- Email: [email protected]
A new thermodynamic model for shaftwork targeting on total sites
Applied Thermal Engineering 25 (2005) 961–972 www.elsevier.com/locate/apthermeng
A new thermodynamic model for shaftwork targeting on total sites M. Sorin *, A. Hammache CANMET Energy Technology Centre-Varennes, 1615 Lionel-Boulet, P.O. Box 4800, Varennes, Que´bec, Canada J3X 1S6 Received 15 January 2004; accepted 12 June 2004
Abstract The purpose of the paper is to introduce a targeting model based on a new thermodynamic insight on cogeneration in general and Rankine cycle in particular. The insight permits to express the ideal shaftwork of a cogeneration unit through the outlet heat load and the difference in Carnot factors between the heat source and heat sink for the given inlet temperature of the heat source. The deviation from the ideal shaftwork to the real one is assessed by using the traditionally turbine isentropic efficiency. Finally the new model allows targeting fuel consumption, cooling requirement and shaftwork production with high accuracy and visualizing them directly as special segments on the T–H diagram. A modified Site Utility Grand Composite Curve (SUGCC) diagram is proposed and compared to the original SUGCC. The shape of the right hand side of the diagram above site pinch is the same, however, below site pinch it is shifted to the left by an amount equal to shaftwork production below site pinch. Above site pinch VHP consumption is also corrected to account for shaftwork production above site pinch that is represented by segments rather than areas on the left hand side of the T–H diagram. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Total sites; Thermodynamic model; Cogeneration targeting; Central utility system
*
Corresponding author. Tel.: +1 450 652 3513; fax: +1 450 652 0999. E-mail address: [email protected] (M. Sorin).
1359-4311/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.06.021
962
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
1. Introduction Many industrial processes operate within ‘‘Total Sites’’, where they are serviced and linked through a central utility system. Established procedures for heat and power integration work well for optimizing each process in isolation. However, greater benefits in terms of energy and capital cost can be obtained by looking at the entire site. Total site integration addresses the task of optimizing each process and the utility system in the context of the overall site. One of the important tasks for the utility systemÕs design is targeting fuel consumption, shaftwork production and cooling requirement ahead of design. A number of pinch analysis models have been proposed for the early estimation of shaftwork production for utility systems using steam turbines. The most important is the temperature enthalpy (T–H) model proposed by Raissi [1]. The model is based on the observation that power is proportional to the heat load of steam and the difference between the inlet and outlet saturation temperatures. Despite the fact that the model provides a powerful graphical tool that helps understand the interactions between site fuel, heat recovery and cogeneration, it results to errors of up to 30% compared with simulation [2]. It is due mainly to the inaccuracy linked to the above-mentioned observation. Moreover given that the power output on the T–H diagram is presented by a rectangular area the model does not allow visualizing the site fuel and cooling targets.
2. Targeting methodology As mentioned in the introduction section, a shaftwork targeting methodology was proposed by Raissi [1]. The method uses the Site Utility Grand Composite Curve (SUGCC), as illustrated in Fig. 1, which represents another form of the site composite curves [3]. Fig. 1 shows the SUGCC of an example that will be illustrated later. It expresses mainly the indirect heat recovery through
Site Utility Grand Composite Curve
Saturation temperature (oC)
350 300 250 200 150 100 50 0 0
5
10 15 Heat Load (MW)
Fig. 1. Site utility grand composite curve.
20
25
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
963
steam mains. The SUGCC are obtained directly from the site composite curves by representing on temperature–enthalpy axes each steam main by its saturation temperature and steam generation and usage loads, respectively from the source and sink profiles of the site composites. The differences between steam generation and steam usage will set the VHP demand or the supply heat available at each main. The SUGCC is generally divided by the ‘‘Total Site Pinch’’ into two separate sections, above and below the site pinch. Above site pinch the system is characterized by heat deficit that should be supplemented by an external hot utility that is VHP steam. Below site pinch the system is characterized by excess heat that should be removed by a cooling agent (cooling water, VLP generation, etc.). The SUGCC proposed by Raissi [1] provides the designer with a powerful tool to determine potentials for cogeneration, fuel and cooling requirement prior to design. As mentioned in the introduction, the targeting method proposed by Raissi [1] is based on an observation that power is proportional to the heat load of steam and the difference between the inlet and outlet saturation temperatures. It is then possible to target shaftwork production from the SUGCC by using the areas defined by heat loads and saturation temperatures and to multiply the areas by a proportionality constant conversion factor obtained through a steam turbine simulation. The proposed method has two drawbacks: first it could be demonstrated that power is not linear to the saturation temperature differences between inlet and outlet pressures; second, the VHP and cooling targets cannot be accurately visualized on the SUGCC. If we express the overall energy and exergy balances for a steam turbine Rankine cycle (neglecting the pump) we have: W_ ¼ Q_ in Q_ out
ð1Þ
Q_ in hH ¼ W_ þ Q_ out hL þ I_
ð2Þ
where W_ is the real shaftwork rate produced from the turbine, Q_ in is the heat supply rate by the boiler, Q_ out is the heat exhaust rate from the turbine, hH is the Carnot factor for the supply heat, hL is the Carnot factor for the exhaust heat and I_ the irreversibility rate due to the so-called ‘‘frictional reheat’’. The Carnot factor is defined as the exergy factor of heat: T0 ð3Þ h¼1 T where T is defined as the average thermodynamic temperature for the supply heat, TH, or the exhaust heat, TL. For simplicity letÕs assume an ideal expansion process so that: I_ ¼ 0, by combining Eqs. (1) and (2) we get an expression for the ideal shaftwork rate, W_ id as hH hL;is _ _ ð4Þ W id ¼ Qout;id 1 hH Eq. (4) expresses the ideal shaftwork rate of a cogeneration unit through the exhaust heat load rate and the difference in Carnot factors between the heat source and heat sink for the given inlet temperature of the heat source. The deviation from the ideal shaftwork to the real one is assessed by using the traditionally turbine isentropic efficiency.
964
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
It is obvious from Eq. (4) and the Carnot factor, Eq. (3), that power is a non-linear function of the input and output temperatures but rather a linear function of the difference in Carnot factors between the heat source and heat sink for the given inlet temperature of the heat source. Taking into account this thermodynamic insight on cogeneration in general and Rankine cycle in particular, we propose a new targeting methodology with two objectives in mind. The first objective is to develop a procedure for a more accurate shaftwork target and the second objective is to modify the SUGCC in order to visualize on the same diagram VHP consumption, thus fuel consumption, shaftwork production and cooling targets. 2.1. Rankine fundamentals As mentioned previously, the ideal shaftwork rate produced by a steam turbine is expressed as: hH hL;is _ _ ) W_ id ð1 hH Þ ¼ ðQ_ out DH_ r ÞðhH hL;is Þ ð5Þ W id ¼ Qout;id 1 hH where subscripts H and L corresponds to states 1 and 2 in Fig. 2, Q_ out the exhaust heat demand rate and DH_ r ¼ H_ 2 H_ 20 represents the frictional reheat rate or the heat exhaust deviation from an ideal expansion. DH_ r ¼ Q_ out Q_ out;id ¼ W_ id W_
ð6Þ
The steam turbine isentropic efficiency is defined as (see Fig. 2): gis ¼
h1 h2 h1 h20
ð7Þ
The real power is expressed as: W_ ¼ gis W_ id
ð8Þ
600 1 500
O
T ( C)
400 2
300 200 100 0
2’ 3
2
4 6 Entropy (kJ/kg/OC)
8
Fig. 2. Steam turbine expansion on a T–s diagram.
10
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
965
The temperatures expressed in the Carnot factors represent average thermodynamic temperatures for inlet and outlet expansion conditions of the steam turbine and defined as (see Fig. 2): TH ¼
h1 h3 s1 s3
T Lis ¼
h 20 h 3 s1 s3
ð9Þ ð10Þ
where h and s are the specific enthalpy and entropy respectively. Combining Eqs. (5), (6) and (8) and the definition expressed by Eq. (3), we derive an expression for the real shaftwork rate, W_ , as a function of the exhaust heat load rate, Q_ out , the isentropic efficiency, gis and the average thermodynamic temperatures given by Eqs. (9) and (10). W_ ¼ Q_ out gis
T H T Lis ð1 gis ÞT H þ gis T Lis
ð11Þ
Q_ out represents also a transiting heat rate at the turbine exhaust, Q_ tr (see Brodyansky et al. [4]), which could be used as process heat demand or further expanded to lower pressure levels for more cogeneration. It is also possible to express the real shaftwork rate as a function of heat supply rate, Q_ supply at the turbine inlet condition that is: T H T Lis W_ ¼ Q_ supply gis TH
ð12Þ
Finally, by using these insights and the SUGCC tool as proposed by Raissi [1], a new model is proposed to allow targeting fuel consumption, cooling requirement and shaftwork production with high accuracy and visualize the targets directly as special segments on a modified SUGCC diagram. It should be noticed that strictly speaking we should use the average thermodynamic temperatures, TH and TLis, on the modified SUGCC diagram and not the saturation temperatures, THsat and TLsat, however we keep the saturation temperatures on the modified SUGCC, but compute TH and TLis as the parameters of THsat and TLsat. The latter are fixed by heat recovery. 2.2. Illustrative example The methodology will be explained more precisely through an illustrative example taken from the literature [2]. It should be noted that the situation below site pinch was added to the example to illustrate a specific characteristic below pinch in the modified SUGCC. The model was tested on a utility steam system comprising of four steam levels above site pinch and one steam level along with a condensing header below site pinch. Above site pinch, the very high pressure (VHP) steam is produced in the boiler house at 500 °C and 90 bar. The high-, medium- and low-pressure levels correspond to 46 bar, 15.5 bar and 2.7 bar, respectively. Below site pinch, steam is generated at 1.5 bar and the condensing header (CW) is at 0.05 bar. Table 1 presents the various data parameters.
966
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
Table 1 Problem data Parameters
Units
Header1
Pressure Tsupply Saturation Temperature, Tsat
Bar °C °C
90 500 303
Header2
Header3
46
Header4
Header5
2.7
1.5
15.5
259
200
130
111
CW 0.05 33
The model distinguishes two separate sections, above site pinch and below site pinch, of the SUGCC diagram as shown in Fig. 1. For both situations, the idea is to start the computations from site pinch and move away to the hot and cold utilities to target VHP steam consumption, thus fuel consumption, shaftwork production and cooling requirements. The first step of the methodology is to divide the SUGCC into several domains characterized by enthalpy and temperature intervals as shown in Fig. 3 for above site pinch. Fig. 3 shows the SUGCC above site pinch. The SUGCC is divided into several enthalpy intervals along the x-axis (I) and temperature intervals along the y-axis (J ). The number of intervals depends on the SUGCC shape as observed in Fig. 3. The outlet heat loads Q_ heat ðI; J Þ that should be satisfied at heat interval I along temperature interval J are respectively Q_ heat ð1;1Þ ¼ 12:5 MW, Q_ heat ð1;2Þ ¼ 3:75 MW and Q_ heat ð2;3Þ ¼ 6:88 MW. The values are extracted directly from the SUGCC. For each domain we simulate an elementary steam turbine in order to target maximum shaftwork production by using Eq. (11) or Eq. (12) as shown in Fig. 4 for above site pinch. Steam flows through the elementary turbines along each enthalpy interval cascading from upper temperature intervals to lower temperature intervals. Fig. 4 shows the methodology used to target maximum shaftwork that could be produced by the utility system above site pinch. Elementary steam turbines are disposed in each domain (I, J ) where steam flows along each enthalpy interval I. 350 Saturation temperature ( C)
12.5 MW
o
300
Header1
Qsupply =10.63 MW
(3,1) 250
200
Header2 (2,1)
(2,2)
(1,1)
(1,2)
Header3
(2,3) Qdemand =6.88 MW
150 J
Header4 12.5 MW Qdemand =16.25 MW
100
50
0
5 I
3.75 MW
10 15 Heat Load (MW)
20
Fig. 3. Site utility grand composite curve above pinch.
25
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
967
Fig. 4. Configuration of the utility system for maximum shaftwork production.
The second step of the methodology is to identify for each enthalpy interval the different inlet and outlet conditions at the various steam turbine ports starting from the supply temperature intervals where inlet parameters are known. To do so, we assume a constant isentropic efficiency for all turbines and use steam tables or simple steam turbine expansion simulations. We could then compute the different average high and low thermodynamic temperatures by using Eqs. (9) and (10) (see Appendix A for details). The third step of the methodology is to calculate starting from site pinch, where heat demands and supplies are known, the shaftwork production using either Eq. (11) or (12) respectively above and below site pinch. Above site pinch and as we move upward to the hot utility level we must compute at each temperature interval a new transiting heat by adding to the heat demand the shaftwork production at lower temperature intervals. Then we compute the shaftwork production for the considered temperature interval using Eq. (11). Below site pinch and as we move downward to the cold utility level we must compute at each temperature level a new supply heat by subtracting to the heat supply of the previous temperature level the shaftwork production at higher temperature intervals. Then we compute the shaftwork production for the considered temperature interval using Eq. (12). The transiting heat and shaftwork generated in each domain are then computed (see Appendix A for details). A special attention should be given to the interpretation of the transiting heat along the temperature and enthalpy intervals. At this point we introduce the modified SUGCC to illustrate how transiting energy is computed above site pinch. The modified SUGCC is introduced to visualize on the same diagram heat recovery through the steam mains, shaftwork production and actual VHP consumption, thus fuel consumption, and cooling requirements. We include to the original SUGCC a new power axis in an opposite direction to heat load axis for visualization convenience. For a simple explanation purpose we consider only a part of a modified SUGCC where heat demand at a specific enthalpy interval should be satisfied by a hot utility cascading steam from VHP to a lower temperature through two temperature intervals as shown in Fig. 5.
968
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972 T-h Diagram Site Utility Grand Composite Curve
Saturation temperature (o C)
Q VHP
1
WI
1
WI
2 4
2’
o
2
T ( C)
(Qtr) I
W II
WII 6
(Q tr )II
5
3 5 6
5’
8 7
Q demand
Power (MW)
Heat Load (MW)
Enthalpy (kJ/kg)
Fig. 5. Illustration of the methodology used to target shaftwork production (Zones (3,1) and (2,1) of Fig. 4).
The problem in hand is to satisfy heat demand 7–8 by the outlet of the component turbine II, then _ 5 h6 Þ ðQ_ tr ÞII ¼ Q_ demand ¼ mðh If (TH)II and the isentropic temperature (TLis)II are known, then shaftwork production by turbine II, W_ II , is given by Eq. (11). The transiting heat for temperature interval I is then: ðQ_ tr ÞI ¼ ðQ_ tr ÞII þ W_ II If (TH)I and the isentropic temperature (TLis)I are known, then shaftwork production by turbine I, W_ I , is given by Eq. (11). Therefore, the overall VHP demand is Q_ VHP ¼ Q_ demand þ W_ I þ W_ II Fig. 4 and Appendix A show detailed computations of the transiting heat above site pinch and supply heat below site pinch. Finally, Fig. 6 describes the modified SUGCC showing the targets directly as special segments on the T–H diagram. We should emphasize that compared to the original SUGCC proposed by Raissi [1], the shape of the right hand side of the diagram above site pinch is the same, however, below site pinch it is shifted to the left by an amount equal to shaftwork production below site pinch. VHP consumption is also corrected to account for shaftwork production above site pinch that is represented by segments rather than areas on a T–W section opposite to the T–H section. Below site pinch we notice that excess heat could also be used for shaftwork production. Therefore, the excess heat is a summation of the potential shaftwork production plus the cooling load as shown in Fig. 6. The original SUGCC proposed by Raissi [1] omits to distinguish this aspect and gives an erroneous target for the cooling load. In order to represent accurately the cooling load requirement the original SUGCC below pinch should be shifted to the left of the T–H diagram by an amount equal to shaftwork production below pinch.
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
969
Site Utility Grand Composite Curve QVHP = 17.174 350 QAP =12.5
WAP = 4.674
Saturation temperature (oC)
300
250
200
150 WBP = 0.672 100
50 Q CW = 4.328 0 10
5
0
5
10
15
20
Power (MW)
25
Heat Load (MW)
Fig. 6. The new representation of the SUGCC.
The problem above site pinch has been simulated using steam properties and steam flows through the various turbines. The results are shown in Fig. 7. Compared to the results obtained through simulation it is observed that the new model based on the transiting heat concept gives a very good estimation of the maximum potential shaftwork production using the SUGCC. The deviation from the simulation is 0.2% for the new T–H model. 350 12.5 MW 21.86 t/h W(3,1) = 0.877
10.63 MW
o
Saturation temperature ( C)
300
21.86 t/h
250
0.82 t/h
6.00 t/h
0.92 t/h W(2,1) = 1.131
12.73 t/h W(2,3) = 0.547
W(2,2) = 0.285 13.56 t/h
200
Q heat = 6.88
20.12 t/h W(1,1) = 1.439
W(1,2) = 0.403
150 20.12 t/h
6.92t/h
Q heat =12.50
100
Q heat =3.75 3.447 MW
0.688 MW
Total Produced Work =4.682MW 50 0
5
10
15
20
Heat Load (MW)
Fig. 7. Results of the simulation (Section above site pinch).
25
970
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
Table 2 Comparison between the existing model [1] and simulation Interval (I, J)
CF
Shaftwork (MW) existing model
Shaftwork (MW) simulation
Deviation(%)
(1,1) (2,3) (3,1)
0.001645 0.001347 0.001595
5.02 4.11 4.87
4.68 4.68 4.68
7.2 12.2 3.9
Table 3 Comparison between the new targeting model and the existing model [1] Target
Existing model [1]
New model
Shaftwork Areas on SUGCC Segments on Modified SUGCC VHP Not accurately Visualized as visualized a segment
Cooling
Not accurately visualized
Visualized as a segment
Fuel
Cannot be visualized
Can be visualized as a segment
Comments The new model exhibits small deviations with simulation compared to the model by Raissi [1]. The new model takes into account the overall amount of VHP consumption for heat requirements and shaftwork production. The model by Raissi [1] visualizes only VHP consumption for heat requirements. The model by Raissi [1] does not visualize accurately cooling requirement if shaftwork production is considered. The new model takes into account this aspect by shifting to the left the SUGCC below the site pinch. The new model gives an accurate VHP consumption, thus fuel consumption can be visualized as a segment by dividing VHP consumption with the boiler house efficiency.
The existing T–H model proposed by Raissi [1] is based on an assumption that power is proportional to the area delimited by the exhaust heat output and saturation temperature difference between inlet and outlet: W_ ¼ CF Area Q_ DT sat where CF is assumed constant for a particular SUGCC. Table 2 summarizes the results for the overall shaftwork production above site pinch for different CF values computed using different intervals (I, J). We notice that the conversion factor in the existing T–H model has an important impact on the shaftwork target and could lead to large discrepancies with simulation if it is not well assessed. Table 3 summarizes the major differences between the existing model and the new model.
3. Conclusion A new model for targeting fuel consumption, cooling water requirement and shaftwork production was developed using an original thermodynamic insight on cogeneration. The model permits to visualize the targets directly as special segments on a modified SUGCC T–H diagram. Compared to the original SUGCC, the shape of the right hand side of the modified SUGCC T–H diagram above site pinch is the same, however, below site pinch it is shifted to the left by an amount
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
971
equal to shaftwork production below site pinch. Above site pinch VHP consumption accounts for heat requirements and shaftwork production. A new characteristic of the modified SUGCC diagram is that segments, on a T–W section opposite to the T–H section, rather than areas represent shaftwork production above site pinch. Compared to the original methodology proposed by Raissi [1], shaftwork targets present small deviations from simulation. Appendix A. Above Site Pinch Assuming a constant isentropic turbine efficiency gis and from using the output heat load of the turbine, the shaftwork production in each enthalpy-temperature domain (I, J) can be computed by using Eq. (11). A special attention should be given to the computation of the transiting heat along each domain starting from the steam level demands to upper levels. For heat interval J ¼ 3;
Q_ tr ð2; 3Þ ¼ Q_ heat ð2; 3Þ
For heat interval J ¼ 2;
Q_ tr ð1; 2Þ ¼ Q_ heat ð1; 2Þ
and Q_ tr ð2; 2Þ ¼ Q_ tr ð1; 2Þ þ W_ ð1; 2Þ
For heat interval J ¼ 1;
Q_ tr ð1; 1Þ ¼ Q_ heat ð1; 1Þ
and Q_ tr ð2; 1Þ ¼ Q_ tr ð1; 1Þ þ W_ ð1; 1Þ
In order to satisfy the required heat demands in heat intervals 2 and 3 and generate the shaftwork in these intervals, it is necessary to import heat from a higher temperature level in heat interval 1. Due to energy balance, the amount of heat imported is exactly equal to the shaftwork generated in intervals 2 and 3. Q_ tr ð3; 1Þ ¼ Q_ tr ð2; 1Þ þ W_ ð2; 1Þ þ W_ ð1; 2Þ þ W_ ð2; 2Þ þ W_ ð2; 3Þ The average high and low isentropic temperatures at each temperature–enthalpy domain are computed from the specific enthalpy and specific entropy values obtained through simple turbine simulation or directly from steam tables. For each heat interval J and temperature interval I, the average hot and low isentropic temperatures are computed by using Eqs. (9) and (10). The values in Table A.1 are obtained by assuming a constant isentropic efficiency gis of 0.7. Table A.1 Transiting heat and shaftwork potential at each temperature–enthalpy domain Case Above Site Pinch I=3 I=2 I=1
J=1 W_ ð3; 1Þ ¼ 0:873 Q_ tr ð3; 1Þ ¼ 15:08 W_ ð2; 1Þ ¼ 1:131 Q_ tr ð2; 1Þ ¼ 13:94 W_ ð1; 1Þ ¼ 1:439 Q_ tr ð1; 1Þ ¼ 12:50
J=2
J=3
W_ ð2; 2Þ ¼ 0:282 Q_ tr ð2; 2Þ ¼ 4:15 W_ ð1; 2Þ ¼ 0:402 Q_ tr ð1; 2Þ ¼ 3:75
W_ ð2; 3Þ ¼ 0:546 Q_ tr ð2; 3Þ ¼ 6:88
972
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
Table A.2 Supply heat and shaftwork potential at each temperature–enthalpy domain Case below site pinch I=1
J=1 W_ ð1; 1Þ ¼ 0:672 Q_ supply ð1; 1Þ ¼ 0:672
The maximum shaftwork production above the total site pinch is then W_ AP ¼ 4:674 MW and the total VHP requirement is Q_ VHP ¼ 12:5 þ 4:674 ¼ 17:174 MW. Assuming a boiler efficiency of 0.9, the targeted fuel consumption will be 19.08 MW.
Appendix B. Below Site Pinch Below the site pinch the SUGCC is characterized only by one enthalpy-temperature domain I = 1 and J = 1, and a heat supply Q_ supply ð1; 1Þ ¼ 5 MW. The shaftwork production is computed by using Eq. (12). The values in Tables A.2 are obtained by assuming a constant isentropic efficiency gis of 0.7. The maximum shaftwork production below the total site pinch is W_ BP ¼ 0:672 MW and the cooling requirement is Q_ CW ¼ 4:328 MW. References [1] K. Raissi, Total Site Integration, PhD Thesis, University of Manchester Institute of Science and Technology (UMIST), Process Integration Department, UK, 1994. [2] S.P. Mavromatis, A.C. Kokossis, Conceptual optimisation of utility networks for operational variations-I. Targets and level optimisation, Chemical Engineering Science 53 (8) (1998) 1585–1606. [3] J. Klemes, V.R. Dhole, K. Raissi, S.J. Perry, L. Puigjaner, Targeting and design methodology for reduction of fuel power and CO2 on total sites, Applied Thermal Engineering 17 (8–10) (1997) 993–1003. [4] V.M. Brodyansky, V.M. Sorin, P. LeGoff, The Efficiency of Industrial Processes: Exergy Analysis and Optimization, Elsevier, 1994.
A new thermodynamic model for shaftwork targeting on total sites M. Sorin *, A. Hammache CANMET Energy Technology Centre-Varennes, 1615 Lionel-Boulet, P.O. Box 4800, Varennes, Que´bec, Canada J3X 1S6 Received 15 January 2004; accepted 12 June 2004
Abstract The purpose of the paper is to introduce a targeting model based on a new thermodynamic insight on cogeneration in general and Rankine cycle in particular. The insight permits to express the ideal shaftwork of a cogeneration unit through the outlet heat load and the difference in Carnot factors between the heat source and heat sink for the given inlet temperature of the heat source. The deviation from the ideal shaftwork to the real one is assessed by using the traditionally turbine isentropic efficiency. Finally the new model allows targeting fuel consumption, cooling requirement and shaftwork production with high accuracy and visualizing them directly as special segments on the T–H diagram. A modified Site Utility Grand Composite Curve (SUGCC) diagram is proposed and compared to the original SUGCC. The shape of the right hand side of the diagram above site pinch is the same, however, below site pinch it is shifted to the left by an amount equal to shaftwork production below site pinch. Above site pinch VHP consumption is also corrected to account for shaftwork production above site pinch that is represented by segments rather than areas on the left hand side of the T–H diagram. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Total sites; Thermodynamic model; Cogeneration targeting; Central utility system
*
Corresponding author. Tel.: +1 450 652 3513; fax: +1 450 652 0999. E-mail address: [email protected] (M. Sorin).
1359-4311/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.06.021
962
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
1. Introduction Many industrial processes operate within ‘‘Total Sites’’, where they are serviced and linked through a central utility system. Established procedures for heat and power integration work well for optimizing each process in isolation. However, greater benefits in terms of energy and capital cost can be obtained by looking at the entire site. Total site integration addresses the task of optimizing each process and the utility system in the context of the overall site. One of the important tasks for the utility systemÕs design is targeting fuel consumption, shaftwork production and cooling requirement ahead of design. A number of pinch analysis models have been proposed for the early estimation of shaftwork production for utility systems using steam turbines. The most important is the temperature enthalpy (T–H) model proposed by Raissi [1]. The model is based on the observation that power is proportional to the heat load of steam and the difference between the inlet and outlet saturation temperatures. Despite the fact that the model provides a powerful graphical tool that helps understand the interactions between site fuel, heat recovery and cogeneration, it results to errors of up to 30% compared with simulation [2]. It is due mainly to the inaccuracy linked to the above-mentioned observation. Moreover given that the power output on the T–H diagram is presented by a rectangular area the model does not allow visualizing the site fuel and cooling targets.
2. Targeting methodology As mentioned in the introduction section, a shaftwork targeting methodology was proposed by Raissi [1]. The method uses the Site Utility Grand Composite Curve (SUGCC), as illustrated in Fig. 1, which represents another form of the site composite curves [3]. Fig. 1 shows the SUGCC of an example that will be illustrated later. It expresses mainly the indirect heat recovery through
Site Utility Grand Composite Curve
Saturation temperature (oC)
350 300 250 200 150 100 50 0 0
5
10 15 Heat Load (MW)
Fig. 1. Site utility grand composite curve.
20
25
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
963
steam mains. The SUGCC are obtained directly from the site composite curves by representing on temperature–enthalpy axes each steam main by its saturation temperature and steam generation and usage loads, respectively from the source and sink profiles of the site composites. The differences between steam generation and steam usage will set the VHP demand or the supply heat available at each main. The SUGCC is generally divided by the ‘‘Total Site Pinch’’ into two separate sections, above and below the site pinch. Above site pinch the system is characterized by heat deficit that should be supplemented by an external hot utility that is VHP steam. Below site pinch the system is characterized by excess heat that should be removed by a cooling agent (cooling water, VLP generation, etc.). The SUGCC proposed by Raissi [1] provides the designer with a powerful tool to determine potentials for cogeneration, fuel and cooling requirement prior to design. As mentioned in the introduction, the targeting method proposed by Raissi [1] is based on an observation that power is proportional to the heat load of steam and the difference between the inlet and outlet saturation temperatures. It is then possible to target shaftwork production from the SUGCC by using the areas defined by heat loads and saturation temperatures and to multiply the areas by a proportionality constant conversion factor obtained through a steam turbine simulation. The proposed method has two drawbacks: first it could be demonstrated that power is not linear to the saturation temperature differences between inlet and outlet pressures; second, the VHP and cooling targets cannot be accurately visualized on the SUGCC. If we express the overall energy and exergy balances for a steam turbine Rankine cycle (neglecting the pump) we have: W_ ¼ Q_ in Q_ out
ð1Þ
Q_ in hH ¼ W_ þ Q_ out hL þ I_
ð2Þ
where W_ is the real shaftwork rate produced from the turbine, Q_ in is the heat supply rate by the boiler, Q_ out is the heat exhaust rate from the turbine, hH is the Carnot factor for the supply heat, hL is the Carnot factor for the exhaust heat and I_ the irreversibility rate due to the so-called ‘‘frictional reheat’’. The Carnot factor is defined as the exergy factor of heat: T0 ð3Þ h¼1 T where T is defined as the average thermodynamic temperature for the supply heat, TH, or the exhaust heat, TL. For simplicity letÕs assume an ideal expansion process so that: I_ ¼ 0, by combining Eqs. (1) and (2) we get an expression for the ideal shaftwork rate, W_ id as hH hL;is _ _ ð4Þ W id ¼ Qout;id 1 hH Eq. (4) expresses the ideal shaftwork rate of a cogeneration unit through the exhaust heat load rate and the difference in Carnot factors between the heat source and heat sink for the given inlet temperature of the heat source. The deviation from the ideal shaftwork to the real one is assessed by using the traditionally turbine isentropic efficiency.
964
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
It is obvious from Eq. (4) and the Carnot factor, Eq. (3), that power is a non-linear function of the input and output temperatures but rather a linear function of the difference in Carnot factors between the heat source and heat sink for the given inlet temperature of the heat source. Taking into account this thermodynamic insight on cogeneration in general and Rankine cycle in particular, we propose a new targeting methodology with two objectives in mind. The first objective is to develop a procedure for a more accurate shaftwork target and the second objective is to modify the SUGCC in order to visualize on the same diagram VHP consumption, thus fuel consumption, shaftwork production and cooling targets. 2.1. Rankine fundamentals As mentioned previously, the ideal shaftwork rate produced by a steam turbine is expressed as: hH hL;is _ _ ) W_ id ð1 hH Þ ¼ ðQ_ out DH_ r ÞðhH hL;is Þ ð5Þ W id ¼ Qout;id 1 hH where subscripts H and L corresponds to states 1 and 2 in Fig. 2, Q_ out the exhaust heat demand rate and DH_ r ¼ H_ 2 H_ 20 represents the frictional reheat rate or the heat exhaust deviation from an ideal expansion. DH_ r ¼ Q_ out Q_ out;id ¼ W_ id W_
ð6Þ
The steam turbine isentropic efficiency is defined as (see Fig. 2): gis ¼
h1 h2 h1 h20
ð7Þ
The real power is expressed as: W_ ¼ gis W_ id
ð8Þ
600 1 500
O
T ( C)
400 2
300 200 100 0
2’ 3
2
4 6 Entropy (kJ/kg/OC)
8
Fig. 2. Steam turbine expansion on a T–s diagram.
10
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
965
The temperatures expressed in the Carnot factors represent average thermodynamic temperatures for inlet and outlet expansion conditions of the steam turbine and defined as (see Fig. 2): TH ¼
h1 h3 s1 s3
T Lis ¼
h 20 h 3 s1 s3
ð9Þ ð10Þ
where h and s are the specific enthalpy and entropy respectively. Combining Eqs. (5), (6) and (8) and the definition expressed by Eq. (3), we derive an expression for the real shaftwork rate, W_ , as a function of the exhaust heat load rate, Q_ out , the isentropic efficiency, gis and the average thermodynamic temperatures given by Eqs. (9) and (10). W_ ¼ Q_ out gis
T H T Lis ð1 gis ÞT H þ gis T Lis
ð11Þ
Q_ out represents also a transiting heat rate at the turbine exhaust, Q_ tr (see Brodyansky et al. [4]), which could be used as process heat demand or further expanded to lower pressure levels for more cogeneration. It is also possible to express the real shaftwork rate as a function of heat supply rate, Q_ supply at the turbine inlet condition that is: T H T Lis W_ ¼ Q_ supply gis TH
ð12Þ
Finally, by using these insights and the SUGCC tool as proposed by Raissi [1], a new model is proposed to allow targeting fuel consumption, cooling requirement and shaftwork production with high accuracy and visualize the targets directly as special segments on a modified SUGCC diagram. It should be noticed that strictly speaking we should use the average thermodynamic temperatures, TH and TLis, on the modified SUGCC diagram and not the saturation temperatures, THsat and TLsat, however we keep the saturation temperatures on the modified SUGCC, but compute TH and TLis as the parameters of THsat and TLsat. The latter are fixed by heat recovery. 2.2. Illustrative example The methodology will be explained more precisely through an illustrative example taken from the literature [2]. It should be noted that the situation below site pinch was added to the example to illustrate a specific characteristic below pinch in the modified SUGCC. The model was tested on a utility steam system comprising of four steam levels above site pinch and one steam level along with a condensing header below site pinch. Above site pinch, the very high pressure (VHP) steam is produced in the boiler house at 500 °C and 90 bar. The high-, medium- and low-pressure levels correspond to 46 bar, 15.5 bar and 2.7 bar, respectively. Below site pinch, steam is generated at 1.5 bar and the condensing header (CW) is at 0.05 bar. Table 1 presents the various data parameters.
966
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
Table 1 Problem data Parameters
Units
Header1
Pressure Tsupply Saturation Temperature, Tsat
Bar °C °C
90 500 303
Header2
Header3
46
Header4
Header5
2.7
1.5
15.5
259
200
130
111
CW 0.05 33
The model distinguishes two separate sections, above site pinch and below site pinch, of the SUGCC diagram as shown in Fig. 1. For both situations, the idea is to start the computations from site pinch and move away to the hot and cold utilities to target VHP steam consumption, thus fuel consumption, shaftwork production and cooling requirements. The first step of the methodology is to divide the SUGCC into several domains characterized by enthalpy and temperature intervals as shown in Fig. 3 for above site pinch. Fig. 3 shows the SUGCC above site pinch. The SUGCC is divided into several enthalpy intervals along the x-axis (I) and temperature intervals along the y-axis (J ). The number of intervals depends on the SUGCC shape as observed in Fig. 3. The outlet heat loads Q_ heat ðI; J Þ that should be satisfied at heat interval I along temperature interval J are respectively Q_ heat ð1;1Þ ¼ 12:5 MW, Q_ heat ð1;2Þ ¼ 3:75 MW and Q_ heat ð2;3Þ ¼ 6:88 MW. The values are extracted directly from the SUGCC. For each domain we simulate an elementary steam turbine in order to target maximum shaftwork production by using Eq. (11) or Eq. (12) as shown in Fig. 4 for above site pinch. Steam flows through the elementary turbines along each enthalpy interval cascading from upper temperature intervals to lower temperature intervals. Fig. 4 shows the methodology used to target maximum shaftwork that could be produced by the utility system above site pinch. Elementary steam turbines are disposed in each domain (I, J ) where steam flows along each enthalpy interval I. 350 Saturation temperature ( C)
12.5 MW
o
300
Header1
Qsupply =10.63 MW
(3,1) 250
200
Header2 (2,1)
(2,2)
(1,1)
(1,2)
Header3
(2,3) Qdemand =6.88 MW
150 J
Header4 12.5 MW Qdemand =16.25 MW
100
50
0
5 I
3.75 MW
10 15 Heat Load (MW)
20
Fig. 3. Site utility grand composite curve above pinch.
25
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
967
Fig. 4. Configuration of the utility system for maximum shaftwork production.
The second step of the methodology is to identify for each enthalpy interval the different inlet and outlet conditions at the various steam turbine ports starting from the supply temperature intervals where inlet parameters are known. To do so, we assume a constant isentropic efficiency for all turbines and use steam tables or simple steam turbine expansion simulations. We could then compute the different average high and low thermodynamic temperatures by using Eqs. (9) and (10) (see Appendix A for details). The third step of the methodology is to calculate starting from site pinch, where heat demands and supplies are known, the shaftwork production using either Eq. (11) or (12) respectively above and below site pinch. Above site pinch and as we move upward to the hot utility level we must compute at each temperature interval a new transiting heat by adding to the heat demand the shaftwork production at lower temperature intervals. Then we compute the shaftwork production for the considered temperature interval using Eq. (11). Below site pinch and as we move downward to the cold utility level we must compute at each temperature level a new supply heat by subtracting to the heat supply of the previous temperature level the shaftwork production at higher temperature intervals. Then we compute the shaftwork production for the considered temperature interval using Eq. (12). The transiting heat and shaftwork generated in each domain are then computed (see Appendix A for details). A special attention should be given to the interpretation of the transiting heat along the temperature and enthalpy intervals. At this point we introduce the modified SUGCC to illustrate how transiting energy is computed above site pinch. The modified SUGCC is introduced to visualize on the same diagram heat recovery through the steam mains, shaftwork production and actual VHP consumption, thus fuel consumption, and cooling requirements. We include to the original SUGCC a new power axis in an opposite direction to heat load axis for visualization convenience. For a simple explanation purpose we consider only a part of a modified SUGCC where heat demand at a specific enthalpy interval should be satisfied by a hot utility cascading steam from VHP to a lower temperature through two temperature intervals as shown in Fig. 5.
968
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972 T-h Diagram Site Utility Grand Composite Curve
Saturation temperature (o C)
Q VHP
1
WI
1
WI
2 4
2’
o
2
T ( C)
(Qtr) I
W II
WII 6
(Q tr )II
5
3 5 6
5’
8 7
Q demand
Power (MW)
Heat Load (MW)
Enthalpy (kJ/kg)
Fig. 5. Illustration of the methodology used to target shaftwork production (Zones (3,1) and (2,1) of Fig. 4).
The problem in hand is to satisfy heat demand 7–8 by the outlet of the component turbine II, then _ 5 h6 Þ ðQ_ tr ÞII ¼ Q_ demand ¼ mðh If (TH)II and the isentropic temperature (TLis)II are known, then shaftwork production by turbine II, W_ II , is given by Eq. (11). The transiting heat for temperature interval I is then: ðQ_ tr ÞI ¼ ðQ_ tr ÞII þ W_ II If (TH)I and the isentropic temperature (TLis)I are known, then shaftwork production by turbine I, W_ I , is given by Eq. (11). Therefore, the overall VHP demand is Q_ VHP ¼ Q_ demand þ W_ I þ W_ II Fig. 4 and Appendix A show detailed computations of the transiting heat above site pinch and supply heat below site pinch. Finally, Fig. 6 describes the modified SUGCC showing the targets directly as special segments on the T–H diagram. We should emphasize that compared to the original SUGCC proposed by Raissi [1], the shape of the right hand side of the diagram above site pinch is the same, however, below site pinch it is shifted to the left by an amount equal to shaftwork production below site pinch. VHP consumption is also corrected to account for shaftwork production above site pinch that is represented by segments rather than areas on a T–W section opposite to the T–H section. Below site pinch we notice that excess heat could also be used for shaftwork production. Therefore, the excess heat is a summation of the potential shaftwork production plus the cooling load as shown in Fig. 6. The original SUGCC proposed by Raissi [1] omits to distinguish this aspect and gives an erroneous target for the cooling load. In order to represent accurately the cooling load requirement the original SUGCC below pinch should be shifted to the left of the T–H diagram by an amount equal to shaftwork production below pinch.
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
969
Site Utility Grand Composite Curve QVHP = 17.174 350 QAP =12.5
WAP = 4.674
Saturation temperature (oC)
300
250
200
150 WBP = 0.672 100
50 Q CW = 4.328 0 10
5
0
5
10
15
20
Power (MW)
25
Heat Load (MW)
Fig. 6. The new representation of the SUGCC.
The problem above site pinch has been simulated using steam properties and steam flows through the various turbines. The results are shown in Fig. 7. Compared to the results obtained through simulation it is observed that the new model based on the transiting heat concept gives a very good estimation of the maximum potential shaftwork production using the SUGCC. The deviation from the simulation is 0.2% for the new T–H model. 350 12.5 MW 21.86 t/h W(3,1) = 0.877
10.63 MW
o
Saturation temperature ( C)
300
21.86 t/h
250
0.82 t/h
6.00 t/h
0.92 t/h W(2,1) = 1.131
12.73 t/h W(2,3) = 0.547
W(2,2) = 0.285 13.56 t/h
200
Q heat = 6.88
20.12 t/h W(1,1) = 1.439
W(1,2) = 0.403
150 20.12 t/h
6.92t/h
Q heat =12.50
100
Q heat =3.75 3.447 MW
0.688 MW
Total Produced Work =4.682MW 50 0
5
10
15
20
Heat Load (MW)
Fig. 7. Results of the simulation (Section above site pinch).
25
970
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
Table 2 Comparison between the existing model [1] and simulation Interval (I, J)
CF
Shaftwork (MW) existing model
Shaftwork (MW) simulation
Deviation(%)
(1,1) (2,3) (3,1)
0.001645 0.001347 0.001595
5.02 4.11 4.87
4.68 4.68 4.68
7.2 12.2 3.9
Table 3 Comparison between the new targeting model and the existing model [1] Target
Existing model [1]
New model
Shaftwork Areas on SUGCC Segments on Modified SUGCC VHP Not accurately Visualized as visualized a segment
Cooling
Not accurately visualized
Visualized as a segment
Fuel
Cannot be visualized
Can be visualized as a segment
Comments The new model exhibits small deviations with simulation compared to the model by Raissi [1]. The new model takes into account the overall amount of VHP consumption for heat requirements and shaftwork production. The model by Raissi [1] visualizes only VHP consumption for heat requirements. The model by Raissi [1] does not visualize accurately cooling requirement if shaftwork production is considered. The new model takes into account this aspect by shifting to the left the SUGCC below the site pinch. The new model gives an accurate VHP consumption, thus fuel consumption can be visualized as a segment by dividing VHP consumption with the boiler house efficiency.
The existing T–H model proposed by Raissi [1] is based on an assumption that power is proportional to the area delimited by the exhaust heat output and saturation temperature difference between inlet and outlet: W_ ¼ CF Area Q_ DT sat where CF is assumed constant for a particular SUGCC. Table 2 summarizes the results for the overall shaftwork production above site pinch for different CF values computed using different intervals (I, J). We notice that the conversion factor in the existing T–H model has an important impact on the shaftwork target and could lead to large discrepancies with simulation if it is not well assessed. Table 3 summarizes the major differences between the existing model and the new model.
3. Conclusion A new model for targeting fuel consumption, cooling water requirement and shaftwork production was developed using an original thermodynamic insight on cogeneration. The model permits to visualize the targets directly as special segments on a modified SUGCC T–H diagram. Compared to the original SUGCC, the shape of the right hand side of the modified SUGCC T–H diagram above site pinch is the same, however, below site pinch it is shifted to the left by an amount
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
971
equal to shaftwork production below site pinch. Above site pinch VHP consumption accounts for heat requirements and shaftwork production. A new characteristic of the modified SUGCC diagram is that segments, on a T–W section opposite to the T–H section, rather than areas represent shaftwork production above site pinch. Compared to the original methodology proposed by Raissi [1], shaftwork targets present small deviations from simulation. Appendix A. Above Site Pinch Assuming a constant isentropic turbine efficiency gis and from using the output heat load of the turbine, the shaftwork production in each enthalpy-temperature domain (I, J) can be computed by using Eq. (11). A special attention should be given to the computation of the transiting heat along each domain starting from the steam level demands to upper levels. For heat interval J ¼ 3;
Q_ tr ð2; 3Þ ¼ Q_ heat ð2; 3Þ
For heat interval J ¼ 2;
Q_ tr ð1; 2Þ ¼ Q_ heat ð1; 2Þ
and Q_ tr ð2; 2Þ ¼ Q_ tr ð1; 2Þ þ W_ ð1; 2Þ
For heat interval J ¼ 1;
Q_ tr ð1; 1Þ ¼ Q_ heat ð1; 1Þ
and Q_ tr ð2; 1Þ ¼ Q_ tr ð1; 1Þ þ W_ ð1; 1Þ
In order to satisfy the required heat demands in heat intervals 2 and 3 and generate the shaftwork in these intervals, it is necessary to import heat from a higher temperature level in heat interval 1. Due to energy balance, the amount of heat imported is exactly equal to the shaftwork generated in intervals 2 and 3. Q_ tr ð3; 1Þ ¼ Q_ tr ð2; 1Þ þ W_ ð2; 1Þ þ W_ ð1; 2Þ þ W_ ð2; 2Þ þ W_ ð2; 3Þ The average high and low isentropic temperatures at each temperature–enthalpy domain are computed from the specific enthalpy and specific entropy values obtained through simple turbine simulation or directly from steam tables. For each heat interval J and temperature interval I, the average hot and low isentropic temperatures are computed by using Eqs. (9) and (10). The values in Table A.1 are obtained by assuming a constant isentropic efficiency gis of 0.7. Table A.1 Transiting heat and shaftwork potential at each temperature–enthalpy domain Case Above Site Pinch I=3 I=2 I=1
J=1 W_ ð3; 1Þ ¼ 0:873 Q_ tr ð3; 1Þ ¼ 15:08 W_ ð2; 1Þ ¼ 1:131 Q_ tr ð2; 1Þ ¼ 13:94 W_ ð1; 1Þ ¼ 1:439 Q_ tr ð1; 1Þ ¼ 12:50
J=2
J=3
W_ ð2; 2Þ ¼ 0:282 Q_ tr ð2; 2Þ ¼ 4:15 W_ ð1; 2Þ ¼ 0:402 Q_ tr ð1; 2Þ ¼ 3:75
W_ ð2; 3Þ ¼ 0:546 Q_ tr ð2; 3Þ ¼ 6:88
972
M. Sorin, A. Hammache / Applied Thermal Engineering 25 (2005) 961–972
Table A.2 Supply heat and shaftwork potential at each temperature–enthalpy domain Case below site pinch I=1
J=1 W_ ð1; 1Þ ¼ 0:672 Q_ supply ð1; 1Þ ¼ 0:672
The maximum shaftwork production above the total site pinch is then W_ AP ¼ 4:674 MW and the total VHP requirement is Q_ VHP ¼ 12:5 þ 4:674 ¼ 17:174 MW. Assuming a boiler efficiency of 0.9, the targeted fuel consumption will be 19.08 MW.
Appendix B. Below Site Pinch Below the site pinch the SUGCC is characterized only by one enthalpy-temperature domain I = 1 and J = 1, and a heat supply Q_ supply ð1; 1Þ ¼ 5 MW. The shaftwork production is computed by using Eq. (12). The values in Tables A.2 are obtained by assuming a constant isentropic efficiency gis of 0.7. The maximum shaftwork production below the total site pinch is W_ BP ¼ 0:672 MW and the cooling requirement is Q_ CW ¼ 4:328 MW. References [1] K. Raissi, Total Site Integration, PhD Thesis, University of Manchester Institute of Science and Technology (UMIST), Process Integration Department, UK, 1994. [2] S.P. Mavromatis, A.C. Kokossis, Conceptual optimisation of utility networks for operational variations-I. Targets and level optimisation, Chemical Engineering Science 53 (8) (1998) 1585–1606. [3] J. Klemes, V.R. Dhole, K. Raissi, S.J. Perry, L. Puigjaner, Targeting and design methodology for reduction of fuel power and CO2 on total sites, Applied Thermal Engineering 17 (8–10) (1997) 993–1003. [4] V.M. Brodyansky, V.M. Sorin, P. LeGoff, The Efficiency of Industrial Processes: Exergy Analysis and Optimization, Elsevier, 1994.