-
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:
[email protected] (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
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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