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Applied Thermal Engineering 37 (2012) 10e18

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate/apthermeng

NLP model of a LiBreH2O absorption refrigeration system for the minimization ofthe annual operating cost

Carlos Rubio-Maya a,*, J. Jesús Pacheco-Ibarra a, Juan M. Belman-Flores b, Sergio R. Galván-González a,Crisanto Mendoza-Covarrubias a

a Faculty of Mechanical Engineering, Edif. W, CU, UMSHN. Morelia, Michoacán, Mexico, CP 58030, MexicobDepartment of Multidisciplinary Studies, Engineering Division, Campus Irapuato-Salamanca, University of Guanajuato, Yuriria, Gto., Mexico

a r t i c l e i n f o

Article history:Received 8 September 2011Accepted 20 December 2011Available online 28 December 2011

Keywords:Absorption refrigerationOptimizationNLPExergyCost analysisLiBreH2O

* Corresponding author. Tel.:þ52 4433223500x1160E-mail address: rmaya@umich.mx (C. Rubio-Maya

1 Code available on request from corresponding au

1359-4311/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.applthermaleng.2011.12.035

a b s t r a c t

In this paper the optimization of a LiBreH2O absorption refrigeration system with the annual operatingcost as the objective function to be minimized is presented. The optimization problem is established asa Non-Linear Programming (NLP) model allowing a formulation of the problem in a simple and struc-tured way, and reducing the typical complexity of the thermal systems. The model is composed of threemain parts: the thermodynamic model based on the exergy concept including also the proper formu-lation for the thermodynamic properties of the LiBreH2O mixture, the second is the economic model andthe third part composed by inequality constraints. The solution of the model is obtained using theCONOPT solver suitable for NLP problems (code is available on request1). The results show the values ofthe decision variables that minimize the annual cost under the set of assumptions considered in themodel and agree well with those reported in other works using different optimization approaches.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Absorption refrigeration systems (ARS) are gaining morepopularity in different cooling and refrigerating applications, due tothe use of refrigerants with almost zero global warming potentialand zero contribution to the ozone layer depletion. The ARS can useseveral low-temperature heat sources as energy input in order toproduce the cooling effect, while conventional systems, i.e.compression systems, utilize mechanical energy for its operation.The heat sources to drive the ARS are wide and include fossil fuel,renewable energy sources and waste heat recovered from primemovers (cogeneration units) or industrial processes, being thelatter source the most cost-effective option when they are inte-grated in multi-product systems from the well-known cogenera-tion to more complex systems of combined production of severaltypes of energy carriers [1e3]. Nevertheless, ARS have two majordrawbacks compared with compression systems, lower efficiencyand higher costs. Therefore, it is necessary to improve its designand operation from several points of view in order to overcomethose drawbacks. So far, several methodologies have been proposedto accomplish the challenge of improving ARS and they can be

; fax:þ52 4433223500x3103.).thor.

All rights reserved.

grouped mainly in second law thermodynamic analysis and ther-moeconomic optimization. However, other works can be found inthe literature, in which improvements to the system are madethrough the examination of the influence of some parametersunder different operating conditions [4e8].

The principle of the second law is a powerful tool because it givesinformation on how, where and howmuch the systemperformanceis degraded. Second law analysis can be based on the concepts ofentropy generation and exergy destruction. When the entropyconcept is used it is possible to identify the components or devicesof a thermal system with higher entropy generation, afterwardsimprovements to the system are achieved minimizing the totalentropy generation. The exergy concept can be utilized in a similarway, determining the exergy destruction of each component andthe total exergy destruction of the system. In this case, in order toincrease the efficiency of the system the exergy destructionmust beminimized. Among the second law analysis, Kaynakli and Yaman-karadeniz [9] studied the performance of lithium bromideewaterARS varying some design parameters through a computationalmodel to determine the entropy generation of individual compo-nents and the total entropy generation. Sencan [10] used a simula-tionprogram todetermine the Coefficient of Performance (COP) andexergetic efficiency of a single effect LiBreH2O ARS finding theoperating conditions that increase the above-mentioned perfor-mance indicators. Exergy analysis can be performed in a moreadvanced fashion, as it is demonstrated byMorosuk and Tsatsaronis

Nomenclature

aC capital recovery factor (dimensionless)A heat transfer area (m2)bC annual cost not affected by optimization ($/year)_B exergy flow (kW)_BD exergy destruction (kW)b specific exergy (kJ/kg)Cp specific heat capacity (kJ/kgK)CT annual cost ($/year)CCk capital cost of the k-st element of the ARS ($)

C 3IN unit cost of input exergy ($/kJ)

COP coefficient of performance or energy efficiency ratio_E energy (kW)h specific enthalpy (kJ/kg)iR interest rate (%)LMTD logarithmic mean temperature (K)_m mass flow rate (kg/s)Ny period of repayment (years)P pressure (kPa)_Q heat flow (kW)s specific entropy (kJ/kg K)top period of operation per year (hours)T temperature (�C, K)

U overall heat transfer coefficient (kW/m2 K)_W work (kW)X concentration of LiBr in the solution (%)Z capital cost of k-st subsystem ($)

Greek symbols3 effectiveness

Subscriptsa coefficientABS absorberci cold inletco cold outletCON condenserCV control volumeEQS element or subsystemEVP evaporatorGEN generatorhi hot inletho hot outleti i-st streamk k-st elementSHX solution heat exchanger0 reference state or value

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e18 11

[11], which consist of splitting the exergy destruction within thecomponents of an ARS and then identifying the potential forimproving each system’s component. Kaushik and Arora [12]carried out the energy and exergy analysis of a single effect andseries flow double effect LiBreH2OARS developing a computationalmodel for the parametric investigation of such systems.

The thermoeconomic analysis, also called exergoeconomics, is anapproach that allows improvements to the performance of thermalsystems by combining the second law and economic considerations.Specifically, thermoeconomics merges both the exergy concept andeconomic analysis into a single framework, which purpose is toachieveabalancebetweenan/theexpenditureoncapital cost and thefuel costs that will result in the minimum cost of the plant product.Cost allocation, cost optimization, and cost analysis are the keyfeaturesof thisdiscipline.Regardingoptimization, thermoeconomicshas provided an alternative tool to the improvements of thermalsystems, mainly when traditionally optimization techniques are toocomplex and time consuming for such systems. Also, refrigerationsystems, including ARS, have been optimized using the thermoeco-nomic approach [13e19], arguing that the reason for using suchapproach is that the optimization of thermal systems cannot alwaysbe carried out using sophisticated mathematical or numerical tech-niques, due to incomplete models, plant complexities, and strongnon-linear nature. The annual operating cost [13,14], the cost perexergy unit of the product (i.e. coolingeffect) [15e17], and theoveralleconomic cost of the final product have been the objective functionsemployed to optimize refrigeration systems, [18,19].

On the other hand, an alternative approach that has beenapplied to the optimization of process industries is the one basedon mathematical programming that can be solved by means ofstandard techniques such as lineal (LP), non-linear (NLP), mixed-integer linear (MILP) and mixed-integer non-linear programming(MINLP). These strategies have been extensively used in the opti-mization of chemical processes but their application to the opti-mization of refrigeration systems is limited and only few researchprojects can be found. Such is the case of Chávez-Islas et al. [20,21]who reported the optimization of an NH3eH2O ARS by the appli-cation of MINLP. Recently, Gebreslassie et al. [22], presented a bi-

criteria NLP optimization of a NH3eH2O ARS minimizing the costand the environmental impact. In the same way Gebreslassie et al.[23], presented a stochastic bi-criteria NLP optimization for theminimization of total cost and the financial risk associated with theinvestment for the same system.

Therefore, based on the above ideas, the objective of the presentwork is to address the optimization of single effect LiBreH2O ARSusing the approach based on mathematical programming, estab-lishing a Non-linear Programming model using the exergy concept(not included in the above works cited), through the annual oper-ating cost as the objective function to beminimized. The NLPmodelproposed is simple and structured and is favorable to reduce thetypical complexity of thermal systems, providing an alternativeapproach to optimize ARS.

2. Single effect LiBreH2O ARS

2.1. System description

Compared to a compression cooling cycle, the basic idea of anabsorption system is to replace the electricity consumption asso-ciated with the vapor compression by a thermally driven system,usually known as thermo-chemical compressor. This is accom-plished by making use of absorption and desorption process thatemploys a suitable working pair (refrigerant and absorbent), [24].Fig. 1 shows the single effect LiBreH2O absorption cycle in a pres-sureetemperature diagram. The system provides chilled water(QEVP) for cooling applications and could be activated using avail-able heat from different sources. The basic components are theabsorber (ABS), condenser (CON), generator (GEN) and evaporator(EVP), solution heat exchanger (SHX), refrigerant expansion valve(REV), solution expansion valve (SEV) and solution pump (SP). Itcan be seen in Fig. 1, when the refrigerant in vapor state comes fromthe evaporator it is absorbed in a liquid forming a weak solution.The liquid is pumped to a higher pressure, where the refrigerant isseparated from the solution by the addition of heat and then therefrigerant is directed to the condenser. Finally, the liquid

Fig. 1. Schematic diagram of a single-effect LiBreH2O ARS.

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e1812

containing less refrigerant (strong solution) is send back to theabsorber [10].

2.2. Thermodynamic modeling and assumptions

A thermodynamic analysis must be performed in order to obtainevery thermodynamic state in the ARS and involves the applicationof mass balance (Equation (1)) as well as the first law of thermo-dynamics (Equation (2)) to relate mass flows, enthalpies, heattransfer and power.

dmCV

dt¼

XIN

_m�XOUT

_m (1)

dECVdt

¼ _QCV � _WCV þXIN

_mh�XOUT

_mh (2)

On the other hand, and exergy analysis is performed using theExergy balance applied to each component using:

dBCVdt

¼ _QCV

�1� T0

T

�� _WCV þ

XIN

_mb�XOUT

_mb� _BD (3)

Not considering chemical exergy, the exergy of a stream can bedetermined according to the following expression:

b ¼ ðh� h0Þ � T0ðs� s0Þ (4)

Heat exchangers can be modeled using the logarithmic meantemperature difference, the heat transfer area, and the overall heattransfer coefficient with:

_Q ¼ U,A,LMTD (5)

The logarithmic mean temperature, which is calculated asa function of the hot and cold end temperature differences:

LMTD ¼ DT2 � DT1

lnDT2DT1

(6)

For the purpose of analysis, the following set of typicalassumptions is considered [9,10,12e16]:

� Steady state operation� Heat exchange between the system and surroundings, otherthan the prescribed by heat transfer at the generator,condenser, and absorber does not occur

� Cooling capacity is known ð _QEVPÞ, used to produce chilledwater

� Heat source is low grade steam� The solution and refrigerant valves are adiabatic� LiBr solutions in the generator and the absorber are in equi-librium at their respective temperatures and pressures

� Refrigerant at the condenser and evaporator exits is in a satu-rated state

� Strong solution of refrigerant leaving the absorber and theweak solution of refrigerant leaving the generator are saturated

� Work input to the solution pump is neglected� Thermodynamic properties of non-equilibrium solutions arethe same as the equilibrium values at the state with the sametemperature and concentration

� Pressure losses in all the heat exchangers and the pipelines areneglected

� The reference environment is defined with T0 ¼ 25 �C andP0 ¼ 101.3 kPa

� To avoid crystallization of the solution, the solution enteringthe throttling valve has the temperature at least 8 �C abovecrystallization

� Temperature of generator, condenser, evaporator, and absorberare uniform throughout the components

� The energy required for running the cooling tower and theassociated water pumps as well as cooling fans are notconsidered in the analysis

2.3. Thermodynamic properties

Thermodynamic properties are an essential part in the thermo-dynamic analysis of LiBreH2O ARS. Properties of wateresteam andthose for lithium bromide solution are needed and a brief review ofliterature shows several correlations [25e30]. However, in order toimplement a determined set of correlations to construct the opti-mization model, they have to satisfy a compromise between beingexpressions in uncomplicated formulation and at the same timeexpressions providing accuracy of results. Under this consideration,the concentration for the lithium bromide solution can be calculatedfrom Lansing [25], enthalpy, heat capacity and entropy from thecorrelations of Kaita [28]. In one hand, enthalpy of wateresteam canbedetermined fromthecorrelationof Lansing [25], in theotherhand,entropy from IAPWS formulation [30]. Finally, the pressure on thesystem can be established with the equation provided by Sun [26].

2.4. Economic model

In order to find the influence of the technical parameters on theeconomic performance of the ARS, it is necessary to perform aneconomic analysis. Asexposedearlier, themost appropriate approachis to join a determined economic parameter with the informationgiven by the thermodynamic analysis, specifically information fromthe second law. For this purpose, the annual operating cost, the costper exergy unit and overall economic cost of the final product are themost appropriated [31,32]. In this work, the annual cost of plantoperation will be investigated, it can be expressed as:

CT ðxÞ ¼ topC 3IN_BINðxÞ þ aC

Xnk

CCk þ bC (7)

In the previous equation (Equation (7)), the first term of theright side represents the cost due to fuel supply of the system

Fig. 2. Flow diagram of the single-effect LiBreH2O refrigeration system under study.

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e18 13

evaluated by the exergy input to the plant (second law infor-mation). The second term is the annual repayment which isnecessary to payback the investment after a specified period,whereas the third term represents the annual cost not affectedby the optimization. The recovery factor, aC, is determinedwith:

aC ¼ iRð1þ iRÞNyð1þ iRÞNy�1

(8)

2.5. Benchmark case

In order to validate the thermodynamic model, thermodynamicproperties and economic model proposed, the work of Misra [13]has been taken to establish a benchmark case. For this purpose,a computationally effective model of the ARS has been developed.Subsequently, the model was coded and solved in MATLAB usingthe set of input data shown in Table 1 and therefore determiningthe values of the benchmark case. Fig. 2 shows the detailed diagramfor the ARS under study.

The values obtained with the computational code are shown inTable 2 and Table 3. Firstly, Table 2 shows the thermodynamicsproperties whereas Table 3 displays the annual cost of plant oper-ation and other related parameters. The information provided inboth tables agrees well with the results of Misra [13]. Therefore, themodel proposed can be considered valid and it can be formulated asa mathematical programming problem to form the ARS non-linearprogramming model.

3. NLP model

In general terms, a NLP model is composed by three elements:objective function, equality constraints and inequality constrains.Mathematically, it can be enunciated according to the followingexpression [33]:

minx

f ðxÞ s:t:

hðxÞ ¼ 0gðxÞ � 0x˛<

(9)

The NLP model of the ARS can be related with each element of(9) as follows: first of all, objective function defined using aneconomical parameter; the equality constrains are related with thethermodynamic model and thermodynamic properties. Finally,

Table 1Set of input data corresponding to the benchmark case [13].

Variable/parameter Symbol Unit Value

Evaporator cooling load [kW] _QEVP kW 201.29Generator temperature T1 �C 80Condenser temperature T2 �C 35Evaporator temperature T4 �C 5Absorber temperature T5 �C 35Generator heating steam temperature T18 �C 100Condenser cooling water outlet temperature T12 �C 33Evaporator chilled water inlet temperature T13 �C 20Evaporator chilled water outlet temperature T14 �C 12Absorber cooling water inlet temperature T15 �C 27Solution heat exchanger effectiveness 3 % 60Reference temperature T0 �C 25Reference pressure P0 kPa 101.3Interest rate iR % 15Period of repayment Ny Years 10Time of operation per year top Hours 5000Unit cost of input exergy CE

IN $/kW h 0.03785

inequality constraints contain equations that allow the operation ofthe ARSwithin safe limits. For the ARS these elements are describedin detail in the following subsections.

3.1. Objective function (economic model)

The objective function to be utilized is the total annual costwhich formulation is:

minx

CT ðxÞ ¼ topC 3IN_BIN þ aC

Pk˛EQS

Zk

ck˛EQSXfEVP;CON;GEN;ABS; SHXg(10)

The investment cost is estimated through equation (11)considering the solution heat exchanger, generator, absorber,condenser and evaporator as simple heat exchangers. In theequation (11) Z0 and A0 represent a reference value and are taken as7900 $/kW and 100 m2, respectively [15,18].

Zk ¼ Z0

�Ak

A0

�0:6

ck˛EQSXfEVP;CON;GEN;ABS; SHXg(11)

3.2. Equality constraints

The thermodynamic model, including the thermodynamic prop-ertiesof theworkingpaircanbeseenas theequalityconstraints in theNLPmodel. In first place, the thermodynamicmodelwill be exposed.

3.2.1. Thermodynamic modelThe mass flow rates in the ARS can be obtained with equations

(12)e(14). Equation (12) determines the refrigerant flow rate. Boththe mass flow rate for the strong solution and the weak solution,are determined with equations (13) and (14), respectively.

_Qk ¼ _m4ðh4 � h2Þck; k ¼ EVP

(12)

_m8ðX8 � X5Þ ¼ _m4X5 (13)

_m5ðX8 � X5Þ ¼ _m4X8 (14)

Table 2Thermodynamic properties of the benchmark case (this work).

Stream Substance m T P x h s B

kg/s �C kPa % kJ/kg kJ/kg K kW

1 Waterevapor 0.085 80.00 5.65 0.00 2650.73 8.609 7.5192 Water 0.085 35.00 5.65 0.00 146.62 0.504 0.0683 Water 0.085 35.00 0.87 0.00 146.62 NC NC4 Waterevapor 0.085 5.00 0.87 0.00 2511.68 9.016 �14.6295 LiBreWater 0.989 35.00 0.87 55.21 115.87 0.223 53.2846 LiBreWater 0.989 35.00 5.65 55.21 NC NC NC7 LiBreWater 0.989 58.49 5.65 55.21 162.93 0.372 56.0228 LiBreWater 0.904 80.00 5.65 60.40 230.54 0.455 89.7649 LiBreWater 0.904 53.00 5.65 60.40 179.06 0.304 83.87310 LiBreWater 0.904 53.00 0.87 60.40 179.06 NC NC11 Water 19.429 30.38 100.00 0.00 127.28 0.441 5.62812 Water 19.429 33.00 100.00 0.00 138.25 0.477 10.70813 Water 6.010 20.00 100.00 0.00 83.82 0.296 0.17314 Water 6.010 12.00 100.00 0.00 50.32 0.182 4.03915 Water 18.447 27.00 100.00 0.00 113.13 0.394 1.27116 Water 18.447 30.38 100.00 0.00 127.28 0.441 5.34417 Waterevapor 0.121 100.00 101.32 0.00 2677.54 7.364 58.76318 Water 0.121 100.00 101.32 0.00 418.76 1.305 4.110

NC e Not calculated.

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e1814

Based on the weak solution side, the effectiveness of the solu-tion heat exchanger can be related with temperatures by usingequation (15):

T9 ¼ T1 � 3SHXðT1 � T5Þ (15)

Based on the strong solution side and combining with equations(13) and (14) the effectiveness can be stated as:

ðT7 � T5ÞðX8Cp5Þ ¼ 3SHXX5Cp8ðT1 � T5Þ (16)

Heat balance of the condenser, absorber and generator give theequality constraints for such devices, equations (17)e(19),respectively.

_Qk ¼ _m4ðh1 � h2Þck; k ¼ CON

(17)

_Qk ¼ h4 _m4 þ h9 _m8 � h5 _m5ck; k ¼ ABS

(18)

_Qk ¼ h8 _m8 þ h1 _m4 � h7 _m5ck; k ¼ GEN

(19)

The logarithmic mean temperature difference is used todetermine the heat transfer characteristics of each heatexchanger. The value for the overall heat transfer coefficient istaken in the range of 0.2e0.5 kW/m2 K, according to [14] and[16]. Equations (20)e(22) are the set of equality constraints tobe satisfied.

_Qk ¼ UkAkLMTDkck˛EQSXfEVP;CON;GEN;ABS; SHXg (20)

Table 3Annual cost of plan operation and other parameters (this work).

Parameter Symbol Unit Value

Coefficient of performance COP adim 0.7376Total irreversibility rate BD,TOT kW 41.634System capital cost ZTOT $ 45,489Annual cost of plan operation CT $/year 16,211

LMTDk ¼ DT2k � DT1k

lnDT2kDT1k

ck˛EQSXfEVP;CON;GEN;ABS; SHXg

(21)

Where:

DTk2 ¼ Thik � Tcok

DTk1 ¼ Thok � Tcikck˛EQSXfEVP;CON;GEN;ABS; SHXg

(22)

The mass flow of chilled water is related with refrigeration loadby means of equation (23) and must satisfy the constraint estab-lished with equation (12).

_Qk ¼ _m13ðh13 � h14Þck; k ¼ EVP

(23)

The mass flow of cooling water in the condenser and theabsorber are determined by using equations (24) and (25), satis-fying the equality constrains of equations (17) and (18),respectively.

_Qk ¼ _m11ðh12 � h11Þck; k ¼ CON

(24)

_Qk ¼ _m15ðh16 � h15Þck; k ¼ ABS

(25)

Finally, the mass flow of steam that must be provided to the ARSin order to activate the system is written with the followingequality constraint:

_Qk ¼ _m17ðh17 � h18Þck; k ¼ GEN

(26)

3.2.2. Thermodynamic propertiesThe pressures on the system are established and determined

with the equation proposed by Sun [26], using the temperatures ofthe condenser and evaporator provided in the set of input data(Table 1):

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e18 15

Pi ¼ exp�9:48654þ 3892:7

42:6776� Ti

�

ci; i ¼ 2;4(27)

The concentration in the system, for the state 5 and 8, isdetermined using the formulation given in Lansing [25]:

X5 ¼ 49:04þ 1:125T5 � T4134:65þ 0:47T5

(28)

X8 ¼ 49:04þ 1:125T1 � T2134:65þ 0:47T1

(29)

Enthalpies for states 1, 2 and 4 can be calculated following theexpressions provided by Lansing [25]. States 2 and 4 are consideredas saturated liquid and saturated vapor, respectively:

h1 ¼ 104:753þ ð2398:19904þ 1:925928T1 � 0:1800324T2Þ(30)

h2 ¼ 104:753þ ð4:1868T2 � 104:67Þ (31)

h4 ¼ 104:753þ ð2398:19904þ 1:7458956T4Þ (32)

The enthalpy of the LiBreH2O mixture is calculated using theexpressions proposed by Kaita [28]. With equation (33), theenthalpy of state 5 is calculated and using equation (34) enthalpiesof states 8 and 9 are determined.

hi ¼ ða0þa1X5ÞTiþ0:5ðb0þb1X5ÞT2i þ�d0þd1X5þd2X

25þd3X

35

�

ci; i¼ 5 (33)

hi ¼ ða0þa1X8ÞTiþ0:5ðb0þb1X8ÞT2i þ�d0þd1X8þd2X

28þd3X

38

�

ci; i¼ 8;9 (34)

Thermal capacity of the weak and strong solution is determinedwith:

Cp5 ¼ ða0 þ a1X5Þ þ ðb0 þ b1X5ÞT5 (35)

Cp8 ¼ ða0 þ a1X8Þ þ ðb0 þ b1X8ÞT1 (36)

In equations (33)e(36) the values for the constants are:a0 ¼ 3.462023, a1 ¼ �2.679895 � 10�2, b0 ¼ 1.3499 � 10�3,b1 ¼ �6.55 � 10�6, d0 ¼ 162.81, d1 ¼ �6.0418, d2 ¼ 4.5348 � 10�3,d3 ¼ 1.2053 � 10�3.

The energy balance applied to the solution heat exchanger,considering that the enthalpy of state 6 has the same value of state5 (since the pump work is negligible), gives the equality constraintto be satisfied and related the enthalpy of state 7:

_m5ðh7 � h5Þ ¼ _m8ðh8 � h9Þ (37)

Enthalpies for reference state as well as states 11 to 19 are ob-tained from the expressions of Lansing [25], equations (38) and (39):

Table 4Constants in the equation of entropy proposed by Kaita [28].

j Bj0 Bj1

0 5.127558 � 10�1 �1.393954 � 10�2

1 1.226780 � 10�2 �9.156820 � 10�5

2 �1.364895 � 10�5 1.068904 � 10�7

3 1.021501 � 10�8 0

hi ¼ 104:753þ ð4:1868Ti � 104:67Þci; i ¼ 0;16 and 18

(38)

hi ¼ 104:753þ ð2398:19904þ 1:7458956TiÞci; i ¼ 17 (39)

The entropy of state 1 is calculated using the IAPWS formulationfor region 2 using part of the code developed by Holmgren [34]through an equation with the form of equation (40). Furtherdetails can be found in [30].

siðp; sÞR

¼ s�g0s þ grs

���g0 þ gr

�

ci; i ¼ 1(40)

Entropies for states 0, 2, 11 to 16 and 18 are determined witha polynomial expression obtained using the software EES [35],which is valid only for saturated liquid, equation (41). A similarexpression for states 4 and 17 that belong to saturated vapor isobtained, equation (42). These expressions are valid only for thetypical range of temperatures of ARS.

si ¼ 0:00513126þ 0:0149286Ti � 0:0000192598T2i

ci; i ¼ 0;2;11� 16;18(41)

si ¼ 9:13788� 0:0247901Ti � 0:0000705193T2i

ci; i ¼ 4;17(42)

The entropy of the LiBreH2O mixture is calculated using theexpressions proposed by Kaita [28], for the states 5, 7, 8 and 9.Values for constants Bjk are given in Table 4.

si ¼P3j¼0

P3k¼0

BjkXkTj

ci; i ¼ 5 and 7;9(43)

Once enthalpies and entropies have been expressed as equalityconstraints, the exergy of states 1 to 18 can be calculated by usingequation (4). Additionally, exergy destroyed in each component canbe obtained with equation (3), resulting in the following set ofequality constraints:

_BD;EVP ¼ �_B2 þ _B13

�� �_B4 þ _B14

�(44)

_BD;CON ¼ �_B1 þ _B11

�� �_B2 þ _B12

�(45)

_BD;GEN ¼ �_B17 þ _B7

�� �_B1 þ _B8 þ _B18

�(46)

_BD;ABS ¼ �_B4 þ _B9 þ _B15

�� �_B5 þ _B16

�(47)

_BD;SHX ¼ �_B5 þ _B8

�� �_B7 þ _B9

�(48)

In the exergy balance of equations (47) and (48) the throttlingvalve and the solution pump are included respectively. Finally, the

Bj2 Bj3

2.924145 � 10�5 9.035697 � 10�7

1.820453 � 10�8 �7.991806 � 10�10

�1.381109 � 10�9 1.529784 � 10�11

0 0

Table 5Comparison of optimal values of this work with [13].

Variable Symbol Unit Value

This work Ref. [13]

Decision variablesGenerator temperature T1 �C 84.8 88.8Condenser temperature T2 �C 39.8 34.8Evaporator temperature T4 �C 8.6 8.8Absorber temperature T5 �C 35.5 31.0Solution heat exchanger

effectiveness3 % 70.7 NRa

Objective FunctionAnnual cost of plant operation CT $/year 14,993.9 15,146.0

Other parametersSystem capital cost ZTOT $ 41,718 60,730Coefficient of performance COP adim 0.7755 NRTotal irreversibility rate BD,TOT kW 39.171 22.38Evaporator heat flow rate _QEVP kW 201.29 201.29Generator heat flow rate _QGEN kW 259.55 NRCondenser heat flow rate _QCON kW 213.37 NRAbsorber heat flow rate _QABS kW 247.47 NRGenerator heat transfer area AGEN m2 133.79 401.58Condenser heat transfer area ACON m2 127.37 249.39Evaporator heat transfer area AEVP m2 144.85 131.43Absorber heat transfer area AABS m2 98.21 190.10Solution HX heat transfer area ASHx m2 54.59 85.80

a NR e Not reported.

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e1816

exergy input to the system can be stated as equation (49) and mustsatisfy the total annual cost established by the objective function ofthe NLP model.

_BIN ¼ Pk

_BD;k þ�_B13 � _B14

�ck˛EQSXfEVP;CON;GEN;ABS; SHXg

(49)

3.3. Inequality constraints

Inequality constraints represent design specifications, such asminimum and maximum equipment capacities and upper andlower limits on design variables, the following equation (50) showsthis:

lb � x � ub (50)

Also, inequality constraints are used in order to prevent negativevalues of the variables. However, an important constraint is the onethat assures that temperature of solution, before entering thethrottling valve, does not reach the crystallization temperature:

T9 � TcðX8Þ � 8 (51)

4. Decision variables

Decision variables are those variables that maximize orminimize the objective function. In this case, the decision vari-ables are the temperature of generator, temperature ofcondenser, temperature of evaporator, temperature of absorberand effectiveness of solution heat exchanger. The appropriate setof these variables will minimize the annual operating cost of theLiBreH2O absorption refrigeration system under the assumptionsconsidered.

5. Results

The NLP model developed was implemented in the modelingsystem GAMS interfacing with CONOPT as optimization package.CONOPT solver is designed to manage NLP problems andemploys the sequential quadratic programming (SQP) methodthat uses exact second derivatives to compute better searchdirections in order to reach faster the solution, more details canbe found in [36]. The optimization problem has 115 equationsand 119 variables. It was solved in 0.031 s using a machine with2 GB RAM and processor with 1.67 GHz. It is worth noting thatglobal optimality of the solution found cannot be guaranteed,since CONOPT is a local optimizer. Thus, these solutions must beregarded as locally optimum unless a global optimizationmethod is employed. However, we consider that a local solutionof the problem is valid for the purpose of the analysis per-formed because the results obtained do agree well with thework being in comparison.

The optimal values of the ARS optimization model of this workare shown in Table 5. This includes the results of the work ofMisra [13] who optimized the ARS using the thermoeconomicalapproach. The decision variables are the temperature of thegenerator, the condenser, the evaporator and the absorberincluding the effectiveness of the solution heat exchanger. It canbe seen that the values are slightly different. For instance, thetemperature of the generator is 84.8. For this work the comparedvalue is 88.8 �C. The condenser and absorber temperature is 5 �Cand 4.5 �C above, respectively. The evaporator remains almostwith a value of 8.6 �C. In the Misra’s research [13] the value of the

solution heat exchanger effectiveness is not reported, but usingtheir own data in the MATLAB code developed in this work itshowed a value of 0.65. The set of values obtained here, leads usto achieve a value of the objective function (the annual cost ofplant operation) with $14,993 (US dollars) per year, this amountwas decreased in about $150 per year compared with the value of$15,146. This decreasing fact is directly related with the lowerinvestment cost of about $41,700 which was originated due to lessheat transfer area of ARS heat exchangers. It can be seen that heattransfer areas are reduced significantly. For instance, the heattransfer area of the generator was reduced from 401 m2 to133 m2. This variation is mainly due to the range of values utilizedfor the overall heat transfer coefficient. From the point of view ofthe optimization procedure, differences in values of the decisionvariables is mainly attributable to the lower value achieved in theobjective function. In other words, the solver finds the set ofvalues that minimize the annual operating cost until reach$14,993 upon the established assumptions. That is, the objectivefunction would not have achieved a lower value compared to thebase case, for the same values of the decision variables in bothcases.

Although the solution of the NLP model proposed found newoptimal values and the value of the objective function wasreduced, it can be seen that the exergy destruction was increasedconsiderably from 22.38 kW to 39.17 kW. In this case, the absorberis responsible of 12.042 kW, the generator of 10.543 kW, in itsturn, the evaporator and the condenser with 8.204 kW and5.519 kW, respectively (not shown in Table 5). From the thermo-dynamic point of view, this fact is not favorable because theenergy resources are not being used efficiently. However, with themodel proposed through uni-objective optimization it is notpossible to assure the minimum values of the majority ofparameters involved.

Finally, Table 6 shows the thermodynamic properties of eachflow in the absorption refrigeration system for the optimumcase. The values in the table are obtained using the optimizeddecision variables and the code developed in MATLAB; whichincludes each property formulation for steam and LiBreH2Omixture.

Table 6Thermodynamic properties of the optimum case.

Stream Material flow m T P x h s B

kg/s �C kPa % kJ/kg kJ/kg K kW

1 Waterevapor 0.086 84.86 7.35 0.00 2659.20 8.513 10.7422 Water 0.086 39.85 7.35 0.00 166.92 0.569 0.1383 Water 0.086 39.85 1.12 0.00 166.92 NC NC4 Waterevapor 0.086 8.67 1.12 0.00 2518.09 8.928 �11.9335 LiBreWater 0.745 35.51 1.12 53.07 108.18 0.239 30.7356 LiBreWater 0.745 35.51 7.35 53.07 NC NC NC7 LiBreWater 0.745 64.32 7.35 53.07 167.45 0.426 33.5438 LiBreWater 0.659 84.86 7.35 59.96 237.55 0.485 64.2419 LiBreWater 0.659 49.97 7.35 59.96 170.59 0.289 58.57110 LiBreWater 0.659 49.97 1.12 59.96 170.59 NC NC11 Water 19.451 30.38 100.00 0.00 127.28 0.441 5.63512 Water 19.451 33.00 100.00 0.00 138.25 0.477 10.72013 Water 6.010 20.00 100.00 0.00 83.82 0.296 0.17314 Water 6.010 12.00 100.00 0.00 50.32 0.182 4.03915 Water 17.488 27.00 100.00 0.00 113.13 0.394 1.20416 Water 17.488 30.38 100.00 0.00 127.28 0.441 5.06617 Waterevapor 0.115 100.00 101.32 0.00 2677.54 7.364 55.89318 Water 0.115 100.00 101.32 0.00 418.76 1.305 3.909

C. Rubio-Maya et al. / Applied Thermal Engineering 37 (2012) 10e18 17

6. Conclusions

In the present work it was addressed the optimization of singleeffect LiBreH2O ARS using the approach based on mathematicalprogramming. We established a Non-linear Programming model,using the exergy concept and defining the annual operating cost asthe objective function to be minimized. In order to validate themodel proposed a benchmark case, it was solved and coded withsimple formulations for the thermodynamic properties of waterand the LiBreH2O mixture. Then, the code was formulated as anoptimization model and the results were compared with a ther-moeconomic method of optimization. The following conclusionsare stated:

� The optimization of single effect LiBreH2O ARS under theapproach based on mathematical programming and using theexergy concept through the annual operating cost as theobjective function to be minimized is possible. Therefore, it canbe stated that the model proposed is simple and structured andalso it reduces the typical complexity of thermal systemsproviding an alternative approach to optimize ARS.

� Modern codes to solve NLP problems allow addressing complexoptimization of thermal systems and the combination withhigh performance computers reduces costs, both computa-tional and economically.

� Exergy is a useful tool in design and optimization stages of ARS,due to its ability to identify which subsystem is less efficient interms of exergy destroyed. The influence of the most relevantdesign variables can be also located.

� The availability of several formulations for the thermodynamicproperties allows implementing simple expressions to formthe equality constraints without losing accuracy.

Acknowledgements

The authors wish to acknowledge the financial support of thisresearch to the Mexican Ministry of Public Education under thePROMEP programwith project’s reference PROMEP/103.5/10/7389.

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