International Journal of Refrigeration 29 (2006) 47–59 www.elsevier.com/locate/ijrefrig
Thermoeconomic evaluation and optimization of an aqua-ammonia vapour-absorption refrigeration system R.D. Misraa, P.K. Sahoob,*, A. Guptab b
a Mechanical Engineering Department, National Institute of Technology Silchar, Silchar 788 010, India Mechanical and Industrial Engineering Department, Indian Institute of Technology Roorkee, Roorkee 247 667, India
Received 30 November 2003; received in revised form 12 May 2005; accepted 25 May 2005 Available online 19 August 2005
Abstract In this paper, the thermoeconomic concept is applied to the optimization of an aqua-ammonia vapour-absorption refrigeration (VAR) system—aimed at minimizing its overall product cost. The thermoeconomic concept based simplified cost minimization methodology calculates the economic costs of all the internal flows and products of the system by formulating thermoeconomic cost balances. The system is then thermoeconomically evaluated to identify the effects of design variables on costs and thereby enables to suggest values of design variables that would make the overall system cost-effective. Based on these suggestions, the optimization of the system is carried out through an iterative procedure. The results show a significant improvement in the system performance without any additional investment. Finally, sensitivity analysis is carried out to study the effect of the changes in fuel cost to the system parameters. q 2005 Elsevier Ltd and IIR. All rights reserved. Keywords: Absorption system; Ammonia-water; Optimization; Exergy; Economic amortization
Evaluation thermoe´conomique et optimisation d’un syste`me frigorifique a` eau/ammoniac a` absorption de vapeur Mots cle´s : Syste`me a` absorption ; Ammoniac-eau ; Optimisation ; Exergie ; Amortisement e´conomique
1. Introduction The energy and environmental norms regarding the ozone layer depletion and the global warming are becoming more and more stringent in almost all countries [1]. In this challenging scenario, absorption refrigeration systems using the binary mixture of ammonia and water are gaining
* Corresponding author. Tel.: C91 1332 85674; fax: C91 1332 85665. E-mail address:
[email protected] (P.K. Sahoo).
0140-7007/$35.00 q 2005 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2005.05.015
popularity, because they use zero global warming and ozone depletion fluids as refrigerants. Moreover, these systems are heat operated and need very little electricity. These systems, however, have lower COP and are costlier than the conventional vapour compression systems and need optimization from thermodynamic as well as economic points of view. In this respect, thermoeconomic concept has emerged to be a better tool as it addresses this multidisciplinary analysis by blending the exergy analysis with economic analysis [2]. During the last 25 years or so, the development of thermoeconomics has been impressive in more than one
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Nomenclature A B c C_ e E_ f h m_ M r R_ s t T Y Y* z Z Z_
heat transfer surface area (m2) constant in cost equations ($ kWK0.8) cost per exergy unit ($ GJK1, $ kJK1) cost flow rate ($ hK1) specific exergy (kJ kgK1) exergy flow rate (kW) thermoeconomic factor (%) specific enthalpy (kJ kgK1) mass flow rate (kg sK1) molecular weight (kg kmolK1) relative cost difference (%) O and M cost invariable to optimization specific entropy (kJ kgK1 KK1) temperature on centigrade scale (C) temperature on Kelvin scale (K) ratio of exergy destruction (or loss) to total exergy supplied to the system ratio of exergy destruction to total exergy destruction of the system concentration investment cost of the system components ($) levelized (annual) investment cost of the system components ($ sK1)
Greek letters b coefficient expressing accounts for the fixed O and M costs depends upon the total investment cost for a system component c effectiveness of the heat exchangers 3 exergetic efficiency (%) h isentropic or mechanical efficiency t annual number of hours of system operations u coefficient expressing variable levelized O and M costs for a system component x capital recovery factor D difference Superscripts Ch chemical component CI capital investment
direction. The recent developments by Tsatsaronis and his coworkers [2,3], Valero and his coworkers [4,5], Frangopoulos [6], von Spakovsky [7], d’Accadia and de Rossi [8], and Wall [9] etc. adequately represent the different directions of development. With regard to thermoeconomic optimization methodologies, Tsatsaronis and his coworkers [2,3] uses an iterative technique of thermoeconomic performance improvement where the analyzer can take part in decision making in the optimization process. Valero and his coworkers [4,5] have used the concept of assigning appropriate cost to each and every exergy flows and thermoeconomic performance improvement of the system
Ph OM OPT x y 0
physical component operation and maintenance optimum efficiency exponent in cost equations capacity exponent in cost equations standard state at normal temperature
Subscripts 0 environmental state 1, 2, 3,.i,.17, 18 system state points a absorber c condenser D exergy destruction e evaporator ea evaporator assembly ex expansion valve F fuel g generator H steam energy H2O water i the ith stream in entering streams k the kth component of the system L exergy losses m electric motor NH3 ammonia out exiting streams p pump P product Q exergy rate due to heat transfer r rectifier rhx refrigerant heat exchanger shx solution heat exchanger tot total system v throttling valve Vector and matrices [A] matrix containing the unknowns [X] vector of unknown variables [Y] vector of known variable
is done through local optimization of the sub-systems. Frangopoulos [6] and von Spakovsky [7] have used the functional decomposition of the system in the thermoeconomic optimization of the systems. The major fields of application of these developments are mainly in the field of large cogeneration and combined power plants, chemical plants, etc. [2–7], whereas the domain of refrigeration and air-conditioning are limited [8–11]. It is because the industrial utilities are probably considered with great interest, as they are capital intensive. However, the performances of refrigeration and air-conditioning have even higher rates of energy consumption and poor
R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
Fig. 1. Schematic diagram of an aqua-ammonia vapour absorption refrigeration system.
performance. These, therefore, deserve greater attention both in the design phase and in everyday handling. In this work, the thermoeconomic optimization of a heatoperated aqua-ammonia VAR system using the thermoeconomic evaluation and optimization method, based on the works of Tsatsaronis and his coworkers [2,3], is presented.
2. Assumptions used in the analysis Fig. 1 shows the schematic diagram of an aqua-ammonia VAR system, which uses ammonia as refrigerant and water as absorbent. For the purpose of analysis the following assumptions are made: † The system is in steady state. † The temperatures of the condenser, evaporator, absorber, generator, and rectifier are constant and uniform throughout the components. † The condenser and the evaporator pressures are the equilibrium pressures corresponding to the temperatures and concentrations in the condenser and evaporator, respectively. † The pressure loss in pipes between the rectifier and the condenser, and between the refrigerant heat exchanger (RHX) and the absorber, respectively, are taken as Dp/ poutZ0.05 and Dp/poutZ0.075 [12]. † The strong solution leaving the absorber and the weak solution leaving the generator are in equilibrium at their respective temperature and pressure.
49
† The refrigerant vapour concentration at the rectifier exit is equal to 0.999 [12]. † The generator temperature and pressure determine the vapour concentration at state 8, while the rectifier temperature and pressure determine liquid concentration at state 9. † The refrigerant vapour at the rectifier and the evaporator exit is equivalent vapour state at their respective temperature and pressure, while refrigerant liquid at the condenser exit is equivalent liquid state at the condenser temperature and pressure. † The rectifier temperature is equal to the average temperature between the generator and the condenser temperature. † The reference environmental state for the system is T0Z 25 8C (environment temperature) and p0Z1 atm (atmospheric pressure). † The system uses high-pressure steam to drive the generator. † The cooling water first enters the absorber at 27 8C, then passes through the condenser, and finally is rejected by the system at 35 8C. † The water to be chilled enters the evaporator at 20 8C and leaves at 10 8C. Based on these assumptions, a workable initial design, referred to as the base case, is conceptualized for the aquaammonia VAR system shown in Fig. 1 with the following data: tgZ150 8C, tcZ40.3 8C, teZK10 8C, taZ39.4 8C, crhxZ60%, cshxZ60%, hpZ75%, hmZ90%, t21Z170 8C, t22Z170 8C. In this work, the thermodynamic properties of the aqua-ammonia mixture in liquid and gas phase are taken from the work of Ziegler and Trepp [13]. The property data for the water/steam are taken from the correlations provided by Irvine and Liley (1984) [14].
3. Exergy analysis Exergy is defined as the maximum possible reversible work obtainable in bringing the state of a system to equilibrium with that of environment [2]. In the absence of magnetic, electrical, nuclear, surface tension effects, and considering the system is at rest relative to the environment, the total exergy of a system becomes the summation of physical exergy and chemical exergy Ph Ch E_ Z E_ C E_
(1)
The physical exergy component is calculated using the following relation: Ph _ K h0 Þ K T0 ðs K s0 Þ E_ Z m½ðh
(2)
The calculation procedure of the chemical exergy of various substances based on standard chemical exergy values of respective species is widely discussed by Bejan et al. [2],
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Table 1 Fuel–product–loss definitions of the aqua-ammonia VAR system Components
Fuels
Products
Loss
Generator Rectifier Evaporator assembly
E_ 20 K E_ 21 E_ 8 K E_ 10 K E_ 9 E_ 10 K E_ 12 C E_ 13 K E_ 15 C E_ 16 C E_ 6 K E_ 1
E_ 5 C E_ 8 K E_ 4 K E_ 9 E_ 3 K E_ 2 E_ 23 K E_ 22
RHX Solution pump SHX Overall system
E_ 15 K E_ 16 W_ m E_ 5 K E_ 6 E_ 20 K E_ 21 C W_ m
E_ 12 K E_ 13 E_ 2 K E_ 1 E_ 4 K E_ 3 E_ 23 K E_ 22
– – E_ 25 K E_ 24 C E_ 26 K E_ 25 – – – E_ 26 K E_ 24
Ahrendts [15], and Szargut et al. [16]. For the aquaammonia system considered here, the chemical exergy of the flows is calculated using the following relation: zi 1 K zi 0 Ch e0Ch;NH3 C eCh;H2 O (3) E_ i Z m_ i MNH3 MH 2 O where, e0Ch;NH3 and e0Ch;NH2 O are the standard chemical exergy of ammonia and water, respectively, and-their values are taken from Ahrendts [15]. A detailed exergy analysis includes calculation of exergy destruction, exergy loss, exergetic efficiency, two exergy destruction ratios, and one exergy loss ratio in each component of the system along with the overall system. Mathematically, these are expressed as follows [2]: E_ D;k Z E_ F;k K E_ P;k K E_ L;k
(4)
E_ P;k E_ D;k C E_ L;k 3k Z Z1K E_ F;k E_ F;k
(5)
YD;k Z
E_ D;k E_ F;tot
valve, absorber, and the throttling valve, the product cannot be readily defined. Therefore, they are considered as a single virtual component, named as evaporator assembly, together with the component(s) they serve, i.e. the evaporator in this case [2]. These considerations lead to the ‘fuel–product– loss’ (F–P–L) definitions for the system and are summarized in Table 1. The system can be considered to be composed of six productive components, namely, generator, rectifier, evaporator assembly, solution pump, RHX, and SHX. The thermoeconomic costs of all the flows that appear in the system’s F–P–L definition are obtained through exergy costing principles. Exergy costing involves formulation of cost balances for each component, which are discussed in the following paragraphs. 4.1. Generator
(6)
The purpose of the generator is to provide a stream of ammonia vapour (stream 8) and a weak solution of refrigerant (strewn 5). Hence, streams 5 and 8 are charged with all costs associated with owning and operating the generator. Thus, the following equations are developed:
E_ Z P D;k YD;k _ k ED;tot
(7)
C_ H C C_ 4 K C_ 5 K C_ 8 C C_ 9 Z KZ_ g
E_ YL;k Z _ L;k Ein;tot
(8)
m_ 8 ðc8 e8 K c4 e4 Þ m_ 8 ðe8 K e4 Þ Z
m_ 5 ðc5 e5 K c4 e4 Þ ðe8 K e5 ÞC_ 4 0 m_ 5 ðe5 K e4 Þ m_ 4 ðe8 K e4 Þðe5 K e4 Þ
4. Thermoeconomic analysis K The prerequisite for the thermoeconomic analysis is a proper ‘fuel–product–loss’ definition of the system to show the real production purpose of its subsystems by attributing a well defined role, i.e. fuel, product or loss, to each physical flow entering or leaving them [2–11]. In the case of aquaammonia system, this is quite easy for the generator, rectifier, evaporator, pump, refrigerant heat exchanger (RHX), and solution heat exchanger (SHX), where exergy of the product stream is increased. However, in the case of heat dissipative devices, e.g. the condenser, expansion
(9)
C_ 5 C_ 8 C Z0 m_ 5 ðe5 C e4 Þ m_ 8 ðe8 K e4 Þ
(10)
m_ 8 ðc8 e8 K c9 e9 Þ m_ 8 ðe8 K e9 Þ Z
C_ 5 m_ 5 ðc5 e5 K c9 e9 Þ 0K m_ 5 ðe5 K e9 Þ m_ 5 ðe5 K e9 Þ C
C_ 8 ðc8 K e5 ÞC_ 9 C Z0 _ _ m8 ðe8 K e9 Þ m9 ðe8 K e9 Þðe5 K e9 Þ
(11)
R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
4.2. Rectifier The purpose of the rectifier is to drag the possible amount of unwanted water vapour from the ammonia vapour leaving the generator (stream 10) by cooling it. The condensed water is returned to the generator (stream 9). The cooling is done by the strong solution leaving the pump (strewn 3), thus increasing its exergy. Hence, streams 9 and 10 are charged with all costs associated with owning and operating the generator. Thus, the following equations are developed: C_ 2 Z C_ 3 C C_ 8 K C_ 10 Z KZ_ r
(12)
m_ 9 ðc9 e9 K c8 e8 Þ m_ 9 ðe9 K c8 Þ m_ ðc e K c8 e8 Þ ðe10 K e9 ÞC_ 8 Z 10 10 10 0 m_ 10 ðe10 K c8 Þ m_ 8 ðe9 K e8 Þðe10 K e8 Þ C_ 9 C_ 10 K C Z0 m_ 9 ðe9 K e8 Þ m_ 10 ðe10 K e8 Þ
(13)
4.3. RHX
E_ 5 _ C_ 5 Z C6 Z 0 E_ 6
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(18)
4.6. Evaporator assembly The evaporator and heat dissipative components— condenser, expansion valve, absorber, and throttling valve—are combined in the evaporator assembly. The cost rates associated with the capital investment for these components and the cost rates associated with the exergy losses are charged to the final product. The cost rates associated with the exergy losses in condenser and absorber are given by the following two equations: C_ 11 K C_ 12 K C_ L;c Z KZ_ c
(19)
KC_ 1 C C_ 7 C C_ 17 K C_ L;a Z KZ_ a
(20)
For the dissipative components and the evaporator, following cost equations are developed: E_ 6 _ C_ 6 K C7 Z 0 (21) E_ 7
The purpose of the RHX is to subcool the liquid refrigerant leaving the condenser (stream 12) by the refrigerant vapour (stream 15) leaving the evaporator. Therefore, stream 13 is charged with all costs associated with owning and operating the RHX. Accordingly, the cost equations become
E_ C_ 10 Z _ 10 C_ 11 Z 0 E11
(22)
E_ 11 _ C 12 Z 0 C_ 11 K E_ 12
(23)
C_ 12 K C_ 13 C C_ 15 K C_ 16 Z KZ_ rhx
(14)
E_ 15 _ C_ 15 K C16 Z 0 E_ 16
(24)
(15)
E_ 13 _ _ C13 K C 14 Z 0 E_ 14 E_ 14 _ _ C 15 Z 0 C14 K E_ 15
(25)
C_ 1 C_ 7 C C_ 17 K Z0 E_ 1 E_ 7 C E_ 17
(26)
E_ 16 _ C_ 16 K C 17 Z 0 E_ 17
(27)
4.4. Solution pump The purpose of the pump is to raise the pressure of the strong solution leaving the absorber (stream 2) to the generator pressure with the help of external work. Thus, stream 2 is charged with all costs associated with purchasing and operating the pump (with motor). Accordingly, the cost equation becomes C_ m C C_ 1 K C_ 2 Z KðZ_ p C Z_ m Þ
(16)
4.5. SHX The purpose of SHX is to transfer the exergy of weak solution (stream 5) coming from the generator to the strong solution moving towards the generator (stream 4). Therefore, stream 4 is charged with all costs associated with owning and operating the SHX Accordingly, the cost equations become C_ 3 K C_ 4 C C_ 5 K C_ 6 Z KZ_ shx
(17)
Since the purpose of the evaporator is to transfer exergy from stream 14 to the product stream 23, therefore, all costs associated with the owning and operating the evaporator is charged to stream 23. Considering all these points the following cost equations are developed: KC_ 1 C C_ 6 C C_ 10 K C_ 12 C C_ 13 K C_ 15 C C_ 16 K C_ L;c K C_ P;tot K C_ L;a Z KðZ_ c C Zex C Z_ e C Z_ a C Z_ v Þ
(28)
The cost per exergy unit of the product of the system is given by,
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Table 2 State properties and cost rates of the streams of the aqua-ammonia VAR system for the base case Flows
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 25–26 24–25 22–23 20–21 19
Temperature (8C)
38 38.15 57.27 98.63 150 94.60 94.75 150 95 95 94.00 40 25.68 K9.94 K10 19.60 19.02 32.31–35 27–32.31 20–10 170 –
Pressure (kPa)
257.77 1563.93 1563.93 1563.93 1563.93 1563.93 257.77 1563.93 1563.93 1563.93 1489.45 1489.45 1489.45 277.10 277.10 277.10 257.77 101.33 101.33 101.33 792.32 –
Mass flow rates (kg sK1)
0.371 0.371 0.371 0.371 0.283 0.283 0.283 0.098 0.010 0.088 0.088 0.088 0.088 0.088 0.088 0.088 0.088 9.811 9.811 2.385 0.112 –
NH3 concentrations (%)
37.64 37.64 37.64 37.64 18.23 18.23 18.23 94.09 42.33 99.90 99.90 99.90 99.90 99.90 99.90 99.90 99.90 0 0 0 0 –
Physical exergy (kW)
Chemical exergy (kW)
2764.23 2764.23 2764.23 2764.23 1020.96 1020.96 1020.96 1826.16 82.89 1743.27 1743.27 1743.27 1743.27 1743.27 1743.27 1743.27 1743.27 0 0 0 0 –
K65.17 K64.71 K62.41 K51.81 K7.40 K24.73 K25.04 11.68 K1.46 6.35 5.77 K0.14 K0.30 K2.10 K15.30 K15.76 K16.66 3.11 3.11 3.49 75.14 0.83
Total exergy (kW)
Costs
2699.05 2699.52 2701.81 2712.42 1013.56 996.23 995.92 1837.84 81.43 1749.61 1749.03 1743.12 1742.97 1741.16 1727.96 1727.50 1726.61 3.11 3.31 3.49 75.14 0.83
Rates C_ ($ hK1)
Per exergy unit, c ($ GJK1)
248.23 248.48 248.93 251.28 130.60 128.36 128.32 131.24 7.23 123.62 123.57 123.16 123.22 123.10 122.16 122.13 122.07 1.90 2.75 1.50 2.69 0.09
25.55 25.57 25.59 25.73 35.79 35.79 35.79 19.84 24.65 19.63 19.63 19.63 19.64 19.64 19.64 19.64 19.64 170.10 230.37 119.23 9.94 31.00
Table 3 Exergy analysis results for the base case of the aqua-ammonia VAR system YD;k (%)
Components
E_ F;k (kW)
E_ P;k (kW)
E_ D;k (kW)
E_ L;k (kW)
YD,k (%)
Generator Rectifier Evaporator assembly RHX Solution pump SHX Overall system
75.14 6.79 46.17
57.55 2.30 3.49
17.59 4.50 36.26
0 0 6.42
23.15 5.92 47.73
26.63 6.81 54.89
0.46 0.83
0.16 0.47
0.30 0.37
0.40 0.48
0.46 0.56
17.33 75.97
10.60 3.49
6.73 66.06
8.85 86.96
10.18 100.00
C_ K C_ 22 C_ cP;tot Z _ P;tot Z _ 23 EP;tot E23 K E_ 22
0 0 0 6.42
(29)
Out of these 22 variables ½XZ fC_ 1 ; .; C_ 17 ; C_ L;c ; C_ L;a ; C_ P;tot ; C_ H ; and C_ m g, the last two are known parameters. They are the costs of steam used to energize the generator ðC_ H Þ and the electrical energy to the solution pump through the motor ðC_ m Þ, and their values are $6.67 per 1000 kg of steam [17] and $31 GJK1 for electrical energy (based on the commercial
YL,k (%)
3k (%)
0 0 8.45
76.59 33.79 7.56
0 0
34.37 55.94
0 8.45
61.19 4.59
power tariffs in India, i.e. Rs.1 5.0 per kW h), respectively. The interest rate of 8% is taken in this analysis. These costs are valid for India. The remaining 20 unknowns are calculated by solving the linear system of equations ([A][X]Z[Y]), where the coefficient matrix [A] contains the terms on the LHS and [Y] contains the terms on the RHS
1 Rupees (Rs.) is the Indian currency. Equivalent approximate US dollar ($) value is 1$yRs. 45.00.
R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
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Table 4 Thermoeconomic analysis results of the aqua-ammonia VAR system—base case Components
cF,k ($ GJK1)
cP,k ($ GJK1)
C_ D;k ($ yK1)
C_ L;k ($ yK1)
Z_ k ($ yK1)
ðC_ D;k C C_ L;k C Z_ k Þ
f (%)
r (%)
($ yK1) Generator Rectifier Evaporator assembly RHX Solution pump SHX Overall system
9.94 16.06 22.75
16.06 54.93 119.23
1133.47 468.28 5345.74
19.64 31.00
119.22 145.16
38.34 73.86
35.79 10.18
61.46 119.23
1560.00 4355.95
0 0 946.52
1148.39 109.94 4260.42
2281.86 578.21 10552.68
50.33 19.01 40.37
61.53 241.97 2051.38
0 0
63.46 271.48
101.80 345.33
62.34 78.61
507.10 368.27
204.02 12115.40
1764.02 16894.71
11.57 71.71
71.73 7342.60
0 423.36
of Eqs. (9)–(28). The cost flow rate C_ and the cost per exergy unit for each flow of the system are calculated based on above formulation. Table 2 summarizes the calculated thermodynamic properties at various state points of the aqua-ammonia VAR system for the base case operating conditions along with the cost flow rates. The exergy analysis of the system and its components are carried out using fuel–product relationships and Eqs. (4)–(8). The results are summarized in Table 3. The results show maximum exergy destruction of 36.3 kW occurs in the evaporator assembly with, YD,ea of 47.7%. The generator is having second largest exergy destruction (17.6 kW) with YD,g of 23.2%. They are followed by SHX (6.7 kW) with YD,shx of 8.9%, rectifier (4.5 kW) with YD,r of 5.9%, and RHX (0.3 kW) with YD,rhx of 0.4%. The exergy loss from the heat dissipative devices is 6.4 kW. Due to these exergy loss and exergy destructions in the evaporator assembly, the 3ea is found to be low (7.6%). The exergy destruction and the exergetic efficiency of the overall system is found to be 66.1 kW and 4.6%, respectively. A similar trend is also observed with YD;k . The exergy analysis suggests the designer to find means for the improvement of the overall exergetic efficiency to maximum possible value. However, the thermodynamic optimum, so obtained, does not guarantee a cost-optimal design. A cost-optimal design can be obtained through the thermoeconomic optimization, which is discussed in the following section.
5. Thermoeconomic evaluation In this section, the aqua-ammonia system is evaluated with the help of the following thermoeconomic variables, namely, the average unit cost of the fuel (cF,k), the average unit cost of the product (cP,k), the cost rate of exergy destruction ðC_ D;k Þ, the cost rate of exergy loss ðC_ L;k Þ, the relative cost difference (rk), and the thermoeconomic factor (fk). These are calculated using the following relations [2]:
cF;k Z
C_ F;k E_ F;k
(30)
cP;k Z
C_ P;k E_ P;k
(31)
C_ D;k Z cF;k E_ D;k ðconsidering E_ P;k as constantÞ
(32)
C_ L;k Z cF;k E_ L;k
(33)
ðconsidering E_ P;k as constantÞ
rk Z
cP;k K cF;k C_ C Z_ k 1 K 3k Z_ k Z D;k Z C _ cF;k 3 cF;k EP;k cF;k E_ P;k k
(34)
fk Z
Z_ k Z_ k C ðC_ D;k C C_ L;k Þ
(35)
Bejan et al. [2] have presented the guidelines for the thermoeconomic evaluation. The thermoeconomic analysis results for the base case operating conditions of the aquaammonia system are presented in Table 4. It is seen that the evaporator assembly has the maximum value of ðC_ D;k C C_ L;k C Z_ k Þwith higher value of rk and lower value of fk. Therefore, improvement of-the exergetic efficiency of this unit should be considered at the cost of capital investment. The generator has the second largest value of ðC_ D;k C C_ L;k C Z_ k Þ with slightly lower values of rk and fk, suggesting the possibility of increasing the generator temperature. The third next component in that order is the SHX, where the lower value of fk suggests improvement in exergetic efficiency through increase in its effectiveness. Next comes the rectifier, where there is no one direct decision variable controlling the exergetic efficiency or the investment cost, rather is controlled mainly by generator and the condenser temperatures. Therefore, the effect of these two variables on rectifier is seen globally. As far as RHX is concerned, there is slight scope for improvement in the exergetic efficiency of the system by increasing its effectiveness. The solution pump, on the other hand, shows a very high value of thermoeconomic factor fk. The
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logical conclusion would be to try to decrease the investment cost of the pump at the cost of its exergetic efficiency. This information for each component is extremely helpful in carrying out the iterative thermoeconomic optimization of the system, which is discussed in the following section.
minimization of the cost per exergy unit of the product of the component. Mathematically, Minimize cP;k Z Z cF;k
cF;k E_ F;k C Z_ k E_ P;k
1 ðx C bk ÞBk 3k R_ C C uk C _ k 1Kx k 1 K 3k 3k EP;k E_ P;k
(40)
subject to the following constraints:
6. Thermoeconomic optimization
E_ P;k Z constant and cF;k Z constant The thermoeconomic optimization of the system requires a thermodynamic model and a cost model. The thermodynamic model, discussed earlier, gives the performance prediction of the system with respect to some thermodynamic variables. The cost model permits detailed calculation of cost values for a given set of the thermodynamic variables, which is discussed below. The annual levelized investment cost associated with the kth component is calculated as CI Z_ k Z xZk
In Eq. (40), cP,k is the sole function of 3k. Thus, differentiating it with respect to 3k and equating to zero provides the minimum cost per unit of product exergy, i.e. !1=ðxkC1Þ dcP;k 1 K 3k xk Bk ðx C bk Þ Z 00 Z Z Fk (41) 1Ky d3k 3k cF;k E_ P;k k Thus, the cost optimal 3k, rk, and fk, respectively, are obtained as
(36)
Z 3OPT k
The term x is known as the capital recovery factor. The calculation for Zk and x are given in Appendix A. The annual levelized operation and maintenance (O and M) cost associated with the kth component is approximately calculated as [2]:
rkOPT Z
OM Z_ k
Z bk Zk C uk E_ P;k C R_ k
(37)
where bk and uk, respectively, accounts for the fixed and variable O and M costs associated with the total investment cost of the kth component, and R_ k includes all other O and M costs that are independent of investment cost and exergetic product. For simplicity, bk is assumed to be 1.25% of total investment cost of each component [2,3]. For every component, it is expected that the investment cost increases with increasing capacity and increasing exergetic efficiency. Therefore, the total investment cost of the kth component can be approximated by the following relationship [2]: xk 3k yk Zk Z Bk (38) E_ P;k 1 K 3k The constants Bk, xk, and yk are found out by plotting the available cost data for the kth component and applying the curve fitting techniques. For simplicity we assume the value of yk as 0.8 for all the components. The calculation procedure for calculating the constants Bk and xk are given in Appendix B. Thus, the total annual levelized costs associated with the kth component is given by: CI OM Z_ k Z Z_ k C Z_ k xk 3k yk Z ðx C bk ÞBk C uk E_ P;k C R_ k E_ P;k 1 K 3k
(39)
The objective function for the kth component is defined as
fkOPT Z
1 1 C Fk
1 C xk Fk xk
1 1 C xk
(42)
(43)
(44)
The optimization procedure in this approach is an iterative one that aims at finding out a better solution for the system, unlike conventional optimization procedure, where the aim is to calculate the global optimum. In this optimization problem, the cost of the exergy destruction has already been included in the product cost term and the cost of the exergy loss is charged to the product cost (as it is a loss for the system). Therefore, the objective function for the overall system is defined as the product cost, which is to be minimized during the optimization process, i.e. Minimize OBF Z C_ P;tot
(45)
In the iterative optimization procedure the following thermoeconomic variables are defined to facilitate the decision-making: D3k Z 100 !
3k K 3OPT k 3OPT k
(46)
Drk Z 100 !
rk K rkOPT rkOPT
(47)
Dfk Z 100 !
fk K fkOPT fkOPT
(48)
These variables express the respective relative deviation actual values from optimal values. In the iterative optimization procedure, engineering judgments and critical
R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
55
Table 5 Variables obtained during the optimization of aqua-ammonia VAR system from base case to optimum case Variable
Base case
tg (8C) tc (8C) te (8C) ta (8C) hP (%) crhx (%) cshx (%)
150.00 40.00 K10.00 38.00 75.00 60.00 60.00
Component
D3 (%)
Generator Evaporator assembly RHX Pump SHX
62.17 K20.45 29.60 90.30 35.48
Dr (%) K65.33 6.47 18.13 K52.38 K37.20
Iterative case-1
Iterative case-2
152.50 39.00 K5.00 37.00 72.50 65.00 65.00 Df (%)
D3 (%)
K30.78 18.79 60.65 17.33 K373.26
61.92 K5.88 39.48 90.22 35.66
Dr (%) K58.17 1.75 28.27 K67.25 K37.83
155.00 38.00 0.00 36.00 70.00 70.00 70.00 Df (%)
D3 (%)
K41.35 K5.26 66.86 18.24 K392.39
61.28 5.70 58.96 90.11 35.86
Dr (%) K47.95 K1.72 17.32 K35.98 K39.54
C_ L;tot ($/yK1) C_ D;tot ($/yK1) C_ P;tot ($/yK1)
423.36 4355.95 17124.80
389.20 3812.81 16469.76
365.35 3434.83 16007.39
Variable
Iterative case-3
Iterative case-4
Iterative case-5
tg (8C) tc (8C) te (8C) ta (8C) hP (%) crhx (%) cshx (%) Component Generator Evaporator assembly RHX Pump SHX
155.00 37.00 5.00 35.00 70.00 75.00 75.00 D3 (%) 59.88 16.38 69.90 90.34 36.07
Dr (%) K35.77 K5.09 25.48 K29.17 K37.70
155.00 37.00 K7.50 35.00 70.00 80.00 80.00 Df (%) K112.47 13.04 69.53 21.18 K452.05
D3 (%) 59.31 20.53 78.44 90.44 37.62
Dr (%) K29.12 K7.57 35.22 K26.57 36.00
Df (%)
D3 (%)
K134.78 15.34 71.34 21.95 K407.38
59.8 20.30 78.99 90.45 37.54
Dr (%) K16.95 K8.33 43.28 K23.00 K35.934
346.60 3138.61 15656.83
337.72 2997.98 15495.04
334.84 2952.23 15444.01
Variable
Iterative case-6
Iterative case-7
Optimum case
Component Generator Evaporator assembly RHX Pump SHX
155.00 37.00 7.50 33.75 70.00 82.50 87.50 D3 (%) 59.13 21.24 28.97 90.45 39.65
Dr (%) K15.87 K9.33 43.05 K19.07 K28.90
155.00 37.00 7.50 33.75 70.00 82.50 90.00 Df (%) K140.62 14.59 71.46 22.12 K287.92
D3 (%) 59.18 21.54 78.96 90.45 40.80
Dr (%) K14.90 K9.78 42.93 K18.07 K28.53
K62.77 4.80 66.37 19.00 K386.18
155.00 37.00 7.50 34.00 70.00 82.50 82.00
C_ L;tot ($/yK1) C_ D;tot ($/yK1) C_ P;tot ($/yK1)
tg (8C) tc (8C) te (8C) ta (8C) hP (%) crhx (%) cshx (%)
Df (%)
Df (%) K140.19 14.31 71.46 22.09 K385.19
155.00 37.00 7.50 33.75 70.00 82.50 90.00 Df (%)
D3 (%)
K140.31 14.61 71.46 22.12 K238.99
59.18 21.54 78.96 90.45 40.80
Dr (%) K14.90 K9.78 42.93 K18.07 K28.53
Df (%) K140.31 14.61 71.46 22.12 K238.99
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R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
Table 5 (continued) Component
D3 (%)
C_ L;tot ($/yK1) C_ D;tot ($/yK1) C_ P;tot ($/yK1)
Dr (%)
Df (%)
D3 (%)
331.60 2900.46 15388.50
Dr (%)
Df (%)
D3 (%)
330.11 2876.54 15363.11
evaluations are used in deciding on the changes made to the decision variables from one iterative step to the next. Also, while taking the decision on the changes of the decision variables, the practical limitations of the system, mentioned earlier, are also considered. The criteria followed in decision-making on the changes of the decision variables from one iterative step to the next are as follows [10]: † The variables D3k, Drk, Dfk, and C_ P;tot are calculated for a change in one decision variable in a certain step, while keeping other decision variables constant. † The effects of the thermoeconomic variables are examined. † If the effect is positive, i.e. C_ P;tot shows reducing trend, then in the next iterative step this variable becomes a candidate for a similar change, otherwise, this variable remains unchanged in the next iterative step. † Repetition of the above three steps for the other decision variables.
7. Result and discussion The decision variables for this problem are tg, (tcKteK ta), hP, crhx, and cshx. The stage-by-stage iteration results from the base case to the optimum case are summarized in
Dr (%)
Df (%)
330.11 2876.54 15363.11
Table 5. Each of these stages is obtained through a series of study of positive or negative effects on OBF, by varying each decision variable at a time. The change in the decision variables are governed by D3k, Drk, and Dfk. Table 6 presents a comparative study of the final cost optimal configuration with the base case. The overall thermoeconomic cost of the product (chilled water) is decreased by about 10.3%. The corresponding decrease in the fuel cost is about 31.4%. Also, the results show a significant reduction in the cost of exergy destruction cost (34.0%) and exergy loss cost (22.0%). The corresponding increase/decrease in the investment costs of the components are also shown in the same table. The greater investment is mainly in the evaporator assembly (19.1%) and the RHX (100.6%). Incidentally, the overall investment cost of the system almost remains same. It is due to the reductions of investment cost in the generator (64.4%), the rectifier (70.5%), and the SHX (30.4%). The exergetic efficiency is increased by about 44.6% and the COP is increased by about 44.2%. In order to remain competitive, it is crucial for any system to be robust with respect to the uncertainty pertaining to frequently changing parameters with time. In general, the economic parameters, such as the fuel costs and the capital investments, are the uncertain terms as they vary with time. Therefore, it is necessary to investigate their effect on the decision variables and the product costs of the
Table 6 Comparative results between the base case and the optimum case of the aqua-ammonia VAR system Properties
Base case
Optimum case
%- Variation
OBF, i.e. C_ P;tot ($ yK1) 3tot (%) COP C_ F;tot ($ yK1) C_ D;tot ($ yK1) C_ L;tot ($ yK1) Investment costs: Z_ tot ($ yK1) Z_ g ($ yK1) Z_ r ($ yK1) Z_ ea ($ yK1) Z_ rhx ($ yK1) Z_ p ($ yK1) Z_ shx ($ yK1)
17124.80 183.73 17.35 5009.40 4355.95 423.36
15363.11 265.61 25.01 3434.73 2876.54 330.11
K10.29 44.57 44.15 K31.43 K33.96 K22.03
12115.40 1148.39 109.94 4260.42 63.46 271.48 204.02
11928.38 408.48 32.39 5076.05 127.33 177.97 141.97
K1.54 K64.43 K70.54 19.14 100.64 K34.44 K30.41
R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
57
† Proper definition of the fuel–product–loss of the system. † Proper selection of local decision variables for various subsystems. † Reliability of the cost information available and how to relate these with thermodynamic parameters. † Proper engineering judgments to relate costs with thermodynamic parameters.
Fig. 2. Sensitivity of the product cost ($ yK1) with respect to the increased fuel cost (%).
Appendix A. Equations for calculating the investment costs
optimal solution. In this work, the variation with respect to capital investment cost is not considered, as once installed the system components are not supposed to alter in response to change in their capital investment. Hence, only the fuel cost is considered. To study the robustness of the optimal solution, analysis is carried out for 20, 40, 60, 80, and 100% increase in the fuel cost for the operating decision variables and the product costs of the system. Fig. 2 shows the variation of the product cost with respect to the increase in the fuel cost. The trend of this increase in the product cost is same as that of the fuel cost variation and Table 7 presents the percentage change in the decision variables with respect to the present optimum. It is seen that the change in the decision variables with respect to the change in the fuel cost is very small; in fact, it is well within G2% for all the variables. Therefore, it can be concluded that the decision variables of the system are not very sensitive to the changes in the fuel cost (uncertainty) in the close vicinity of the optimal solution space.
The investment cost of the aqua-ammonia system is taken from the technical report of Transparent Energy systems Pvt. Ltd. [18]. For the purpose of analysis, the generator, rectifier, condenser, evaporator, absorber, RHX, and SHX of the system are considered as simple heat exchangers. The investment costs of these heat exchangers are calculated based on weighted area after duly deducting the investment costs of the pump and the motor and using the following power law relation [2,19]: Zk Z ZR;k
Ak AR
0:6 (A1)
where kZg, r, c, e, a, rhx, shx. The subscript ‘R’ represents the reference component of a particular type and size. The investment cost of the pump and the motor can be written respectively as [2,19]: Zp Z ZR;p
8. Conclusions
W_ P W_ R;P
The analysis presented in this paper amply demonstrates the application of the thermoeconomic concept in optimization of an aqua-ammonia VAR system. Moreover, due to its iterative nature the designer can incorporate the operator’s suggestions during the optimization, which is not possible through the conventional optimization techniques. However, the designer should pay adequate attention to the following points:
Zm Z ZR;m
mp
W_ m W_ R;m
1 K hip hip
mm
np (A2)
1 K him him
(A3)
These costs are then updated to the processing year with the help of Marshall and Swift (M and S) equipment cost index (available in the ‘Chemical Engineering’ journal) [19,20]. The capital recovery factor x, for a system corresponding to an interest rate of ir, life span of Ny years, and t hours of operation per year is given by [2,3,19]:
Table 7 Sensitivity of the decision variables of aqua-ammonia VAR system with respect to increase in the fuel cost (%) Increase in fuel cost 20 40 60 80 100
Changes in decision variables (%) tg
tc
te
ta
crhx
cshx
hp
K2.12 K0.72 0.03 K1.92 K0.29
K0.04 0.14 K0.96 K0.45 K1.23
0.73 1.08 K1.23 K2.80 0.55
0.41 K1.29 K0.60 K0.04 K1.74
K0.20 1.70 1.63 K0.89 1.11
0.18 0.20 0.21 0.10 0.15
0.31 0.44 0.42 0.49 0.85
58
R.D. Misra et al. / International Journal of Refrigeration 29 (2006) 47–59
Fig. B1. Plot of investment cost equation (Eq. (38)) for generator (140%tg%170 8C).
ir ð1 C ir ÞNy yrK1 ð1 C ir ÞNy K 1 ir ð1 C ir ÞNy 1 Z sK1 Ny t !3600 ð1 C ir ÞNy K 1
which three decision variables tc, te, and ta art together. The variations of the product cost of this component with respect to a combination of decision variables tc, te, and ta are plotted. It is found that for the permissible limit of evaporator temperature, i.e. 08%te%7.5 8C, for each combination of tc and ta, a particular value of te gives the minimum value of the product cost. Thus, a set of combination of the decision variables tc and ta with the minimum value of te for each combination of tc and ta are formed. These combinations are then used to obtain the parameter Bea and the exponent xea of the Eq. (38) for the evaporator assembly as shown in Fig. B2.
References
xZ
(A4)
An interest rate of 8%, 20 years of life span and 1800 h of operation per year are used in the analysis presented in this paper.
Appendix B. Determination of constants of Eq. (40) Eq. (38) predicts the investment cost of a system component as a function of its exergetic product and exergetic efficiency. Now, to use this equation directly in the optimization calculation (Eq. (40)), the constants of that equation (Bk, xk, and yk) has to be pre-calculated based on the investment cost data. Considering the case of the generator, for each data set E_ P;g ; E_ D;g ; 3g , and Zg are calculated. Now, with yk as 0.8 0:8 (see text), ½Zg =E_ P;g is plotted against [3g/(1K3g)], as shown in Fig. B1. Curve fitting with power law provides Bg $211.73 kWK0.8 and xgZ0.5194. The evaporator assembly has to be considered in isolation as it is composed of several components, in
Fig. B2. Plot of investment cost-equation (Eq. (38)) for evaporator assembly.
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