ab

s b

h i g h l i g h t s

Comparison of four empirically based models Experimental data of 12 kW absorption chille DDt0 , MPR, ANN methods are suitable for com

w a slig

Performance analysisModeling

sorption chiller performance is presented. Four empirically based models: the adapted Gordon-Ng model(GNA), the characteristic equation model (DDt0), the multivariable polynomial model (MPR) and the

models in complete energy supply and demand models included insimulation software packages. These simple chiller models, char-acterized by a low number of input parameters, can serve to facil-itate the annual simulations of complex building systems providingat the same time an adequate level of performance prediction. Also,

which may help in

both physical andd in the literature.were reported byost recent or rele-eloped a modular

simulation tool for absorption systems called ABSIM. With thissoftware is possible to study various absorption cycle congura-tions using different working uids. ABSIM calculates the cycleinternal state points and thermal loads in each component using acycle conguration build by the user graphically and for givenworking uid specications and operating conditions. This isenabled through the governing equations for each component ofthe cycle contained in the software subroutines. However, thecalculation convergence is not always easy. Silverio and Figueiredo

* Corresponding author. Tel.: 34 977 297068; fax: 34 977559691.

Contents lists available at

Applied Therma

sev

Applied Thermal Engineering 58 (2013) 305e313E-mail address: juancarlos.bruno@urv.cat (J.C. Bruno).1. Introduction

The main aim of this paper is to present a comparative evalua-tion of different modeling approaches for predicting the perfor-mance of small absorption chillers. The comparative evaluation canserve as a reference when there is a need for simple, but accuratemodels of absorption chillers, for example to integrate these

this paper aims to provide a statistical approachselecting the appropriate model.

With respect to absorption chiller modeling,empirical approaches were many times presentePhysical or more precise thermodynamic modelsmany authors. Here just a brief review of the mvant will be given. Grossman and Zaltash [1] devonly the variables of external water circuits as model input parameters. 2013 Elsevier Ltd. All rights reserved.Statistical indicators articial neural networks model (ANN) were applied using the experimental data and thoroughlyexamined. The paper also presents statistical indicators and tests which might assist in selection of themost appropriate model.The excellent statistical indicators such as coefcient of determination (>0.99) and coefcient of

variation (

l En[2] used a thermodynamic approach for steady-state simulation ofan ammonia-water absorption system. The thermodynamic staterelations, the pressure drop equations and the heat transfer co-efcients were solved by using an algorithm based on the Substi-tution Newton Rapshonmethod. Kaynakli and Kilic [3] performed atheoretical study on the performance of a H2OeLiBr absorptionsystem using a thermodynamic analysis of the absorption cycle.These authors investigated the inuences of the driving tempera-ture and heat exchanger effectiveness on the thermal loads of thecomponents and COP. Yin et al. [4] developed a detailed thermo-dynamic model of a 16 kW double-stage H2OeLiBr absorptionchiller. The steady-state model was based on the working uidsproperty relations, detailed mass and energy balances, and the heatand mass transfer relationships for each chiller component. One ofthe most recent application of the thermodynamic approach inabsorption system modeling can be found in the paper of Wu et al.[5]. The authors developed thermodynamic models of differentabsorption heat pump cycles to test their applicability withdifferent heat sources, working pairs and in different cold regions.All these thermodynamic models are very demanding since theyrequire comprehensive knowledge of the absorption cycleincluding some internal state points. These models need lots ofinput parameters such as heat transfer coefcients (U) and heattransfer areas (A) of heat exchangers, the rich solution ow rate,working uid properties and water side ows and temperatures aswell as some additional assumptions for the convenience ofmodeling. A more complete explanation on all these degrees offreedom in the modeling of absorption chillers can be found inDereje et al. [6]. In practice, however, especially with commercialunits, the internal parameters are not available. This is the reasonwhy thermodynamic models are more adequate during the designstage of absorption equipment as explained in the paper of Florideset al. [7]. Also, the computation time in simulation software pack-ages using these models is very long since they require a lot ofsimultaneous iterations. The annual simulation of absorptionchillers under different ambient and operating conditions on anhourly time step basis is a clear example of this.

Thus, there is a need for simple models which can providesufciently good representation of the absorption machinebehavior based only on available external parameters (experi-mental measurements or manufacturer catalog data). Simplemodels can be more easily incorporated in simulation programs orused for fault detection and control. Contrary to the physicalmodels, the empirical and semi-empirical models require less timeand effort to develop and computation time is much shorter whenthey are built into complete energy management simulation pro-grams. The parameters and tting coefcients in these models aredetermined by using a regression method or a minimization algo-rithm applied to a dataset obtained performing experimentalmeasurements or using a manufacturer catalog.

The studies about development of empirically based models forabsorption chillers have been reported by several authors. Gordonand Ng [8] developed a general model for predicting the absorptionchillers performance. The model lays both on physical and empir-ical principles. The physical principles that govern the performanceof the absorption chiller are tted to the experimental or manu-facturer data by using a regression method. Ziegler et al. [9,10]developed a model (Characteristic equation method) which pre-dicts the performance of the absorption chiller by using two simplealgebraic equations: one to calculate the cooling capacity andanother for the driving heat input. These two previous modelsbelong to semi-empirical (gray-box) category of models, in whichthe tted parameters can be interpreted under the actual physicalprinciples which govern the absorption chiller performance. Labus

J. Labus et al. / Applied Therma306et al. [11] used a completely empirical approach to modelabsorption chillers based on manufacturers curves in order toinvestigate the energy savings when different absorption chillercongurations were considered for their integration in a completechiller plant.

The Articial neural networks approach has been also used forabsorption chiller modeling. ANN models belong to the black-boxmodel category, that unlike gray-box models, the estimated pa-rameters of the model have no physical interpretation. Szen [12]used the ANN to determine thermodynamic properties of analternative working pair for absorption systems. The study alsodemonstrated that ANN can replace mathematical models in thesimulation of absorption systems. In the paper of Szen andAkayol [13] the ANN approach was proposed for performanceanalysis of an absorption chiller. The ANN model used only theworking temperatures in the four main components as input pa-rameters in order to predict the performance of the chiller. Man-ohar et al. [14] applied ANN for the modeling of steam red doubleeffect absorption chiller. Later, a similar work was carried out byRosiek and Batlles [15], who used ANN to model solar-assisted airconditioning systemwith hotwater driven double effect absorptionchiller. The last approach considered in this paper is the simplemultivariable polynomial regression which also belongs to theblack-box category of models.

Regardless the numerous studies on the modeling of absorptionequipment, literature review shows that there is a lack of infor-mation with respect to comprehensive comparative studies ondifferent modeling techniques for predicting absorption equipmentperformance in a similar way as Swider [16] or Lee et al. [17] did forthe case of vapor-compression chillers.

The main aim of this paper is to present a comparative evalua-tion of different modeling approaches for predicting the perfor-mance of absorption systems. In the next section are presented theexperimental data and a brief description of the evaluated types ofmodels. Later the application of these models to the experimentaldata is evaluated with the help of statistical indicators and statis-tical tests to select the best modeling approach.

2. Experimental data and absorption chiller models

Four different types of absorption chiller models were devel-oped and examined:

Adapted GordoneNg model, Adapted characteristic equation model, Multivariate polynomial regression model and Articial neural networks model.

The experimental data required for the models developmentwere obtained in the state-of-the-art test bench of the Rovira iVirgili University in Tarragona (Spain). The test bench is fullyequipped to test under controlled operation conditions a variety ofunits commonly used in HVAC systems. A more detailed explana-tion about the functionality of the test bench can be found in Labuset al. [18] and Labus [19]. For the models described in this researchthe data were collected in a series of experiments with a 12 kWabsorption chiller Pink Chilli PSC12. The measured variables in theexperiments were inlet and outlet temperatures of hot, chilled, andcooling water circuit; volumetric ow rates and pressure drop ineach circuit; and electric consumption of the chiller. The raw datawere processed using a comprehensive test procedure which in-cludes several techniques: data reduction, development of steady-state detector with additional ltering and uncertainty estimation.Based on external measurable parameters only, this procedure al-lows the creation of the complete performance map for absorption

gineering 58 (2013) 305e313machines based on highly accurate data. In data reduction, the

collected data were used to calculate thermal loads and efciencycoefcients. Off-line steady-state detector for absorption chillerswas developed based on analogy with steady-state detector forvapor compression chillers using the moving window average.Additional data ltering was engaged to eliminate remainingtransient periods caused by time delays when changing from onesteady-state to another. The evaluation of experimental uncertaintywas carried out by judgment based on available information on thepossible variability of input quantities. When the uncertainties ofheat loads were evaluated, the following input quantities were

modeling. Models developed with small datasets are not reliable

1COP

Tac TevaTout

$gen

T in T in 1: $

gen

T in T in

J. Labus et al. / Applied Thermal Enand statistically correct, since small datasets are insufcient to formstrong relationships within the models. On the other hand, themodels created with large dataset which completely covers theoperating range of the absorption chiller show very high level ofpredicting capabilities.

2.1. Adapted GordoneNg model (GNA)

The general thermodynamic model for absorption chillersdeveloped by Gordon and Ng [8] is actually a combination ofphysical and empirical approaches. According to the authors, thedominant irreversibility of the absorption chillers is nite-ratemass transfer. The losses due to the nite-rate mass transfer cantherefore be approximated as temperature independent. Theoriginal model was based on external input parameters of the fourmain components (generator, condenser, evaporator and absorber)assuming that manufacturers catalogs provide the operating con-ditions for each of them. However, the current manufacturerspractice is to provide operating curves based on three circuits, i.e.treating absorber and condenser as one component. The mainreason for that is the arrangement in series of the absorber andcondenser in the majority of the commercial absorption chillers.Therefore, in our case the original model was modied considering

Table 1Experimental operation range conditions.

Variable Range

T ineva [C] [4.98e12.1]

Toutac [C] [26.95e35.01]

T ingen [C] [79.9e100.12]

Q:

eva [kW] [0.49e15.23]Q:

ac [kW] [5.99e41.98]Q:

gen [kW] [4.64e24.04]COP [0.11e0.76]

3 : :taken into account: inlet and outlet temperatures of external cir-cuits, volumetric ow rate, density, and specic heat capacity.Uncertainty contribution for temperature and volumetric ow ratewas calculated as a combination of different sources of uncertainty:repeatability, accuracy of the instrument and resolution of the in-strument. In order to be in accordance with international stan-dards, time length for the steady-state tests was not shorter than30min collected in 5s intervals. The experimental database used formodeling consists of 138 steady-state points and covers thefollowing temperatures ranges: inlet hot water temperature 80e100 C, inlet cooling water temperature 27e35 C and outlet chilledwater temperature 5e12 C, as presented in Table 1. The main cri-terion to select these three temperatures as input variables for theempirical models was their availability to the operating engineersin practical applications. In detail description of the test procedureas well as the experimental results can be found online in LabusPhD thesis [19].

Also, it is important to explain the inuence of database size onFlow rates [m /h] xed at: meva 1:7; mac 4:8;m:gen 2:2eva gen ac Qeva gen ac

$

"a1 a2$

T inacT ingen

# (1)

where a1 and a2 are the regression parameters to be tted withexperimental data and, at the same time, the constants whichcharacterize the entropy generation of particular chiller.

Considering that a plot ofT inacT ingen

against

"T ingen T inacT ingen$COP

T inac ToutevaTouteva

#$Qeva

:

leads to a straight line, is it possible to calculate a1

and a2 as the intercept and slope of this line using linear regression.Bearing in mind that the purpose of this analysis is to comparedifferent modeling approaches bymeans of the deviations betweenexperimental and modeled heat loads, the nal equation of theGNA model (eq. (1)) was adapted to obtain the chiller capacity (Eq.(2)). The heat input can be derived from the COP (Eq. (3)).

Q:

eva B

1=COP A (2)

Q:

gen B

1 A$COP (3)

where:

A "T inac Touteva

Touteva

#$

"T ingen

T ingen T inac

#;

B "

T ingenT ingen T inac

#$

"a1 a2$

T inacT ingen

# (4)

2.2. Adapted characteristic equation model (DDt0)

For the modeling of absorption chillers, Ziegler et al. [9] devel-oped an approximate method which is able to represent bothcooling capacity and driving heat input by simple algebraic equa-tions. These equations are expressed as a function of so-calledcharacteristic temperature difference (DDt), which depends on theaverage temperature of the external heat carrier uids. One of themain assumptions is that the heat transfer processes in absorptionchillers dominate their performancebehavior. In thisway, a complexresponse to all external heat carrier temperatures is reduced to alinear function of heat ow and the external temperatures.

A simple linear correlation is very convenient, but it has beenfound that the predicted performance of the cooling capacity de-the absorber and condenser as a single source of heat at mediumtemperature.

According to Gordon and Ng approximation, the nite-ratemasstransfer is roughly temperature independent. With respect to that,the losses in the evaporator can be neglected, while the losses inother two heat exchangers (generator, absorber/condenser) can beviewed as a constant characteristic of each particular chiller. Thegeneral equation for the GNA model can be obtained after series oftransformation, starting from the First law of thermodynamics andusing the entropy balance which takes into account the dominantirreversibility [8,19]. The GNA model calculates the inverse of COPusing the following equation (Eq. (1)):

"in out

# "T in

# " # "T in

#

gineering 58 (2013) 305e313 307viates considerably from the linear behavior, for instance, at high

To test the robustness and the prediction ability of the models, the

l Endriving temperatures, due to higher internal losses. With respect tothat, an adapted characteristic equation method was proposed byKuhn and Ziegler [20]. This improved model uses a numerical t ofcatalog or experimental data to improve the characteristic equa-tion. The adapted characteristic temperature function (DDt0) takesthe form (Eq. (5)):

DDt0 tgen a$tac e$teva (5)And the linear characteristic equation for each component loads

k (Eq. (6)):

Q:

k s0$DDt0 r (6)Combining Eqs. (5) and (6) yields one correlation which repre-

sents the thermal performance of the components as a function ofthe external arithmetic mean temperatures of the generator (tgen),absorber-condenser (tac) and evaporator (teva), when the externalow rates are constant.

Q:

k s0$tgen s0$a$tac s0$e$teva r (7)The four parameters (s, a, e and r) are estimated by using a

multiple linear regression algorithm to t the experimental data.This algorithm chooses regression coefcients to minimize the re-sidual sum of squares. The analyses of Puig-Arnavat et al. [21]conrmed the capability of the DDt0 method to obtain good re-sults and also better accuracy than the original method DDt. Finally,the combination of the obtained characteristic functions with theequations of the external arithmetic mean temperatures and withthe external energy balances, results in a system of six equationswith six unknowns which can easily be solved. The developedmodel requires only three temperatures (one from each of theexternal circuits) at constant ow rates of external heat carriers topredict the performance of the absorption chiller.

2.3. Multivariate polynomial regression model (MPR)

The MPR models belong to the black-box category of models,which do not carry the information about the physical processesincorporated in the model structure. MPR models are a veryeffective tool for describing complex non-linear relationships be-tween input and output variables without disregarding what oc-curs within the system. The parameters for the MPR model arecalculated by tting the experimental data minimizing the sum ofsquares of the residuals using a polynomial function. Due to theirsimple structure, MPR models have been applied in various eldssuch as forecasting, control, optimization, fault detection anddiagnosis [17,22]. Lee et al. [17] proved that MPR model of vapor-compression water chiller can have a high prediction accuracy,with the coefcient of variation of 0.61%. Similarly, the paper of Kimet al. [22] conrmed that MPR models are acceptable for faultdetection and diagnosis of residential heat pump systems. A typicalpolynomial regression model contains the squared and higher or-der terms of the estimator variable. Normally, the higher orderMPRmodels offer better accuracy of prediction. However, high-orderMPR can become impractical due to its excessive number of pa-rameters. One of the common techniques in the case of the highorder MPR models with large number of parameters is to reducethe model by retaining only those parameters that are statisticallysignicant. Also, excessive polynomial order for a relatively smalldatabase may worsen data interpolation. These are some of thereasons why it has been decided to apply only second order poly-nomials to predict the absorption chiller performance. Thus, theMPR models were developed to calculate the thermal loads of the

J. Labus et al. / Applied Therma308absorption chiller by using the measurements of external circuits:experimental dataset was split into three parts: 70% of data wasused for the model training, 20% for the model validation and theremaining 10% for the model testing.

The ANN absorption chiller model to calculate the thermal loadin each component is given by the general equation (Eq. (9)):

Q:

k Xji

"LW1;j$

2

1exp2PR

1IWj;RIRb1j1

!#b2

(9)

where I is the input, R is the number of the inputs (R 3), b aregenerator inlet temperature, absorber/condenser inlet tempera-ture, and evaporator outlet temperature. The generalized secondorder model in case of absorption chillers can be represented usingEq. (8):

Q:

k b0;k b1;kT ingen b2;kToutac b3;kT ineva b4;kT ingenToutac b5;kT ingenT ineva b6;kToutac T ineva b7;k

T ingen

2 b8;k

Toutac

2 b9;kT ineva2(8)

2.4. Articial neural network model (ANN)

ANN models also belong to the group of black-box models. Anarticial neural network is an adaptive systemwhich can be trainedto perform a particular function or behavior on the basis of inputand output information that ows through the network. ANN havefound their place in the elds of modeling, identication, optimi-zation and control in steady state and dynamic systems due to theirability to model complex relationships between inputs and outputsor to nd patterns in data. Various applications of neural networksin renewable energy problems such as energy prediction andoptimization of energy consumption in building service systemswere presented in the review of Kalogirou [23]. The review ofMohanraj et al. [24] covers the applications of ANN in energy andexergy analysis of refrigeration and air-conditioning systems, theircontrol and in prediction of refrigerant properties. Moon et al. [25]developed an adaptive control method using the ANN model toenhance thermal comfort in buildings. Yabanova and Keebas [26]developed ANN-based PID controller for geothermal district heat-ing system in Turkey, which increased energy efciency and costsaving of the system by 13%. The most common ANN architectureapplied in the eld of absorption systems and their applications arefeed-forward neural networks with back-propagation [27].

In this research, ANN models of the absorption chiller weredeveloped by using MatLab Neural Network toolbox. Since there isno explicit rule to determine the topology of ANN (the number ofneurons in the hidden layer or the number of hidden layers) thetrial and error method is usually applied to nd the best solution.Thus, the adopted topology for ANN models was (3e7e1), asillustrated in Fig. 1. Each model consists of one input layer withthree variables, one hidden layer with seven neurons and oneoutput layer with one output: a component load (three differentANN models are built for thermal power exchanged in the evapo-rator, absorber/condenser and generator). The training of the ANNwas based on the error back propagation technique using theLevenbergeMarquardt algorithm of optimization. The input pa-rameters were normalized in the [0.2, 0.8] range. A hyperbolictangent sigmoid function (tansig) was used in the hidden layer andthe linear transfer function (purelin) was used in the output layer.

gineering 58 (2013) 305e3131

biases in the hidden layer, b2 are biases in the output layer, J is the

AngelResaltado

AngelSubrayado

AngelSubrayado

AngelSubrayado

AngelSubrayado

AngelSubrayado

AngelSubrayado

AngelSubrayado

AngelSubrayado

AngelSubrayado

J. Labus et al. / Applied Thermal EnFig. 2 shows the comparison of the measured and calculatedcooling capacity of the chiller with a generator and condenser/absorber temperature of 85 C and 27 C, respectively. As can beThe developed models require different parameters in order tocalculate the absorption chiller performance. All these parameterswere estimated according to themethods explained above and theyare listed in the appendix.

3.2. Evaluation of the models

3.2.1. Simple comparisonnumber of neurons in the hidden layer (J 7), and IW and LW arethe weights in the input and output hidden layer, respectively.

3. Results and discussion

3.1. Model parameters

Fig. 1. ANN topology.seen from the selected dataset, GNA model prediction shows aconsiderable deviation when compared to the other three models.On the other hand, DDt0, MPR and ANN models show very closeagreement with experimental data. The discrepancy between themodel predictions and experimental data in the worst case is lessthan 5%. Unfortunately, cross validation of the models with datareported by other authors was not possible due to lack ofinformation.

Fig. 2. Comparison of the experimental data with the data obtained by simulation.3.2.2. Comparison through statistical indicatorsThe goodness-of-t of a model is usually evaluated in terms of

statistical indicators. The statistical performance analysis of theevaluated models was conducted including several statistical in-dicators: the residual sum of squares (SSres), the coefcient ofdetermination (R2), the root mean square error (RMSE) and thecoefcient of variation (CV).

The most common parameters to check how close the predictedvalues are to observed data are the residual sum of squares and thecoefcient of determination. Residual is unexplained variation aftertting a model and is the difference between the value predicted bythemodel and the associated observed value. The sum of squares ofthese differences is called the residual sum of squares and can beunderstood as a measure of the discrepancy between the data andan estimation model. A smaller SSres indicates better t to theobserved data.

The coefcient of determination is another parameter whichquanties the goodness of t. The R2 can be calculated from theresidual sum of squares and the total sum of squares (SStot) by theEq. (10) and can be interpreted as a statistical measure of how wella model prediction approximates the observed data.

R2 1 SSresSStot

(10)

An R2 of 1.0 would indicate that model prediction perfectly tsthe observed data. However, the statistical analysis cannot rely onlyon R2, no matter how reasonable the t is. It should be interpretedtogether with other indicators.

RMSE is used to obtain the condence interval (CI) which is away to visualize the precision of each model. Narrower CI indicatesbetter precision since RMSE is lower.

Normally, CI is constructed by using the standard deviation:

CI y z$s (11)

where y is the mean value of the measurement, s is the standarddeviation of the measurement, and z is the score of the standardnormal distribution. Using the RMSE instead of s and assuming acondence level of 95% the CI can be estimated as:

CI Qk 1:96$RMSE (12)We also use the coefcient of variation of the root-mean-square

error (CV) in order to compare the models in terms of predictingcapabilities. CV is dened as RMSE divided by the dependent var-iable average (Eq. (13)).

CV RMSEQk

$100% (13)

The predicted cooling capacity and driving heat input using thefour models of the absorption chiller are compared in Fig. 3applying the R2 and CI (in the form of dashed lines) indicators.The comparison was performed using the entire experimentaldatabase. The solid line represents the ideal match of the modelwith experiment, while the dashed lines limit the 95% condencearea. A smaller distance between the two lines indicates a moreaccurate prediction of the model. As it could be expected in thisgure is shown that much better accuracy is obtained by pureblack-box modeling methods.

GNA model shows the poorest performance, with the lowest R2,0.9 in case of the cooling capacity and 0.83 in case of the generatorheat input, as illustrated in Fig. 3(a, e). Also, the widest CI rangeamong all the models clearly indicates that GNA has the lowest

gineering 58 (2013) 305e313 309accurate prediction. The other three methods (DDt0, MPR and ANN)

l Engineering 58 (2013) 305e313J. Labus et al. / Applied Therma310had much better statistical indicators. Excellent t with theobserved data is visible through high coefcient of determination(R2w 0.99), while the narrow CIs indicate very accurate predictionof all three modeling methods. Among them, the narrower CIranges and highest R2 values (>0.998) were obtained with the ANNmodeling method.

Another statistical indicator is the coefcient of variation (CV) ofthe root mean square error. CV indicator is a normalizedmeasure ofdispersion of the probability distribution and is dened as a per-centage of the RMSE divided by the dependant variable mean(Eq. (14)).

CV RMSEQk

$100% (14)

The CV values for the different modeling methods are illustratedin Fig. 4. If 10% deviation of CV is assumed to be acceptable to obtaina satisfactory prediction, it is clear that the developed GNA modelcannot pass this threshold.

On the other hand, the calculated CV values of DDt0, MPR andANN are lower than 5% which is more than satisfactory. Actually,

Fig. 3. Comparison between the measured and praccording to Hydeman et al. [28], the models with CV values in therange of 3e5% are supposed to have good accuracy for performanceprediction in practical applications. The best CV indicators (

mance with high accuracy, it is still used in some cases due to itssimplicity. This is justied by a fact that on a whole year hourlysimulation, these deviations will most probably equal out to a greatextent. However, the statistical analysis indicates that would bemore appropriate to use one of the other three methods in order toobtain better accuracy. Excellent statistical indicators (R2 around0.99, CV lower than 5% and narrow CI) clearly show that any of thethreemethods (DDt0, MPR and ANN) is suitable for the performanceprediction of absorption systems, and could be used for the chillercontrol and monitoring, fault detection or optimization. Never-theless, the best prediction was obtained with the ANN methodwith R2 > 0.998 and CV

.368549.449.5354851.32.868

.069719199ses i_Qac9447

J. Labus et al. / Applied Thermal Engineering 58 (2013) 305e313312Nomenclature

a GNA regression coefcientb MPR coefcient3 efciencyb1, b2 biasA heat transfer areas [m2]CL condence limitsCOP coefcient of performance [-]I input variableIW, LW matrix weightsJ number of neurons in the hidden layerQ:

heat ow [kW]R number of neurons in the input layerR2 coefcient of determinationRMSE root mean square error

Table A.3ANN coefcients.

Input weights

IW _Qeva IW _Qac2.6842 26.2414 40.2259 1.3136 427.994 37.8786 151.8244 1.6348 3.54.5898 1.0661 1.0163 3.1738 00.4744 2.2865 7.902 3.85 12.0043 14.9552 27.8277 0.6074 1.23.0631 3.4876 3.0904 1.0057 10.6453 1.4337 0.9214 0.2914 0Output weightsLW _Q eva 0.6507 0.4582 1.1899 1LW _Qac 3.5228 3.8286 1.8483 0.8LW _Qgen 0.4252 18.2436 0.4097 9.6Biases in input layer Biab1 _Qeva b1 _Qac b1 _Qgen b2 _Qeva b27.5266 2.1264 22.9476 6.993 23.73.789 0.5695 1.07472.7049 1.3703 1.38443.5951 11.9184 2.064733.3159 0.1494 3.67321.984 9.4941 10.91560.0916 3.7636 5.1815SS sum of squaresT temperature [C]

Sub-indexac absorber/condensereva evaporatorgen generatorin inputk absorber/condenser, evaporator or generator thermal

loadout outputres residualshx solution heat exchangertot total

References

[1] G. Grossman, A. Zaltash, ABSIM e modular simulation of advanced absorptionsystems, International Journal of Refrigeration 24 (2001) 531e543.

[2] R.J.R. Silverio, J.R. Figueiredo, Steady state simulation of the operation of anevaporative cooled watereammonia absorption scale ice maker with experi-mental basis, Journal of the Brazilian Society of Mechanical Sciences and En-gineering 28 (2006) 413e421.

[3] O. Kaynakli, M. Kilic, Theoretical study on the effect of operating conditions onperformance of absorption refrigeration system, Energy Conversion andManagement 48 (2007) 599e607.[4] H. Yin, M. Qu, D.H. Archer, Model based experimental performance analysis ofa microscale LiBreH2O steam-driven double-effect absorption chiller, AppliedThermal Engineering 30 (2010) 1741e1750.

[5] W. Wu, X. Zhang, X. Li, W. Shi, B. Wang, Comparisons of different workingpairs and cycles on the performance of absorption heat pump for heating anddomestic hot water in cold regions, Applied Thermal Engineering 48 (2012)349e358.

[6] D.S. Ayou, J.C. Bruno, A. Coronas, Steady-state operational degrees of freedomin absorption chillers and heat pumps: methodology and case study, Inter-national Journal of Refrigeration 35 (2012) 1570e1582.

[7] G.A. Florides, S.A. Kalogirou, S.A. Tassou, L.C. Wrobel, Design and constructionof a LiBr-water absorption machine, Energy Conversion and Management 44(2003) 2483e2508.

[8] J.M. Gordon, K.C. Ng, A general thermodynamic model for absorptionchillers: theory and experiment, Heat Recovery Systems and CHP 15 (1995)73e83.

[9] F. Ziegler, H.M. Hellmann, C. Schweigler, An approximative method formodeling the operating characteristics of advanced absorption chillers, in:29th International Congress of Refrigeration, 1999, Sydney.

[10] H.-M. Hellmann, F.F. Ziegler, Simple absorption heat pump modules for sys-tem simulation programs, ASHRAE Transactions 105 (1999).

IW _Qgen2.9482 4.0929 36.7107 28.43316.8414 0.617 2.0084 4.4218

4 3.5448 14.3784 4.0429 11.22276 7.2797 0.8131 2.9104 7.6695

0.0293 14.7601 29.0476 43.484397 5.0261 3.305 1.8499 13.82696 6.4158 0.6262 9.0005 9.7543

3 0.616 0.9387 10.067427.8663 1.233 4.17510.3298 1.5636 1.9776

n hidden layer b2 _Qgen

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J. Labus et al. / Applied Thermal Engineering 58 (2013) 305e313 313

Performance analysis of small capacity absorption chillers by using different modeling methods1. Introduction2. Experimental data and absorption chiller models2.1. Adapted GordonNg model (GNA)2.2. Adapted characteristic equation model (t)2.3. Multivariate polynomial regression model (MPR)2.4. Artificial neural network model (ANN)

3. Results and discussion3.1. Model parameters3.2. Evaluation of the models3.2.1. Simple comparison3.2.2. Comparison through statistical indicators3.2.3. Statistical tests

4. ConclusionsAcknowledgementsAppendixNomenclatureReferences