R-Curve Concept and Its Application for Industrial Energy

Ind. Eng. Chem. Res. 2000, 39, 2315-2335

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R-Curve Concept and Its Application for Industrial Energy Management Hiroyuki Kimura and (Frank) X. X. Zhu* Department of Process Integration, UMIST, P.O. BOX 88, M60 1QD, Manchester, United Kingdom

A new method for analyzing industrial energy systems is presented in this paper. The major part of this method is two graphical tools, namely, “retrofit R-curve” and “grassroots R-curve”. The new method can be applied for three major scenarios. The retrofit R-curve is dedicated for the operational management and retrofit of utility systems. The R-curve will indicate how the existing operation could be improved without capital investment. For retrofit, the analysis of this scenario will denote, through steam marginal prices, which steam is the most valuable steam worth saving and how much can be saved. The steam marginal prices determined from the method can be used as the basis for steam level switching and heat-exchanger network retrofit. The grassroots R-curve is built on the basis of a hypothetical ideal utility system without imposing capacity limits for equipment; thus, it can indicate ideal fuel utilization. This curve is used for debottlenecking the existing utility system when new processes are added and/or the throughput needs to be increased. The analysis of this scenario will indicate the most economic modifications to cope with significant changes in heat and power demand. These two curves are easy to construct and simple to understand, but they are powerful in providing insights and perspectives for improving existing energy systems. 1. Introduction Energy is a major contributor to the site operational costs for many industrial plants. To maximize the profit for a plant, reduction of energy cost should be examined. The reduction of energy use will also reduce the gaseous emissions and contribute to the conservation of the environment. A graphical method for analysis of a total site was first presented by Dhole and Linnhoff1 and later modified by Raissi.2 In their works, graphical tools based on a temperature-enthalpy diagram were developed to establish targets for cogeneration and fuel saving. Nishio et al.3 and Chou and Shih4 presented methods for designing utility systems using thermodynamic principles. These methods aim to achieve the maximum overall thermal efficiency. Nishio and Johnson5 developed a linear-programming formulation to optimize a utility system for fixed utility loads. Colmenares and Seider6 used nonlinear programming for grassroots design of a utility system. Mavromatis7 presented a methodology for designing steam turbine networks based on integer-programming methods. The most recent work for total site energy management was proposed by Makwana et al.,8 in which Makwana et al. introduced the top-down philosophy for site energy retrofit and developed a utility system analysis method called “top-level analysis”. The top-down method investigates the utility system first and considers process changes in the last stage, which is in the opposite direction of analysis compared to the traditional bottomup method (Figure 1). In this paper, a systematic analysis method using a graphical tool, called the “R-curve”, is proposed for total site analysis. The R-curve concept was first introduced by Kenney9 for analyzing the cogeneration potential of a site. However, this original concept is difficult to apply * To whom correspondence should be addressed. E-mail: [email protected].

Figure 1. Top-down philosophy and bottom-up philosophy.

Figure 2. Energy transformation.

to complex systems because the original R-curve is generated on the basis of a simple configuration of a utility system and does not take into account capacities and efficiencies of the existing equipment. To overcome

10.1021/ie9905916 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/08/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000

Figure 3. Simplest steam turbine network.

these limitations, new R-curves, namely, “retrofit Rcurve” and “grassroots R-curve”, have been developed. The new R-curves indicate the maximum energy utilization in the form of cogeneration efficiency versus the variation of site heat and power demand and take into account constraints and configuration of complex utility systems and performance of existing equipment. The R-curve analysis can identify the most effective way to reduce operating cost with a better management of energy systems. Furthermore, the debottlenecking options for the site can also be determined by the R-curve analysis. Overall, optimal options for energy cost reduction and for a debottlenecking utility system can be determined with minimal engineering effort. 2. R-Curve Concept

Figure 4. Typical R-curve.

In an industrial plant, the fuel energy is converted into heat and power, and part of the energy is lost in the transformation (Figure 2). The cogeneration efficiency (ηcogen), which is the fuel utilization efficiency, is defined as a ratio of the useful part of energy and the fuel consumption (eq 1). A power-to-heat ratio is defined in eq 2 which specifies the operating condition of a site. The net steam heat demand (Qheat) can be

ηcogen )

(W + Qheat) Qfuel

R)

W Qheat

(1) (2)

calculated on the basis of steam generation and steam usage for a site (Figure 3). Qheat includes the heating demand in steam heaters and also process steam demand (e.g., stripping steam and reforming steam). Here, process steam is treated as a heat source, although it is sometimes given merely as mass. However, generally speaking, the temperatures of steam mains are kept approximately constant using a de-superheating system. Therefore, the heat balance around a steam main can take the steam supply as mass into account. The shaded area in Figure 3 indicates the cogeneration potential. In this situation, a back-pressure turbine (BPT) can generate power (W) until the cogeneration potential is exhausted. After the cogeneration potential is exhausted, a condensing turbine (CT) could be used

for additional power demand (Figure 3). The fuel consumption (Qfuel) in a boiler can be calculated on the basis of the amount of steam generation in the boiler. Then, cogeneration efficiency (ηcogen) and R-ratio (R) can be calculated using eqs 1 and 2. For a fixed steam heat demand, by varying power generation, we can obtain different R-ratios and corresponding cogeneration efficiencies. As a result, we can generate the so-called R-curve as shown in Figure 4. The R-curve shows the relation between the maximum cogeneration efficiency (ηcogen) and power-to-heat ratio (R). It also indicates the optimal configuration for a simple utility system for the required R-ratio (the horizontal arrows in Figure 4). The R-curve constructed above has two major limitations. First, it is developed based on a simple utility system (Figure 3), and cannot deal with a complex system, which includes multiple steam distribution levels and complex steam turbine configuration. Secondarily, it assumes constant isentropic efficiencies for steam turbines, which leads to nonrealistic results. The above two limitations prevent this R-curve from practical applications. 3. Grassroots R-Curve Concept Total Site Composite Curves. It has been realized that the original R-curve9 cannot be applied for analyzing complex utility systems. To overcome the problem, the concept of site composite curves2 is applied to represent the steam heat demand, power generation potential, and fuel consumption for a complex site. The

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2317

Figure 5. Heat demand, power generation potential, and fuel consumption.

Figure 6. Ideal configuration for the grassroots R-curves.

total site profiles can be constructed by gathering data relating to all the hot and cold streams. Furthermore, site composite curves can be obtained by placing the steam levels as given in Figure 5 a,b, where VHP, HP, MP, LP, and VLP are very-high-, high-, medium-, low-, and very-low-pressure steam, respectively. The shaded area indicates power generation potential. Figure 5a corresponds to a full cogeneration operation. When the condensing power generation is increased, power generation can be increased with more fuel consumption (Figure 5b). The site grand composite curve (Figure 5c) is used to better represent steam heat demand (Qheat) and power generation (W) from each expansion zone. At each level, the steam flow rate (m1, m2, and m3 in Figure 5c) corresponding to the heat demand (Qheat) can be calculated. For example, the following equation is used to calculate the steam flow rate (m1) at the HP level related to the heat demand in HP (Qheat(HP)),

m1 )

Qheat(HP) q(HP)

(3)

where q(HP) is the specific heat content of HP steam.

Similarly, m2 and m3 can be calculated using Qheat(MP), Qheat(LP), q(MP), and q(LP), respectively. If we set the steam flow rate for a condensing turbine (m4 in Figure 5c), the power generation (W) for each expansion zone and the fuel consumption in a boiler (Qfuel) can be calculated using these steam flow rates (m1-m4). Then, we can calculate the overall cogeneration efficiency (ηcogen) related to a certain R-ratio. By changing the value for m4, we obtain different cogeneration for each expansion zone and we can calculate the different overall cogeneration efficiencies for different R-values and thus construct the R-curve for the complex utility system. A detailed discussion will be given later. It should be noted that the site grand composite curves (Figure 5c) can be constructed on the basis of the steam heat demand in each steam level which can be obtained by measuring steam flow rates leaving from a utility system to the production processes. This is much easier than measuring the individual stream data. The former case of measurement is also more accurate because it takes into account the immeasurable steam such as steam for tracing and steam distribution losses.

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New R-Curve for an Ideal Utility System. A “grassroots R-curve’” is named such that, for a given steam heat demand, it indicates the maximum cogeneration efficiency, irrespective of the configuration of the existing utility system. The grassroots R-curve is constructed on the basis of the ideal configuration of the utility system shown in Figure 6. This configuration is postulated on the basis of the following assumptions: (a) Only one steam turbine is placed in each expansion zone, which covers all the power generation potential of the zone. (b) All steam turbines operate at full loads. These two assumptions are made to obtain the highest isentropic efficiency for steam turbines and then the maximum cogeneration efficiency for a certain R-ratio. For a fixed steam heat demand and variation of power demand, the R-ratio and the corresponding maximum cogeneration efficiencies are calculated and then the grassroots R-curve is built. The construction procedure is summarized in Figure 7, and the details for calculating power and heat for both steam and gas turbine are explained as follows. Modeling of Steam Turbines. To make the new Rcurve realistic, the change in isentropic efficiency relating to the steam turbine size and load is taken into account using a model proposed by Mavromatis.7 This model is developed on the basis of the Willans line (eq 4) which relates power generation (W) to steam mass flow (m) for a simple steam turbine in a linear fashion. The calculation for the specific power generation (n) and the internal loss (Wloss) are provided for each expansion zone as follows.

W ) n × m - Wloss n)

Wloss )

(

A 6 1 × × ∆His 5 B mmax

(4)

)

(5)

1 1 × × (∆His × mmax - A) 5 B

(6)

∆Ts 1854 - 1931 × qin

(7)

∆His )

A ) a1 + a2 × Ts.in B ) b1 + b2 × Ts.in (a1, a2, b1, b2) ) (-0.928, 0.006 23, 1.120, 0.000 47) for Wmax g 1.2 MW (8) where mmax is the maximum steam flow rate for each expansion zone. For example, mmax for expansion zones 4 and 1 are m4 and m1 + m2 + m3 + m4, respectively (Figures 5c and 6). ∆His is the isentropic enthalpy change between the turbine inlet and the outlet. A and B are regression parameters. qin is the specific steam heat content in the turbine inlet steam. Ts.in is the saturation temperature of the turbine inlet steam and ∆Ts is the saturation temperature difference between the turbine inlet and outlet. It should be noted that, for construction of the grassroots R-curve, mmax is used for m as well in eq 4 according to assumption (b).

Figure 7. Grassroots R-curve construction.

Singh et al.10 provided equations to estimate the heat content in the turbine exhaust (qout). For each expansion zone,

qout ) qin + cp∆Ts cp )

W m

(9)

3.38 + 0.006 123Ts.ave 3.6 (100 < Ts.ave < 300 °C) (10)

where cp is the specific heat of saturated water and Ts.ave is the average saturation temperature between the inlet and outlet steam. The automatic procedure of calculating power generation and steam flow rates is iterative. Initially, steam flow rates (m1-m3) are calculated using eq 3 by assuming initial values of q’s (q(HP), q(MP), q(LP)), and then the maximum power generation (W) and the heat content (q) in exhaust steam are calculated using eqs 4 and 9, respectively. The newly calculated q’s may be different from the initial q values. Therefore, the steam flow rates are recalculated using eq 3 on the basis of the new values of the q’s and then W and q by eqs 4 and 9. This procedure is repeated until the calculated q’s at every stage become close enough to those calculated at the previous stage. Modeling of Gas Turbine. A gas turbine (GT) is taken into account in constructing the grassroots R-curve. The model for industrial gas turbines proposed by Manninen and Zhu11 is adopted and given as follows:

Qfuel.GT ) 2.8412WGT.max + 7.3291

(11)

Texhaust.GT ) 0.4WGT.max + 493.42

(12)

mexhaust.GT ) 2.9WGT.max

(13)

Using these equations, the fuel gas consumption (Qfuexhaust temperature (Texaust.GT), and flue gas flow rate of a gas turbine (mexhaust.GT) are estimated by the

el.GT),


Figure 8. Example of a utility system (Example 1).

maximum capacity of the gas turbine (WGT.max). The heat content in the generated steam from a heat recovery steam generator (HRSG) (Qheat.HRSG(j)) can be expressed as the function of WGT.max:

Qheat.HRSG(j) ) a × WGT.max + b

(14)

Coefficients a and b can be determined by fitting the results from HRSG simulation using Texhaust.GT and mexhaust.GT. To maximize the heat recovery from the GT exhaust, a multiple-level steam generator is assumed and the generated steams are supplied to steam mains according to pressure levels. As we can see in Figure 6, there are tow effects by generation of HP, MP, and LP steam through a HRSG. Although it can reduce VHP steam generation from boilers, it reduces, on one hand, cogeneration potential in steam turbines in the same time. On the other hand, generation of VHP steam from a HRSG can reduce VHP generation in boilers without losing cogeneration potential in a steam turbine. To make VHP steam through a HRSG, it usually requires a supplementary or fully fired HRSG. Modeling of a Boiler. Calculation of boiler duty is site specific, which relates to condensate losses and the boiler feedwater (BFW) preheating system. Basically, the calculation can be expressed as a function of the VHP steam requirement. For simplicity, the boiler efficiency is treated as constant in this work. Calculation of the cogeneration efficiency is done on the basis of one fuel source. It is possible to take into account multiple fuel supplies. In this case, fuel consumptions (Qfuel) for different fuel sources are converted into the monetary equivalent fuel consumption using the fuel price ratio:

Qfuel(fuel1/base) ) Qfuel1 + Qfuel2

Cfuel2 Cfuel3 + Qfuel3 + ... Cfuel1 Cfuel1 (15)

Example of a Grassroots R-Curve. The utility system for Example 1 (Figure 8) is used for illustration

Figure 9. Grassroots R-curve for Example 1.

and the grassroots R-curves for this example are given in Figure 9. These grassroots R-curves are constructed on the basis of the existing heat demand (38.75(HP) + 21.60(MP) + 58.60(LP)). It should be noted that an unfired HRSG is used in the gas turbine case. As shown in Figure 9, in the range of low R values, use of CT is better than a small GT (WGT.max e 5 MW in particular). However, in the high R-range, the configuration which includes GT can achieve much higher cogeneration efficiency. Remarks. The grassroots R-curve is mainly developed for debottlenecking scenarios where major modifications to the existing utility system is required. These kinds of problems are treated more toward grassroot design for exploiting great opportunities to obtain a much improved design. That is the reason the ideal utility system configuration is used for constructing the grassroots R-curve. More detailed discussions will be given later. 4. Retrofit R-Curve Concept Although the grassroots R-curve can provide an opportunity for maximum improvement of a utility

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 Table 1. Steam Turbine Model for Example 1 (refer to Figure 11) original turbine T1 T1 T1 T3 T3 T3 T2

VHP/HP HP/MP MP/LP VHP/HP HP/MP MP/LP MP/COND

(MW h)/t

MW

na ) 0.1080 nb ) 0.0354 nc ) 0.0718 nd ) 0.1063 ne ) 0.0346 nf ) 0.0713 ng ) 0.2051

Wloss.a ) 4.4253 Wloss.b ) 1.0422 Wloss.c ) 0.8779 Wloss.d ) 3.5651 Wloss.e ) 0.7883 Wloss.f ) 0.8616 Wloss.g ) 2.6898

Wa ) na × (ma + mb + mc) - Wloss.a. Wb ) nb × (mb + mc) - Wloss.b. Wc ) nc × mc - Wloss.c. Wd ) nd × (md + me + mf) - Wloss.d. We ) ne × (me + mf) - Wloss.e. Wf ) nf × mf - Wloss.f. Wg ) ng × mg - Wloss.g.

Figure 10. Retrofit R-curve construction.

load, this serial configuration requires an iterative optimization7 and the solutions could be trapped at the local optimal points. To overcome these deficiencies, a new decomposition is proposed. This method tries to decompose the network into a simpler configuration where all simple turbines are placed in parallel and every steam turbine links to the VHP main (Figure 12). The new decomposition attempts to decompose the steam network on the basis of the power-generation-path concept.8 For example, the power-generation paths linking MP main are represented as follows (Figure 12):

(1) VHP f T1 f HP f letdown valve f MP (path P2, steam flow rate M2) (2) VHP f T1 f MP (P3, M3) (3) VHP f T3 f HP f letdown valve f MP (P8, M8)

Figure 11. Decomposition of Example 1 by Chou’s method.

system, it cannot usually be achieved because of the constraints in the existing system. To take the existing constraints into account, the R-curve for existing systems is developed. The construction of the retrofit R-curve is summarized conceptually in Figure 10. As seen in Figure 10, optimization is done under a given heat and power demand. Therefore, from the definition of cogeneration efficiency (eq 1), maximizing cogeneration efficiency is equivalent to minimizing fuel consumption. In the optimization, operating conditions of the utility system and opportunities for power importing are optimized. Decomposition of a Steam Turbine Network. To model a complex steam turbine network, it must be decomposed into a combination of simple turbines. Then, simple turbines are modeled using the Willans linear equation (eq 4). Chou and Shih4 proposed a decomposition scheme for complex steam turbines in which simple turbines are arranged in series. Using this decomposition, the steam turbine network in Figure 8 can be converted to Figure 11, and the steam turbine models for Figure 11 are given in Table 1. This decomposition is useful for optimization of a steam turbine network under fixed conditions of steam mains. However, if we try to take into account the changes in heat content in the steam turbine exhaust, which varies with the turbine working

(4) VHP f T3 f MP (P9, M9) (5) VHP f letdown valve f HP f letdown valve f MP (D2, ML2) Similarly, we can identify power generation paths linking HP and LP mains. Furthermore, we can identify the following paths for condensing power generation.

(1) VHP f T1 f MP f T2 f COND (P5, M5) (2) VHP f T3 f MP f T2 f COND (P11, M11) If we provide a mathematical formulation for these paths, the optimization can be done by optimizing the steam flow rates (M1-M12 and ML1-ML3). Steam Turbine Models for the New Decomposition. Figure 13 shows a graphical interpretation of the new steam turbine models. The shaded areas represent power generation from steam turbines (a-g) in Chou’s decomposition (Wa-Wg) (Figure 11), and the steam turbine models for Wa-Wg are given in Table 1. Each power generation (Wa-Wg) in each row in Figure 11 is divided into several elements according to the steam flow rates on each path (M1-M12). The elements on the same column for each path are added to give overall power generation for this path. For example, the power generated from path P1 occupies a certain proportion


Figure 12. New decomposition of Example 1.

Figure 13. Graphical interpretation of the new decomposition.

of Wa, which can be expressed as

Wa ×

M1 6

{

) na ×

∑1 M(i) n a × M1 -

M1 6

}

6

∑1

M1

M(i) - Wloss.a ×

)

6

∑1 M(i)

× Wloss.a ) na × M1 - WL1 (16)

∑1 M(i) where WL1 is the internal loss of steam turbine path P1. For path P3, the power generation consists of two elements on the same column, which is given as

Wa ×

M3 6

∑1 M(i)

+ Wb ×

)(

M3 6

(

) n a × M3 -

∑3 M(i)

Wloss.a + nb × M3 -

M3 6

∑3 M(i)

M3 6

×

∑1 M(i)

)

× Wloss.b )

(na + nb) × M3 - WL3 (17)

where WL3 is the internal loss of steam turbine path P3.

Finally, steam turbine models for the new decomposition are obtained as in Table 2, where WL(i) is the summation of the terms relating to the internal losses for path i. Advantage of This New Decomposition. With this new decomposition, any complex steam turbine systems can be treated as a set of parallel simple turbines. As a result, the task of mathematical formulation becomes straightforward. Furthermore, because every path directly links to the VHP main, it is straightforward to calculate the required boiler duty. The steam turbine models in Table 2 are mathematically the same as the steam turbine models in Table 1 and both are intertransferable. However, this parallel configuration allows optimization to be done without the need for iteration. It also makes optimization possible to find the global optimal solution more easily. This is because optimization based on the new decomposition can optimize the steam distribution simultaneously over all the expansion zones. Mathematical Formulation. Boiler Duty and Fuel Consumption of a Path. An example of Figure 14 is used for illustration. The contribution of the HP path toward 0 ) can be calcuthe specific boiler heat duty (Qboiler(HP) lated as 0 ) (hVHP - hBFW) Qboiler(HP)

QEX(HP)

× (1 Qheat(HP) LMASS(HP)) × (1 - LHEAT(HP)) × (hsat(HP) - hBFW) (18)

The second term denotes the heat return to the boiler

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Table 2. Steam Turbine Model for Example 1 (refer to Figure 12) from power VHP W(i) to (MW)

path i

path type

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 D1 D2 D3

BPT BPT + LD BPT BPT + LD CT BPT BPT BPT + LD BPT BPT + LD CT BPT LD LD LD

HP HP/MP MP MP/LP COND LP HP HP/MP MP MP/LP COND LP HP MP LP

W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12

∆W n(i) (MW h)/t) ) na × ) na × ) (na + nb) × ) (na + nb) × ) (na + nb + ng) × ) (na + nb + nc) × ) nd × ) nd × ) (nd + ne) × ) (nd + ne) × ) (nd + ne + ng) × ) (nd + ne + nf) ×

steam M(i) (t/h) M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 ML1 ML2 ML3

loss WL(i) (MW) -

WL1 WL2 WL3 WL4 WL5 WL6 WL7 WL8 WL9 WL10 WL11 WL12

where

Figure 14. Mass and heat balance around HP steam main.

It should be noted that the boiler efficiencies would vary with the operational loads of the boilers, especially at very low partial loads. Therefore, to handle the boiler efficiencies as constant, reasonable operational ranges should be added as constraints. Alternatively, the boiler efficiency could be given as a function of the operational loads, although it introduces additional nonlinearity to the optimization. Exhaust Heat Content in a Steam Turbine Path. The heat content in a steam turbine exhaust at the jth steam j ) can be calculated on the basis of eq 9 main (Qexhaust(i) as j Qexhaust(i) ) qout(i) × M(i)

(

) qin + cp(i) × ∆Ts(i) M(i) associated with condensate. Specific fuel consumption 0 ) is then expressed as of the HP path (Qfuel(HP) 0 Qfuel(HP)

)

0 Qboiler(HP)

ηb

(19)

In practice, all the parameters in eqs 18 and 19 are known. Thus, the specific boiler duty and specific fuel consumption defined in these two equations become constant. When the steam flow rate is multiplied with them, the total heat duty (Qboiler(HP)) and fuel consumption (Qfuel(HP)) for the HP path is obtained.

Qboiler(HP) )

0 Qboiler(HP)

× m(HP)

0 × m(HP) Qfuel(HP) ) Qfuel(HP)

(20) (21)

Similarly, boiler heat duty and fuel consumption for the other paths (MP, LP, CT) can be calculated.

)

n(i) × M(i) - WL(i)

× M(i)

) {qin + cp(i) × ∆Ts(i) - n(i)} × M(i) + WL(i) (22) where n(i) is the specific power generation for path i (e.g., n3 ) na + nb). cp(i) and ∆Ts(i) are the specific heat and the saturation temperature difference between the inlet and outlet of steam turbine path i. Heat Content in a Letdown Path. The heat content of steam passing through letdown path i at the jth steam j ) can be calculated as main (QLD(i) j ) {hVHP - hsat(j)} × ML(i) QLD(i)

(23)

where hsat(j) is the specific enthalpy of the saturated liquid in the jth level steam and ML(i) is the steam flow rate through letdown path i. Heat Content in a (Steam Turbine + Letdown) Path. If steam expands to steam level j through a steam turbine i and then further to level k through a letdown


valve (e.g., path P2,4,8,10 in Figure 12), the exhaust heat of these kinds of paths can be calculated by the following equation:

Qexhaust(i) ) [{qin + cp(i) × ∆Ts(i) - n(i)} × M(i) + WL(i)] + {hsat(j) - hsat(k)} × M(i) (24) This equation is the general form for calculating exhaust heat, which is the extension of eq 22. In the case of eq 22, there is no letdown valve following a steam turbine; thus, j ) k and the second term in eq 24 vanishes. Consideration of Power Importing. The effect of importing power can be taken into account in the following way. It is assumed that the money spent for importing a certain amount of power (Wimport) can be converted to the monetary equivalent amount of fuel (Qfuel.import) (Makwana et al.).13 Thus,

Cp.import Qfuel.import ) Wimport × CF

(25)

where CP.import and CF are the prices for importing power and fuel. Then, we can consider importing power in calculating R and ηcogen:

R)

ηcogen )

∑i W(i) + Wimport

(26)

Qheat

∑i W(i) + Wimport + Qheat ∑i Qfuel(i) + Qfuel.import

Q(j) heat )

∑ i∈I

(34)

W ˜ L(i) e WL(i) - WLmin(i) × {1 - Z(i)}

(35)

where Mmin(i), Mmax(i), WLmin(i), and WLmax(i) are minimum and maximum limits of the steam flow rate and internal loss for path i. W ˜ L(i) is the newly defined internal loss which is used to control path on/off options. In calculations of power generation and exhaust heat duty (Table 2 and eqs 22 and 24), W ˜ L(i) is used instead of WL(i). Objective Function. Objective function for this overall model is minimizing the total fuel consumption (Qfuel):

Minimize Qfuel )

∑i Qfuel(i) + Qfuel.import

(27)

(36)

Handling of Nonlinear Terms. Bilinear Terms. Nonlinear terms are involved in the internal losses (Table 2). If the inverse of the summation of steam flow rates are substituted by new variables S(t) (Table 2), internal losses for the new models can be rewritten as in Table 3, where $1-$22 represent bilinear terms of S(t) and M(i) (e.g., $1 ) S1 × M1, $2 ) S1 × M2, $10 ) S2 × M6, $22 ) S5 × M11, etc). To make the searching range for S(t) smaller, the maximum and minimum limits are set as Smin(t) ) 0 and Smax(t) ) 0.3 for most cases. Then,

Smin(t) e S(t) e Smax(t)

Heat Balance. The heat demand for steam main j should be balanced against the heat from the paths of steam turbines (P) and letdown valves (D) connecting with this steam main. P1-P12

W ˜ L(i) g WL(i) - WLmax(i) × {1 - Z(i)}

(37)

Mass Balance for Turbines. Because each bilinear term ($) represents a steam mass ratio for a path versus the total steam flow rate of a turbine (a-g) (Figure 11), the summation of these bilinear terms through a turbine should be equal to 1:

for turbine a, $1 + $2 + $3 + $4 + $5 + $6 ) 1 × ZX1 (38)

D1-D3 j Q exhaust(i) +

j

∑ i∈I

j Q LD(i)

(28)

$7 + $8 + $9 + $10 ) 1 × ZX2

j

Power Balance. The power demand has to be satisfied by generated power and importing power:

Wdemand )

∑i W(i) + Wimport

for turbine b,

(29)

(39)

for turbine d, $12 + $13 + $14 + $15 + $16 + $17 ) 1 × ZX3 (40) for turbine e,

Mass Limit. The steam turbine capacity limits are set as constraints. Consider mp steam from T1 as an example:

$18 + $19 + $20 + $21 ) 1 × ZX4

(41)

for turbine g,

M1 + M2 e 80 (max)

(30)

$11 + $22 ) 1 × ZX5

M3 + M4 g 20 (min)

(31)

It should be noted that turbines c and f are not decomposed as shown in Figure 13; therefore, they do not require the above mass-balance equations. However, if paths P1-P6 are not used, this implies that the M1-M6 values are zero and thus the left-hand side of eq 38 is set to zero. In this case, to force the righthand side to be zero, binary variable ZX1 is introduced. Similarly, ZX2-ZX5 are introduced for eqs 39-42. Therefore, ZX(t), defined as such, can represent the on/ off position for turbine t (t ) a, b, d, e, g). Because each turbine can consist of several paths, the following

Path On/Off Control. If steam flow of a path turns out to be zero from optimization, this path is regarded as being in the “off” position. Otherwise, this path is in the “on” position. This on/off option can be dealt with using binary variables Z(i):

M(i) g Mmin(i) × Z(i)

(32)

M(i) e Mmax(i) × Z(i)

(33)

(42)

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Table 3. Rewritten Internal Losses for Example 1 Using Bilinear Variables WL1 ) $1 × Wloss.a WL2 ) $2 × Wloss.a WL3 ) $3 × Wloss.a + $7 × Wloss.b WL4 ) $4 × Wloss.a + $8 × Wloss.b WL5 ) $5 × Wloss.a + $9 × Wloss.b + $11 × Wloss.g WL6 ) $6 × Wloss.a + $10 × Wloss.b + Wloss.c WL7 ) $12 × Wloss.d WL9 ) $13 × Wloss.d WL9 ) $14 × Wloss.d + $18 × Wloss.e WL10 ) $15 × Wloss.d + $19 × Wloss.e WL11 ) $16 × Wloss.d + $20 × Wloss.e + $22 × Wloss.g WL12 ) $17 × Wloss.d + $21 × Wloss.e + Wloss.f

$(k) g Mmin(i) × S(t) + Smin(t) × M(i) - Mmin(i) × Smin(t) (49)

constraints are used to link each turbine with corresponding paths:

ZX1 g Z1, ..., Z6

(43)

ZX2 g Z3, ..., Z6

(44)

ZX3 g Z7, ..., Z12

(45)

ZX4 g Z9, ..., Z12

(46)

ZX5 g Z5, Z11

(47)

Note that binary variable Z(i) represents the on/off position for path i. These ZX(t) values are added to the objective function as the penalty term (eq 48). Minimizing the objective function also minimizes this penalty term, which effectively forces ZX(t) to be zero for the case when all relevant Z(i) values are zero. It should be noted that this penalty term should be subtracted from the optimal objective value.

Minimize Qfuel )

∑i Qfuel(i) + Qfuel.import + c × ∑ ZX(t) t

where c , Qfuel(i), Qfuel.import.

Searching for the Global Optimal Solution. Basically, the above-mentioned optimization program is mixedinteger nonlinear programming (MINLP) where all nonlinear terms are bilinear terms. In theory, this optimization could be trapped in local optimal points. To find the global optimal point, a relaxed linear programming using “underestimator/overestimator”12 is applied. In this relaxed linear program, every bilinear term is relaxed to several linear equations:

(48)

$(k) g Mmax(i) × S(t) + Smax(t) × M(i) - Mmax(i) × Smax(t) (50) $(k) e Mmin(i) × S(t) + Smax(t) × M(i) - Mmin(i) × Smax(t) (51) $(k) e Mmax(i) × S(t) + Smin(t) × M(i) - Mmax(i) × Smin(t) (52) The searching of the global optimal point proceeds according to a scheme shown in Figure 15. Variables S(t) are selected as key variables because variables S(t) appear more frequently in the binary terms than other variables. By assuming Smin(i), Smax(i) at each node of a binary tree where binary variables are fixed, the relaxed linear problem (LP) and then the original nonlinear problem (NLP) are solved, respectively. If the difference between the optimal solution from the relaxed LP problem and that from the original NLP problem satisfies a specific tolerance, the global optimal is found. r values from the relaxed LP Otherwise, one of the S(t) problem is selected to adjust the lower and upper b bounds (S(t.selected) ). This procedure is repeated until the global solution for this node is found. This searching

Figure 15. Searching the global optimal point for a node in the binary tree.


Figure 16. Retrofit R-curve for Example 1.

method is used for every node of a binary tree to give the global solution for the original MINLP. Construction of Retrofit R-Curve. Using the above mathematical programming, we can obtain the retrofit R-curve (Figure 16) for Example 1 (Figure 8), which is constructed using the procedure in Figure 10. This retrofit R-curve is based on the current steam demand. The retrofit R-curve indicates the maximum cogeneration efficiency for different R-ratios for this site. Horizontal arrows show the corresponding optimal configuration of the utility system. As we can see in this curve, the back-pressure turbine (BPT) is used for low power demand (low R), and the condensing turbine (CT) needs to be used for increasing power demand. After the capacity of power generation is exhausted, the option of imported power has to be considered. In the middle region (R ) 0.22-0.24) of the curve, it indicates that the importing power could be beneficial. Otherwise, T2 could be used at a low running load but at a low efficiency. This could be much more expensive than importing power. For comparison, the grassroots R-curve for the same site is plotted in Figure 16. Obviously, the grassroots R-curve indicates higher cogeneration efficiency than the retrofit R-curve over the whole range of R-ratios because it does not take existing constraints into account. 5. Case Studies Case Study 1: Operational Management Using Retrofit R-Curve. Fuel Saving and Financial Benefits by Steam Redistribution. A current operating point for Example 1 (Figure 8) is plotted on the retrofit R-curve (Figure 16). We can see the difference between the maximum achievable cogeneration efficiency and the current operating point. The difference can be eliminated by the operational changes. The three steam turbines (T1, T2, and T3) are currently running; however, the R-curve indicates that the optimal configuration corresponding to the current R-ratio (0.3384) is (T2 + T3). This implies that T1 should be switched off and steam distribution needs to be adjusted to the optimum. When this is done as such, the cogeneration efficiency can be improved from 0.647 to 0.671, which leads to fuel savings (∆Qfuel) by 8.736 MW (eq 53). The financial benefit of the fuel savings (Bfuel) is about 1.4 MM$/year if we assume 8000 h/year of operation (eq 54).

∆Qfuel ) (Rcurrent + 1) × Qheat ×

) (0.3384 + 1) × 119 ×

[

1 ηcogen.current 1 ηcogen.improved

]

1 1 [0.6472 0.6710]

) 8.736 MW

(53)

Bfuel ) ∆Qfuel × CF × (h/year) )

(8.736 × 1000) × 0.02 × 8000 106

) 1.39 MM$/year (54)

where Rcurrent is the current R-ratio and ηcogen.current and ηcogen.improved are the cogeneration efficiency of the current operation and improved operation, respectively. It should be noted that the cost savings is achieved by the operational changes without any capital investment. Exploitation of Other Opportunities. After improvement by the steam redistribution, the site operation cost could be further improved by exploiting other options. The impact of different options on operation cost can be analyzed using the retrofit R-curve. (a) Power Saving Option. Let us first assume that it is possible to reduce the power demand by 0.5 MW. In this case, Rcurrent changes from 0.3384 to 0.3342 power ), and the maximum cogeneration efficiency can (Rsave power ). As a be improved from 0.6710 to 0.6760 (ηcogen.save result, 0.4 MM$/year of financial benefit can be expected for 0.5 MW of power savings.

∆Qfuel )

(Rcurrent + 1) × Qheat (Rpower save + 1) × Qheat ηcogen.improved ηpower cogen.save

)

(0.3384 + 1) × 119 (0.3342 + 1) × 119 0.6710 0.6760 ) 2.496 MW

(55)

Bfuel ) ∆Qfuel × CF × (h/year) ) 0.399 MM$/year (56) (b) Power Exporting Issue. Figure 17 is a retrofit R-curve for the same example, which has been enlarged for the R-range of 0.3 and 0.5. The current operation

2326


Figure 19. Financial benefit of power exporting for Example 1. Figure 17. Power exporting potential of Example 1.

Figure 20. Different retrofit R-curves relating to steam saving for Example 1.

Figure 18. R-curve for LP venting for Example 1.

does not fully utilize the capacity of power generation. To generate more power for exporting, more VHP steam has to be generated, thereby increasing fuel consumption. The financial benefit for the power exporting (Bexport) can be calculated as

[{

Bexport ) Wexport × CP.export -

}

]

× (Rexport + 1) (Rcurrent + 1) × Qheat × CF ηcogen.export ηcogen.improved (h/year) (57)

For Example 1, more power exporting can be achieved by LP steam venting or by increasing LP steam demand in the processes. Figure 18 shows the performance improved by the LP venting. The financial benefit of power exporting is shown in Figure 19. It should be noted that the financial benefit of power exporting is very sensitive to the price ratio of exported power and fuel. On one hand, when the price ratio of power to fuel is 5 (CP.export ) 0.10 $/(kW h)), power exporting is definitely beneficial. If, on the other hand, the price ratio is 3.5 (CP.export ) 0.07 $/(kW h)), exporting power is marginally beneficial in the first 5 MW and then it becomes nonbeneficial beyond 5 MW of power exporting. Energy Retrofit Using the Retrofit R-Curve. In many cases, it is beneficial to consider steam saving and

Figure 21. Retrofit R-curves for steam saving.

steam level switching; that is, use lower pressure steam to replace higher pressure steam currently in use. To assess these two options, we need to know which steam is worth saving or switching with how much. The R-curve analysis can be used for this purpose. R-Curve for Steam Saving. So far, the retrofit R-curve (Figure 16) is constructed for the current heat demand. If the heat demand changes, the retrofit R-curve will also change. For different heat demands, we can generate different retrofit R-curves (Figure 20). A locus curve linking the points on different R-curves can be obtained which indicates the relationship between cogeneration efficiency and R-ratio for steam saving with different amounts. This locus curve is then defined as the R-curve for steam saving. In this way, we can generate R-curves for saving HP, MP, and LP steam, respectively (Figure 21).


Steam Level Switching. Similar to steam saving, steam level switching gives different R-curves compared to the original retrofit R-curve. Although steam level switching could improve cogeneration efficiency, it does not change the steam demand (Qheat) and R-ratio. The fuel savings for steam switching is calculated by eq 61. The financial benefit and marginal prices are calculated using eqs 59 and 60, respectively.

∆Qfuel ) (Rcurrent + 1) × Qheat × 1 1 ηcogen.improved ηsteam

[

Figure 22. Financial benefit of steam saving for Example 1.

Figure 23. Marginal prices of saved steam for Example 1.

Financial Benefit of Steam Saving and Steam Marginal Prices. Steam saving changes both the R-ratio and cogeneration efficiency. This change can be translated into fuel savings (∆Qfuel) using eq 58, which can be further translated into financial benefits (Bfuel) and steam marginal prices (Mprice) by using eqs 59 and 60, respectively. Here, the steam marginal price is defined as the marginal benefit change for reduction of 1 unit of steam. In Figures 22 and 23, Bfuel and Mprice are plotted versus the steam savings (∆Qheat). When considering the bene-

∆Qfuel ) heat.save (Rcurrent + 1) × Qheat (Rsteam save + 1) × Q (58) ηcogen.improved ηsteam cogen.save

Bfuel ) ∆Qfuel × CF × (h/year) Mprice )

∂Bfuel ∂Qheat

fit of steam saving, the current operation is first improved toward the maximum cogeneration efficiency for the same heat demand using operational optimization. This improved case is then used as the base case to consider the benefit of steam saving. That is the reason ηcogen.improved appears in eq 58 instead of ηcogen.current. Surprisingly Figures 22 and 23 show that the MP and LP are more valuable than HP in the first 20 MW of steam savings. This is because, in the optimal operation at the current R-ratio, parts of the MP and LP are supplied through the letdown valves. Therefore, reduction of the MP and LP contribute to the financial benefits directly. However, beyond 20 MW of savings, further LP saving will give penalty. The steam marginal prices determined in this way can be used as steam prices for energy retrofit projects.

]

(61)

Case Study 2: A More Complex System A site shown in Figure 24 has four levels of steam mains (VHP, HP, MP, and LP) and only one extraction turbine (T1). This site has a gas turbine installed, and the power deficit is satisfied by importing power (16.8 MW currently). VHP is supplied from three boilers which combust three types of fuel (gas, fuel oil, and asphalt). These three boilers have the same design originally; however, the no. 2 boiler’s maximum capacity is limited to 135 t/h to satisfy environmental regulation. Currently, too much LP steam is generated from processes and part of the LP is being vented (24.4 t/h currently). Furthermore, hot condensate is not returned to the boilers at all; instead, boiler feedwater (BFW) is preheated by a BFW preheating system, which recovers heat from some of the processes. Total boiler heat requirement (Qboiler) is calculated by the following equation:

Qboiler ) 0.8082 × (hVHP - hBFW) × m(VHP) (62) where hVHP and hBFW are the specific enthalpies of VHP and cold BFW and m(VHP) is the VHP steam requirement. The coefficient is obtained by analysis of the existing system, which takes into account the BFW preheating. This site utility system features one gas turbine and can burn multiple fuels in three boilers. We need to account for them in the optimization model. Modeling of the Gas Turbine. The following equations are given to calculate the fuel consumption in the GT (Qfuel.GT) and the heat content in HP steam generated at the HRSG (Qheat.HRSG(HP)),

(59) (60)

cogen.switch

Qfuel.GT ) 2.75 × WGT + 8.01

(63)

Qheat.HRSG(HP) ) 0.37 × WGT + 2.55

(64)

where WGT is generated power from the GT. Then, the GT system can be linked with the rest of the utility system by using the following equations (compare with eqs 28, 29, and 36): P

Qheat(HP) )

∑i

D

HP Qexhaust(i) + Qheat.HRSG(HP) +

(65)

∑i W(i) + WGT + Wimport

(66)

∑i Qfuel(i) + Qfuel.GT + Qfuel.import

(67)

Wdemand ) Qfuel.total )

HP ∑i QLD(i)

Modeling of Multiple Boilers and Multiple Fuels. Boiler efficiencies for different fuels are usually different. Generally, it is high for gas fuel and low for oil fuel.

2328


Figure 24. Example of a complex utility system (Example 2).

For these three boilers, boiler efficiency for gas fuel (ηb(GAS)), for fuel oil (ηb(FO)), and for asphalt (ηb(AS)) are 93.1%, 91.8%, and 88.0%, respectively. For simplicity, it is assumed that these efficiencies are constant within the normal operational range of the boilers. With this assumption, the various aspects of the multiple boilers and multiple fuels are addressed in the optimization program as follows. Objective Function. Fuel consumption is calculated by using asphalt as the reference (Qfuel(AS/base)):

Minimize Qfuel(AS/base) )

{∑

}

Qfuel(FO.k) ×

k

Qfuel.GT ×

{

∑k Qfuel(GAS.k)

CF(FO)

+

CF(AS)

CF(GAS) CF(AS)

{∑

}

×

CF(GAS)

+

CF(AS)

}

Qfuel(AS.k) +

k

+ Wimport ×

CP.import CF(AS)

Figure 25. Retrofit R-curve for Example 2.

(68)

where Qfuel(GAS.k), Qfuel(FO.k), and Qfuel(AS.k) are new variables introduced to represent the consumption of fuels (gas, fuel oil, and asphalt) in boiler k. It should be noted that the asphalt price is corrected by the operational cost of the flue gas desulfurization unit because the amount of combusted asphalt affects the desulferization cost significantly. Constraints around the Boilers. The overall boiler duty required in all paths (∑i Qboiler(i)) should be equal to the summation of heat duties generated by burning the three fuels:

{ }

∑i Qboiler(i) ) ∑k Qfuel(GAS.k)

{∑ k

Qfuel(FO.k) × ηb(FO) +

}

× ηb(GAS) +

{∑ k

}

Qfuel(AS.k) × ηb(AS) (69)

In addition to this, the burner and boiler capacity limits (min/max) need to be considered. Additional Constraints. For this system, the following constraints should be included. (a) The amount of asphalt combustion is limited by the capacity of the flue gas desulfurization unit. (b) A certain amount of gas fuel has to be combusted to keep total site gas balance. (c) Fuel oil and asphalt cannot be combusted simultaneously in one boiler because of the arrangement of the fuel header. Retrofit R-Curve for Example 2. Th retrofit Rcurve for Example 2 (Figure 24) based on current heat demand is given in Figure 25. For the low power demand, that is, low R-ratio, the R-curve indicates that the back-pressure turbine (BPT) should be used. Ideally, the cogeneration efficiency curve in the BPT configuration should be virtually flat. For this case, however, a certain amount of gas fuel, which is the most expen-


Figure 26. Operational improvement for Example 2.

sive, has to be combusted to keep the total site gas balance. The gas fuel ratio in the total fuel consumption at a lower R-ratio is much higher than that at a high R-ratio. This results in a much lower ηcogen at low R values. ηcogen increases with R-ratio because of the increased amount of asphalt burnt in the boilers. When the BPT cogeneration potential is exhausted, normally the gas turbine (GT) should be used. However, it is found in this case that use of importing power is more efficient than the GT at low operation load (R ) 0.150.16). For further power demand, the GT should be used. When the GT capacity is reached, power is imported to satisfy the increasing power demand. It might be surprising to see that the cogeneration efficiency within the BPT + GT range goes down at R ) 0.18 as steeply as the power importing. This is because the amount of fuel oil burnt in the boilers increases with increased GT utilization to satisfy constraints associated with the

burners in the boilers. An increase in burning fuel oil, which is more expensive than asphalt, causes the decrease in cogeneration efficiency. Operational Management for Example 2. The current operating point is plotted on the retrofit R-curve (Figure 26). In this case, the steam turbine and the gas turbine are being operated optimally. The difference in cogeneration efficiency (0.023 ) 0.391-0.386) mainly comes from nonoptimal composition in the fuel mixture used in the boilers. The financial benefit of optimizing the type and amount of fuels to be burnt in each boiler is 0.594 MM$/year. Obviously, there is a tradeoff between fuel prices and boiler efficiencies. Energy Retrofit Analysis for Example 2. The financial benefits for steam saving and steam switching are plotted in Figure 27. A significant degree of financial benefit can be obtained by 0.6-0.7 MM$/year for the first 4 MW of the HP or MP saving. The benefit diminishes beyond 4 MW for HP saving. However, MP saving beyond 4 MW is not beneficial because the steam saving decreases the power generation and the increasing cost for imported power is much larger than the fuel saving. Switching from HP to MP is marginally beneficial before 10 MW of switching is reached. Between 10 and 11 MW, the switching produces a large benefit, and afterward the benefit declines. Switching from MP to LP has initially the same behavior as MP saving because LP is vented to the atmosphere by 24.4 t/h (14.6 MW). Before the venting reaches zero, this switching is equivalent to the MP saving (i.e., no contribution for power generation). The above-mentioned behaviors can be seen more clearly in marginal price charts (Figure 28). From Figure 28, one might be curious about the sudden changes in the financial benefits and the mar-

Figure 27. Financial benefits of steam saving and steam level switching for Example 2.

Figure 28. Marginal prices of saved and switched steam for Example 2.

2330


Figure 29. Example for site integration problem (Example 3). Table 4. Steam Turbine Models for Site 2 original turbine T4 T4 T4 T5 T5 T5 T6

VHP/HP HP/MP MP/LP VHP/HP HP/MP MP/LP MP/COND

(MW h)/t na nb nd ne nf ng

0.0723 0.0619 0.0784 0.0695 0.0734 0.2057

MW Wloss.a Wloss.b Wloss.d Wloss.e Wloss.f Wloss.g

Table 5. Summary of Operational Improvement for the Site Integration Problem integrated

1.2956 0.5553 4.5522 3.5196 2.0775 1.5111

ginal prices for steam saving and switching. These sudden changes are caused by changes in fuel composition in the boilers to satisfy the constraints relating to the maximum and minimum limits for burning a certain type of fuel in the boilers and burners. These step changes indicate that steam saving and switching could be difficult to achieve because a large amount of heat has to be saved or switched before the large benefits are obtained. The step changes also indicate that bottlenecks may exist in the boilers and burners. Actually, if the asphalt combustion limit can be expanded by 10 MW by modifying the flue gas desulfurization, the financial benefit can be improved by 0.5 MM$/year. Case Study 3: Site Integration Problem The R-curve analysis can be applied for investigating the benefit achieved by integration of two utility systems. A case study of two sites located next to each other is considered (Figure 29). Site 1 is the same as that in Example 1 (Figure 8). Site 2 has a similar configuration to Site 1; however, the pressure level of HP in Site 2 is higher than that of Site 1. Steam turbine models for Site 2 are given in Table 4. In Site 2, the power is balanced by the LP venting. Integrating these two sites was considered by connecting two LP mains and/or the MP mains. It is because LP and MP mains are operated at the same pressure between Site 1 and Site 2, and integration of these steam mains will not affect the process heat transfer. The following three assumptions were made for simplicity.

site 1 site 2 total R-ratio (current) ηcogen (current) (improved) steam shift MP: site 1w2 LP: site 2 w1 (t/h) power shift site 1 w 2 (MW) ∆Qfuel (by operation) (MW) fuel saving benefit (MM$/year)

LP

MP

LP and MP

0.338 0.335

0.337

0.337

0.337

0.647 0.619 0.671 0.625

0.632 0.661

0.632 0.659

0.632 0.665

11 13 0

0

8.7

2.7

1.4

0.4

32 13

0.8

3.9

4.0

11.5 23.5

21.4

26.3

3.4

4.2

1.8

3.8

(a) Only one fuel source (natural gas) is used. (b) Internal power shifting between Site 1 and Site 2 is possible. (c) Fuel and power prices are the same between Site 1 and Site 2. Generally speaking, integration of VHP mains is not practical. Because a VHP main usually runs only within a boiler house, connecting VHP pipe lines would be very long and costly. Furthermore, integration of the condensate mains could be difficult because of the contaminant problem. Three integration opportunities were examined. They are (a) LP main integration, (b) MP main integration, and (c) integration of both LP and MP mains Table 5 shows the financial benefits of fuel saving for each integration case. In the individual improvement for the two sites separately, one of the steam turbines is switched off in each site (T1 off in Site 1 and T4 off in Site 2) (Figure 30). For Site 2 about 4 MW of power is imported to satisfy the power demand. The total


Figure 30. Individual improvement.

Figure 33. LP and MP integration.

Figure 31. LP integration.

Figure 34. Grassroots R-curves for Example 2 (with the retrofit R-curve).

steam and power shifting. It is noticed that LP integration can achieve 80%-90% benefit of the LP and MP integration case. Therefore, the LP integration case could be the most promising option. However, detailed examination, which takes into account capital investment, is necessary to determine the final choice. Figure 32. MP integration.

financial benefit from the separate improvement (Site 1 + Site 2) is 1.8 MM$/year (Table 5). In contrast, the financial benefits from integrating two sites in the above three cases are 3.8, 3.4, and 4.2 MM$/year, respectively (Table 5). The site integration in terms of the steam mains can be examined by modifying eq 28 to obtain eq 70

Qheat.integrated(j) )

(

P

∑s ∑ i∈I j

D

j Q exhaust(i,s) +

j Q LD(i,s) ∑ i∈I j

)

(70)

where Qheat.integrated(j) is the heat demand at the intej is the exhausted heat grated steam main j. Qexhaust(i,s) of the steam turbine path i to j level main in site s. j is the heat content in the letdown steam path i Q LD(i,s) to j level main in site s. As seen in Figure 31, in the LP integration case, a part of LP demand in Site 1 is satisfied by the shifted LP from Site 2. Furthermore, T3 is switched off while T1 is on, which is opposite to the case of individual improvement. This is because T1 has a larger capacity in the MP extraction and a smaller capacity in the LP extraction. The deficit in LP for Site 1 is satisfied by importing LP from Site 2 while excess power generated in Site 1 is exported to Site 2. As a result, purchased power (4 MW) can be reduced to zero. Similarly, for the case of MP integration and LP&MP integration, the operation changes are shown in Figures 32 and 33. The main reason for greater benefit achieved by integration of two sites is that the steam turbines in Site 1 and Site 2 are utilized at more efficient loads by

Case Study 4: Screening of Debottlenecking Options R-Curve analysis can also be applied to screening of debottlenecking options. So far, the retrofit R-curve, which is based on the capacity limits and efficiencies of the existing system, was used for analysis of operational management and energy retrofit. For debottlenecking cases, the grassroots R-curve is more suitable, which is developed on the basis of the ideal configuration of a utility system. Example 2 (Figure 24) is used again for illustration. In this example, the capacities of the power generation equipment (the steam turbine and the gas turbine) are not big enough. Thus, expensive power is imported with a large amount. Therefore, debottlnecking this utility system by installing new equipment for power generation is considered. Grassroots R-Curve for Example 2. The grassroots R-curves for Example 2 are shown in Figure 34. These R-curves are developed on the basis of the ideal configuration shown in Figure 35. The following three cases are considered. Case 1: BPT and CT configuration (without GT). Case 2: BPT configuration with the existing GT. Case 3: BPT configuration with a new GT (the existing GT is replaced). The models for calculating the steam heat content of the HRSG placed after the new GT are given as follows:

Qheat.HRSG(HP) ) 0.735 × WGT.max - 0.585

(71)

Qheat.HRSG(MP) ≈ 0

(72)

Qheat.HRSG(LP) ) 0

(73)

2332


it indicates the possibility of installing a new steam turbine only for this expansion zone. Similarly, if the steam extraction capacity for a certain steam level is smaller than the ideal extraction capacity (∆ in Table 6), it may be possible to improve the cogeneration efficiency of the site by considering additional extractions on these steam levels. Generation of the Reduced Superstructure. The following rules can be used for building the reduced superstructure. Rule 1: The new steam turbine should cover all the expansion zones where the existing capacities are smaller than the ideal capacities. Rule 2: The extraction should be placed at every steam level where the existing extraction capacities are smaller than the ideal capacities. Using the above rules, the reduced superstructure for the three cases can be constructed (Figure 36). Models for the New Equipment. Models for the new steam turbine are implemented in the optimization program using the newly proposed decomposition (Figure 37). These models are obtained on the basis of the turbine models proposed by Mavromatis7 (Figure 37). For the new gas turbine, eqs 11 and 71 are used. New Objective Function and Constraints. The final configuration and the capacities of the new equipment required are determined through optimization. In this case, the objective function of the optimization is minimizing the total annual cost (TAC ) annual capital investment + operation cost), which is expressed as

Figure 35. Ideal configuration for Example-2.

The existing boilers are not considered candidates for modifications. The boiler duty requirements are calculated using eq 62. Furthermore, the additional constraints mentioned in case study 2 are assumed to be satisfied. These grassroots R-curves (Figure 34) show the maximum cogeneration efficiencies relating to a variety of power demands (at the fixed heat demand) for the above three cases. These curves indicate that all of the above three cases have great improvement potential compared to the existing system. For the current R-ratio (Rcurrent), Case 3 can achieve the highest cogeneration efficiency which could bring the biggest savings in operating cost. However, the decision for selecting the most economic option should be made on the basis of the tradeoff between capital cost and operating cost. This tradeoff is determined using superstructure-based mathematical optimization which is explained below. Generation of Reduced Superstructure. If a general superstructure is constructed to cover all the possible options for debottlenecking, the number of paths in a utility system with their combinations to be considered for optimization could become prohibitive. Therefore, it is necessary to generate a reduced superstructure. Assessment of the Existing Steam Turbine Capacity and Extraction. The maximum capacities of the existing steam turbine are compared to the ideal capacities determined from the grassroots R-curve (Table 6). If an existing steam turbine capacity in a certain expansion zone is smaller than the ideal capacity (∆ in Table 6),

Minimize TAC )

{∑ k

}

{∑

}

Qfuel(GAS.k) × CF(GAS) +

k

Qfuel(FO.k) × CF(FO) +

{∑

}

Qfuel(AS.k) × CF(AS) +

k

Qfuel.GT × CF(GAS) + Wimport × CP.import + f × (CST + CGT) (74)

where CST and CGT are the capital investments for a new steam turbine and a gas turbine, respectively. CST and CGT include the installation costs and the initial start-up cost. f is an annualizing factor. In this examination, the capital investments are calculated as shown in Table 7. Furthermore, eq 69 was modified to give eq 75.

{ {∑

new ) ∑ Qfuel(GAS.k) ∑i Qboiler(i) + ∑i Qboiler(i) k

{∑

}

Qfuel(FO.k) × ηb(FO) +

k

}

}

× ηb(GAS) +

Qfuel(AS.k) × ηb(AS) (75)

k

Table 6. Comparison of the Existing Capacity to the Ideal Capacity existing capacity

case 1 BPT + CT ideal capacity

GT (MW) 8.4 BPT VHP w HP (MW) 14.5 HP w MP (MW) 8.4 MP w LP (MW) 0.0 LP w COND (MW) 0.0 extract HP (t/h) 124.0 MP (t/h) 135.0 LP (t/h) 0.0 COND (t/h) 0.0 VHP required (t/h) 340.0 a ∆ ) Ideal capacity - existing capacity.

case 2 BPT + existing GT ∆a

ideal capacity

case 3 BPT + new GT

∆a

ideal capacity

∆a

0.0

-8.4

8.4

0.0

18.2

9.8

17.7 14.1 2.8 6.4

3.2 5.8 2.8 6.4

15.5 12.2 1.4 3.5

1.0 3.8 1.4 3.5

12.9 10.0 0.0 0.0

-1.6 1.6 0.0 0.0

108.6 145.1 0.0 57.5 311.2

-15.4 10.1 0.0 57.5 -28.8

98.3 144.9 0.0 31.0 274.2

-25.7 9.9 0.0 31.0 -65.8

85.3 144.6 0.0 0.0 229.9

-38.7 9.6 0.0 0.0 -110


Figure 36. Narrowed down superstructure for Example-2.

Figure 37. Decomposition of the new steam turbine for Example 2. Table 7. Calculation of Capital Investment steam turbine gas turbine HRSG

annualizing factor

0.73 (1 + 0.15X) + 0.0634W + cost (install) 100% MM$ (W in MW, X is number of extractions) 0.90WGT.max MM$ (includes installation cost) (WGT.max in MW) 0.013 × QSH/LMTDSH + 0.0065QEVA/LMTDEVA + 0.0065 × QPRE/LMTDPRE + 0.00591mHRSG + 0.001184mexhaust.GT + cost (install) 100% MM$ (QSH,EVA,PRE in MW, LMTDSH,EVA,PRE in °C, mHRSG in t/h, mexhaust.GT in kg/s) SH ) super heater, EVA ) evaporator, PRE ) preheater, LMTD ) logarithmic mean temperature difference f ) i/(1 + i)Life - 1 (f ) 0.1638 at interest (i) 10%, unit life 5 years)

Screening Results. The optimizations have been done for Cases 1-3 (Table 8). For all the cases, a common option was identified. This option indicates that the TAC is improved significantly by installing an extraction steam turbine. The option of installing a new GT (Case 3) is rejected because it is too expensive. It should be noted that the existing gas turbine (GT) is not used in Case 2 because compared to the new steam turbine, the existing GT is too small to be used ef-

ficiently. As a test, a little larger existing GT (WGTMW) was assumed (Table 8). In this test case, the use of the hypothetical larger GT is a better option.

.max)18

6. Conclusions The R-curve analysis has been developed for analyzing the energy system in process industries. The applications of the R-curve analysis are summarized in

2334


Table 8. Screening Summary of Debottlenecking for Example 2

Figure 38. The R-curve indicates maximum cogeneration efficiencies for different power-to-heat ratios, and it also indicates the optimal use of equipment in the utility system. First, the retrofit R-curve is developed on the basis of the existing utility systems, and it shows how the utility system should be operated and which steams are worth saving or switching. Because the retrofit R-curve reflects the capacities and the efficiencies of the existing utility system, it is possible to achieve the optimal operating points on the R-curve without capital investment for the utility systems. For steam saving and switching, the marginal prices can be calculated using the R-curves. These marginal prices can be provided as the basis for energy retrofit projects (e.g., heat exchanger network retrofit). Second, the grassroots R-curve is developed on the basis of the ideal configuration of the utility system, and

it shows possible debottlenecking options, which are used for generating reduced superstructure. With optimization of the superstructure, promising debottlenecking options can be identified effectively. Furthermore, the R-curve analysis can deal with highly complex systems. Problems with multiple boilers, multiple fuels, site integration, and so forth can be handled effectively by using the R-curve analysis. Overall, the R-curve analysis can provide insights and solutions for many problems encountered in managing industrial energy systems with little engineering effort. Nomenclature Abbreviations AS ) asphalt fuel BFW ) boiler feedwater BPT ) back-pressure turbine CT ) condensing turbine FO ) fuel oil GAS ) gas fuel GT ) gas turbine HRSG ) heat recovery steam generator Mathematical Symbols

Figure 38. Summary of R-curve analysis.

ηb(i) ) boiler efficiency for fuel i ηcogen.export ) cogeneration efficiency for power exporting ηcogen.improved ) cogeneration efficiency for improved operation power ηcogen.save ) cogeneration efficiency for power saving steam ηcogen.save ) cogeneration efficiency for steam saving steam ηcogen.switch ) cogeneration efficiency for steam switching $(k) ) bilinear term of M(i) and S(t) Bexport ) financial benefit of power exporting (MM$/year) Bfuel ) financial benefit of fuel saving (MM$/year) cp(i) ) specific heat of saturated water for path i ((MW h)/ (t °C)) CF(i) ) price of fuel i ($/(kW h)) CGT ) capital investment for a gas turbine system includes HRSG (MM$) CP.export ) price of exporting power ($/(kW h)) CP.import ) price of importing power ($/(kW h)) CST ) capital investment for steam turbine (MM$) f ) annualizing factor h(j) ) specific enthalpy at condition j ((MW h)/t) hsat(j) ) specific enthalpy of saturated water at level j (MWh/t) ∆His ) isentropic enthalpy change between the steam turbine inlet and outlet ((MW h)/t)

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2335 LMASS(j) ) mass loss ratio versus the whole condensate in level j LHEAT(j) ) heat loss ratio versus the condensate return from level j mexhaust.GT ) flow rate of gas turbine exhaust (kg/s) mmax ) maximum steam flow rate of steam turbine (t/h) M(i) ) steam flow rate of steam turbine path i (t/h) Mmax(i) ) maximum limit of steam flow rate for path i (t/h) Mmin(i) ) minimum limit of steam flow rate for path i (t/h) Mprice ) marginal price of steam (MM$/MW) n(i) ) specific power generation of steam turbine path i ((MW h)/t) na-g ) specific power generation of steam turbines a-g ((MW h)/t) qin(i) ) heat content in the inlet of path i including super and latent heat ((MW h)/t) qout(i) ) heat content in the exhaust of path i including super and latent heat ((MW h)/t) Qboiler(i) ) boiler duty required for path i ((MW h)/t) new ) boiler duty required for new steam turbine path Qboiler(i) i ((MW h)/t) QEX(j) ) heat demand in steam heater at level j (MW) Qfuel(i) ) fuel consumption in boiler for path i ((MW h)/t) Qfuel(i,k) ) consumption of fuel i in boiler k (MW) Qfuel.GT ) fuel consumption of gas turbine (MW) Qfuel.import ) monetary equivalent amount of fuel for importing power (MW) Qheat(j) ) heat demand of steam in level j (MW) Qheat.current ) current total heat demand (MW) Qheat.HRSG(j) ) heat content in generated steam from HRSG (MW) Qheat.integrated(j) ) heat demand at the integrated steam main j (MW) Qheat.save ) heat demand after saving (Qheat.current - ∆Qheat) (MW) QPR(j) ) heat demand of steam in level j for processes (MW) 0 ) specific boiler duty for path i ((MW h)/t) Qboiler(i) 0 ) specific fuel consumption in boiler for path i Qfuel(i) ((MW h)/t) j Q exhaust(i,s) ) heat content in turbine exhaust for path i linking level j in site s (MW) j Q LD(i,s) ) heat content of a letdown path i linking level j in site s (MW) r Qfuel ) total fuel consumption from relaxed problem (MW) ∆Qheat ) steam heat saving (MW) ∆Qfuel ) fuel saving (MW) Rcurrent ) current R-ratio Rexport ) R-ratio for power exporting power Rsave ) R-ratio for power saving steam ) R-ratio for steam saving Rsave S(t) ) inverse of summation of steam flow rates (h/t) Smax(t) ) maximum limit of S(t) (h/t) Smin(t) ) minimum limit of S(t) (h/t) b ) new boundary of S(t) (h/t) S(t) r ) S(t) determined from relaxed problem (h/t) S(t) Texhaust.GT ) exhaust temperature of gas turbine (°C) Ts.ave ) average saturation temperature between the turbine inlet and outlet (°C) Ts.in ) saturation temperature of the steam turbine inlet (°C)

Ts.out ) saturation temperature of the steam turbine outlet (°C) ∆Ts(i) ) saturation temperature difference between the inlet and outlet of path i (°C) W(i) ) power generation of steam turbine path i (MW) Wa-g ) power generation of steam turbines a-g (MW) Wexport ) exporting power (MW) WGT ) power generation of gas turbine (MW) WGT.max ) maximum power generation of gas turbine (MW) Wimport ) importing power (MW) Wloss.a-g ) internal loss of steam turbines a-g (MW) WL(i) ) internal loss of steam turbine path i (MW) WLmax(i) ) maximum limit of internal loss for path i (MW) WLmin(i) ) minimum limit of internal loss for path i (MW) W ˜ L(i) ) newly defined internal loss which is controlled by binary variable Z(i) (MW) Z(i) ) binary variable for path on/off selection ZX(t) ) binary variable for turbine on/off selection

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Received for review August 5, 1999 Revised manuscript received April 10, 2000 Accepted April 17, 2000 IE9905916

R-Curve Concept and Its Application for Industrial Energy

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