correct duty and limiting temperatures for the additional heat exchanger. All of the parameters associated with the

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Figure 10. Steam system with dedicated preheater.

additional heat exchanger have been designated as i* to distinguish it. The inlet stream to the i* heat exchanger is the stream that has just been pumped, thus the limiting inlet temperature is Tpump with the predetermined minimum approach temperature shown in constraint 40. Constraint 41 then shows the subsequent limiting outlet temperature, which is the boiler return stream temperature Tboil and the minimum approach temperature. Constraint 42 then shows the duty for the i* heat exchanger which is made up of latent and sensible heat. TLin,i* ) Tpump + ∆Tmin

∀i* ∈ I

(40)

TLout,i* ) Tboil + ∆Tmin

∀i* ∈ I

(41)

Qi* ) SSi*λ +

∑ (c SL p

j,i*Tsat)

j∈I

+

∑ (c L

p j,i*Tout,j)

j∈I

(cpFout,i*(Tboil + ∆Tmin))

(42)

The boiler efficiency is fixed in this formulation. Thus only the boiler efficiency constraints from case 1 in section 6.2.1 will be used, along with constraint 43, which is used to calculate the new boiler return temperature Tboil. Tboil ) Tpump +

Qi* (FS + slack+ + Fturb)cP

stream no.

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 4. Relevant Data for Turbine Portion of Steam System turbine condensate mass flow rate turbine condensate return temperature

7.28 kg/s 113 °C

perform a cost evaluation on these formulations before implementing anything in practice. 7. Case Study: Steam Reduction and Maintained Boiler Efficiency

-

∀i* ∈ I

Table 3. Hot Stream Data

The case study presented by Coetzee and Majozi3 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 3 contains the hot utility stream data for the case study. Table 4 contains information about the turbine component of the steam system.

(43)

This formulation contains many nonlinear terms that originate from the variable nature of the i* heat exchanger’s duty and limiting temperatures. Some are products of continuous variables that are treated with the technique of Quesada and Grossmann.11 The others are products of continuous variables and binary variables, and as such, Glover12 transformations are made. This formulation uses steam from the boiler to reheat the return stream and as such may not be as appealing as using sensible heat as in section 6.2.1. There is, however, no reason why the formulations from sections 6.2.1 and 6.2.2 cannot be combined into a single formulation to guarantee the boiler efficiency is maintained with as low a steam flow rate as possible. The additional heat exchangers do add a considerable capital cost to the system, and as such, it would be wise to

Figure 11. Minimum flow rate network layout.


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Figure 12. Minimum flow rate network layout.

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 29 the original boiler efficiency was calculated as 63.5%. This is for a parallel HEN and the system saturated steam information as given in Table 1. Figure 11 shows an example of this parallel HEN. After steam reduction using the formulation of Coetzee and Majozi,3 the steam flow rate was reduced from 10.90 to 7.69 kg/s, a 29.3% reduction. Figure 12 shows the network layout using the minimum steam flow rate. This, however, has the effect of reducing the overall combined process outlet temperature to 46 °C. This stream is the outlet to process 1 in Figure 5. With the turbine data from Table 4 and the new HEN flow rate and outlet temperature, the new boiler return temperature was calculated as 78.6 °C. This reduction in return temperature and flow rate corresponds to a new boiler efficiency of 59.9%, a 5.6% reduction. This loss of boiler efficiency can be remedied by reheating the boiler feed. Clearly from this figure it can be seen that the process stream associated with heat exchanger 4 requires heating from both saturated steam and saturated condensate. Thus the HEN requires 1 extra heat exchanger. A split heat exchanger is indicated by the main heat exchanger number being accompanied by a subscript. Heat exchanger 4 in Figure 12 is made up of 41 and 42, representing two separate physical heat exchangers. 7.1. Maintaining Boiler Efficiency Using Sensible Heat. Using the first premise of maintained boiler efficiency, we 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 steam is used to preheat the boiler feed. This was successfully accomplished using 79.2% of the available sensible heat. With the reduction in steam flow rate, the return temperature had to be increased from 113.0 to 117.7 °C. Given that the steam flow rate was not changed, the HEN as seen in Figure 12 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 79.2% of the available sensible heat once again. As not all of the available sensible heat was used for preheating the boiler feed, there exists the possibility of

Table 5. Results for Varying Fractions of Sensible Heat Use fraction

return temp (°C)

new efficiency

increase (%)

1 0.95 0.9 0.85 0.8

128.0 125.5 123.1 120.6 118.1

0.6460 0.6430 0.641 0.6380 0.6350

1.7 1.3 1.0 0.5 negligible

increasing the boiler efficiency by further preheating the feed. Although not all of the sensible heat should be used for this purpose, five different calculations were completed using the following sensible heat fractions: 1, 0.95, 0.9, 0.85, and 0.8. Table 5 shows the results of these simulations. In Table 5 it can be seen that if 100% of the available sensible heat could be used safely without the risk of condensation, the boiler efficiency could be increased by up to 1.7%. As mentioned in section 5, this is merely the increase according to constraint 4. Both formulations resulted in the same HEN and were able to maintain the boiler efficiency while achieving the minimum steam flow rate 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. 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 relaxed solution was found to be 8.25 kg/s. The exact solution then resulted in a flow rate of 8.32 kg/s, which still yielded a 23.7% 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 13 below. As seen in Figure 13, some saturated liquid is returned to the boiler from heat exchangers 3 and 5. 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 92.3 °C. For the change in return flow rate the return temperature to the boiler is raised to 117.3 °C. This demonstrates how the boiler efficiency can indeed be maintained, even with low amounts of sensible heat but still show a fairly large saving in steam flow rate.

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Figure 13. New network layout to maintain boiler efficiency for reduced flow rate.

Figure 14. HEN with extra heat exchanger. Table 6. Comparison of Model Numerical Results model

boiler efficiency reduction (%)

flow rate reduction (%)

return temp reduction (°C)

Coetzee and Majozi sensible energy preheater

29.6 29.6

5.9 0

preheater 30% sensible heat (case 1) preheater 30% sensible heat (case 2) dedicated preheater in HEN

23.7

37.4 -1.6 (return temperature had to be increased) -1.3

29.6

22.6

3.5

19.6

-1

0

3

0

comments boiler efficiency not considered 79.2% of sensible heat required to preheat return steam reduction compromised to maintain efficiency boiler efficiency compromised to achieve minimum steam flow rate saving made without sensible heat


With 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 61.3%, a decrease of 3.5% from the original boiler efficiency. The process outlet temperature remained at 46 °C, while the boiler return temperature dropped to 93.4 °C. The network required for this flow rate is the same as that of Figure 12, since there is no change in flow rate. 7.2. Maintaining Boiler Efficiency with a Dedicated Preheater. Using this formulation, we found that the boiler efficiency could be maintained by using the additional heat exchanger. The steam flow rate had to be increased to 8.76 kg/s, which is 13.9% higher than the minimum flow rate for the network. This is, however, still a savings of 19.6% on the parallel HEN. Figure 14 shows the new HEN arrangement with the additional heat exchanger being distinguished with an asterisk. It was found that the process outlet temperature was 60.4 °C, resulting in a pumping temperature of 84.2 °C when combined with the turbine stream. The duty of the additional heat exchanger was calculated as 2256.4 kW. The return stream to the boiler thus had a temperature of 117.0 °C. Table 6 summarizes the numerical results from the various models. From this table it can be seen that sensible energy can aid in steam flow rate reduction by allowing the boiler efficiency to be maintained. In the event that sensible heat cannot be used, considering a holistic optimization framework allows a steam flow rate savings to be made while still maintaining the boiler efficiency by including a dedicated boiler return preheater. 8. Conclusions From reducing the steam flow rate while maintaining boiler efficiency for single steam pressure levels the following conclusions can be made: • Reducing steam flow rate affects the boiler efficiency by lowering the 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. • The system let down valve can be replaced by a heat exchanger that will utilize the sensible heat of high pressure steam to preheat the boiler feed. • 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, unless another preheating source is used. • It is occasionally possible to increase boiler efficiency by further preheating the boiler feedwater by utilizing as much of the sensible heat as possible. Nomenclature Sets I ) {i or j|i or j ) 1, 2, ..., I} is the set of heat exchangers Parameters a ) regression parameter b ) regression parameter cp ) heat capacity [(kJ/kg) · k] FU ) maximum boiler capacity (kg) Fturb ) flow rate of turbine condensate (kg/s) hsup ) enthalpy of superheated HP steam (kJ/kg) hsat ) enthalpy of saturated HP steam (kJ/kg) Qi ) duty of heat exchanger i (kW) q ) latent and superheated sensible heat of HP steam (kJ/kg)

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Tin,iL ) limiting utility Tout,iL ) limiting utility

inlet temperature for heat exchanger i (°C) outlet temperature for heat exchanger i (°C) Tturb ) temperature of turbine condensate (°C) λ ) latent heat of steam (kJ/kg) θ ) fraction of sensible heat used for preheating Binary Variables

{ {

1 if heat exchanger i receives heat from condensate 0 otherwise 1 if heat exchanger i receives heat from steam yi ) 0 otherwise Continuous Variables xi )

F ) mass flow to the boiler (kg) FB ) return flow to the boiler from the process HEN (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 form 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 to the heat exchanger network (kg/s) Lj,i ) subcooled condensate reuse from heat exchanger j to heat exchanger i (kg/s) nb ) boiler efficiency slack ) slack variable used in objective functions SLj,i ) saturated condensate reuse/recycle from heat exchanger j to heat exchanger i (kg/s) SSi ) saturated steam flow rate to heat exchanger i (kg/s) Tboil ) temperature of boiler return flow (°C) Tproc ) outlet temperature from HEN (°C) Tpump ) combined temperature of process and turbine condensates (°C) ∆Tsat ) temperature difference between boiler return and saturated HP steam (°C)

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ReceiVed for reView March 21, 2010 ReVised manuscript receiVed July 25, 2010 Accepted August 3, 2010 IE1007008

On Synthesis and Optimization of Steam System Networks. 1

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