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Performance analysis of small capacity absorption chillers by using different modeling methods
Applied Thermal Engineering 58 (2013) 305e313
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Performance analysis of small capacity absorption chillers by using different modeling methods Jerko Labus a, Joan Carles Bruno b, *, Alberto Coronas b a b
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia Universitat Rovira i Virgili, CREVER, Av. Països Catalans 26, 43007 Tarragona, Spain
h i g h l i g h t s Comparison of four empirically based models: GNA, DDt0 , MPR, ANN. Experimental data of 12 kW absorption chiller are used for modeling. DDt0 , MPR, ANN methods are suitable for complex simulation environments. The statistical indicators and tests show a slight advantage of the ANN method.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2013 Accepted 12 April 2013 Available online 2 May 2013
This paper presents a review and comparison of simple, yet accurate steady-state models of small capacity absorption chillers using highly reliable experimental data obtained with an absorption chiller of 12 kW in a state-of-the-art test bench. These models can potentially be used in complete modeling and simulation tools or in supervisory control strategies for air-conditioning systems using absorption chillers. With respect to that, a comparative evaluation of different modeling methods for predicting the absorption 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 artificial neural networks model (ANN) were applied using the experimental data and thoroughly examined. The paper also presents statistical indicators and tests which might assist in selection of the most appropriate model. The excellent statistical indicators such as coefficient of determination (>0.99) and coefficient of variation (<5%) clearly indicate that it is possible to develop highly accurate empirical models by using only the variables of external water circuits as model input parameters. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Absorption chillers Performance analysis Modeling Statistical indicators
1. Introduction The main aim of this paper is to present a comparative evaluation of different modeling approaches for predicting the performance of small absorption chillers. The comparative evaluation can serve as a reference when there is a need for simple, but accurate models of absorption chillers, for example to integrate these models in complete energy supply and demand models included in simulation software packages. These simple chiller models, characterized by a low number of input parameters, can serve to facilitate the annual simulations of complex building systems providing at the same time an adequate level of performance prediction. Also,
* Corresponding author. Tel.: þ34 977 297068; fax: þ34 977559691. E-mail address: [email protected] (J.C. Bruno). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.04.032
this paper aims to provide a statistical approach which may help in selecting the appropriate model. With respect to absorption chiller modeling, both physical and empirical approaches were many times presented in the literature. Physical or more precise thermodynamic models were reported by many authors. Here just a brief review of the most recent or relevant will be given. Grossman and Zaltash [1] developed a modular simulation tool for absorption systems called ABSIM. With this software is possible to study various absorption cycle configurations using different working fluids. ABSIM calculates the cycle internal state points and thermal loads in each component using a cycle configuration build by the user graphically and for given working fluid specifications and operating conditions. This is enabled through the governing equations for each component of the cycle contained in the software subroutines. However, the calculation convergence is not always easy. Silverio and Figueiredo
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[2] used a thermodynamic approach for steady-state simulation of an ammonia-water absorption system. The thermodynamic state relations, the pressure drop equations and the heat transfer coefficients were solved by using an algorithm based on the Substitution Newton Rapshon method. Kaynakli and Kilic [3] performed a theoretical study on the performance of a H2OeLiBr absorption system using a thermodynamic analysis of the absorption cycle. These authors investigated the influences of the driving temperature and heat exchanger effectiveness on the thermal loads of the components and COP. Yin et al. [4] developed a detailed thermodynamic model of a 16 kW double-stage H2OeLiBr absorption chiller. The steady-state model was based on the working fluids property relations, detailed mass and energy balances, and the heat and mass transfer relationships for each chiller component. One of the most recent application of the thermodynamic approach in absorption system modeling can be found in the paper of Wu et al. [5]. The authors developed thermodynamic models of different absorption heat pump cycles to test their applicability with different heat sources, working pairs and in different cold regions. All these thermodynamic models are very demanding since they require comprehensive knowledge of the absorption cycle including some internal state points. These models need lots of input parameters such as heat transfer coefficients (U) and heat transfer areas (A) of heat exchangers, the rich solution flow rate, working fluid properties and water side flows and temperatures as well as some additional assumptions for the convenience of modeling. A more complete explanation on all these degrees of freedom in the modeling of absorption chillers can be found in Dereje et al. [6]. In practice, however, especially with commercial units, the internal parameters are not available. This is the reason why thermodynamic models are more adequate during the design stage of absorption equipment as explained in the paper of Florides et al. [7]. Also, the computation time in simulation software packages using these models is very long since they require a lot of simultaneous iterations. The annual simulation of absorption chillers under different ambient and operating conditions on an hourly time step basis is a clear example of this. Thus, there is a need for simple models which can provide sufficiently good representation of the absorption machine behavior based only on available external parameters (experimental measurements or manufacturer catalog data). Simple models can be more easily incorporated in simulation programs or used for fault detection and control. Contrary to the physical models, the empirical and semi-empirical models require less time and effort to develop and computation time is much shorter when they are built into complete energy management simulation programs. The parameters and fitting coefficients in these models are determined by using a regression method or a minimization algorithm applied to a dataset obtained performing experimental measurements or using a manufacturer catalog. The studies about development of empirically based models for absorption chillers have been reported by several authors. Gordon and Ng [8] developed a general model for predicting the absorption chillers performance. The model lays both on physical and empirical principles. The physical principles that govern the performance of the absorption chiller are fitted to the experimental or manufacturer data by using a regression method. Ziegler et al. [9,10] developed a model (Characteristic equation method) which predicts the performance of the absorption chiller by using two simple algebraic equations: one to calculate the cooling capacity and another for the driving heat input. These two previous models belong to semi-empirical (gray-box) category of models, in which the fitted parameters can be interpreted under the actual physical principles which govern the absorption chiller performance. Labus et al. [11] used a completely empirical approach to model
absorption chillers based on manufacturers curves in order to investigate the energy savings when different absorption chiller configurations were considered for their integration in a complete chiller plant. The Artificial neural networks approach has been also used for absorption chiller modeling. ANN models belong to the black-box model category, that unlike gray-box models, the estimated parameters of the model have no physical interpretation. Sözen [12] used the ANN to determine thermodynamic properties of an alternative working pair for absorption systems. The study also demonstrated that ANN can replace mathematical models in the simulation of absorption systems. In the paper of Sözen and Akçayol [13] the ANN approach was proposed for performance analysis of an absorption chiller. The ANN model used only the working temperatures in the four main components as input parameters in order to predict the performance of the chiller. Manohar et al. [14] applied ANN for the modeling of steam fired double effect absorption chiller. Later, a similar work was carried out by Rosiek and Batlles [15], who used ANN to model solar-assisted air conditioning system with hot water driven double effect absorption chiller. The last approach considered in this paper is the simple multivariable polynomial regression which also belongs to the black-box category of models. Regardless the numerous studies on the modeling of absorption equipment, literature review shows that there is a lack of information with respect to comprehensive comparative studies on different modeling techniques for predicting absorption equipment performance in a similar way as Swider [16] or Lee et al. [17] did for the case of vapor-compression chillers. The main aim of this paper is to present a comparative evaluation of different modeling approaches for predicting the performance of absorption systems. In the next section are presented the experimental data and a brief description of the evaluated types of models. Later the application of these models to the experimental data is evaluated with the help of statistical indicators and statistical tests to select the best modeling approach. 2. Experimental data and absorption chiller models Four different types of absorption chiller models were developed and examined:
Adapted GordoneNg model, Adapted characteristic equation model, Multivariate polynomial regression model and Artificial neural networks model.
The experimental data required for the models’ development were obtained in the state-of-the-art test bench of the Rovira i Virgili University in Tarragona (Spain). The test bench is fully equipped to test under controlled operation conditions a variety of units commonly used in HVAC systems. A more detailed explanation about the functionality of the test bench can be found in Labus et al. [18] and Labus [19]. For the models described in this research the data were collected in a series of experiments with a 12 kW absorption chiller Pink Chilli PSC12. The measured variables in the experiments were inlet and outlet temperatures of hot, chilled, and cooling water circuit; volumetric flow rates and pressure drop in each circuit; and electric consumption of the chiller. The raw data were processed using a comprehensive test procedure which includes several techniques: data reduction, development of steadystate detector with additional filtering and uncertainty estimation. Based on external measurable parameters only, this procedure allows the creation of the complete performance map for absorption machines based on highly accurate data. In data reduction, the
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collected data were used to calculate thermal loads and efficiency coefficients. Off-line steady-state detector for absorption chillers was developed based on analogy with steady-state detector for vapor compression chillers using the moving window average. Additional data filtering was engaged to eliminate remaining transient periods caused by time delays when changing from one steady-state to another. The evaluation of experimental uncertainty was carried out by judgment based on available information on the possible variability of input quantities. When the uncertainties of heat loads were evaluated, the following input quantities were taken into account: inlet and outlet temperatures of external circuits, volumetric flow rate, density, and specific heat capacity. Uncertainty contribution for temperature and volumetric flow rate was calculated as a combination of different sources of uncertainty: repeatability, accuracy of the instrument and resolution of the instrument. In order to be in accordance with international standards, time length for the steady-state tests was not shorter than 30 min collected in 5s intervals. The experimental database used for modeling consists of 138 steady-state points and covers the following temperatures ranges: inlet hot water temperature 80e 100 C, inlet cooling water temperature 27e35 C and outlet chilled water temperature 5e12 C, as presented in Table 1. The main criterion to select these three temperatures as input variables for the empirical models was their availability to the operating engineers in practical applications. In detail description of the test procedure as well as the experimental results can be found online in Labus’ PhD thesis [19]. Also, it is important to explain the influence of database size on modeling. Models developed with small datasets are not reliable and statistically correct, since small datasets are insufficient to form strong relationships within the models. On the other hand, the models created with large dataset which completely covers the operating range of the absorption chiller show very high level of predicting capabilities.
the absorber and condenser as a single source of heat at medium temperature. According to Gordon and Ng approximation, the finite-rate mass transfer is roughly temperature independent. With respect to that, the losses in the evaporator can be neglected, while the losses in other two heat exchangers (generator, absorber/condenser) can be viewed as a constant characteristic of each particular chiller. The general equation for the GNA model can be obtained after series of transformation, starting from the First law of thermodynamics and using the entropy balance which takes into account the dominant irreversibility [8,19]. The GNA model calculates the inverse of COP using the following equation (Eq. (1)):
# " # " # " # " in in Tgen Tgen 1 T in T out 1 þ $ ¼ ac out eva $ in : in in T in Teva COP Tgen Tac Tgen Qeva ac # " in Tac $ a1 a2 $ in Tgen
and a2 as the intercept and slope of this line using linear regression. Bearing in mind that the purpose of this analysis is to compare different modeling approaches by means of the deviations between experimental and modeled heat loads, the final equation of the GNA model (eq. (1)) was adapted to obtain the chiller capacity (Eq. (2)). The heat input can be derived from the COP (Eq. (3)). :
B 1=COP A
(2)
:
B 1 A$COP
(3)
2.1. Adapted GordoneNg model (GNA)
Table 1 Experimental operation range conditions. Variable
Range
in [ C] Teva out [ C] Tac in [ C] T:gen
[4.98e12.1] [26.95e35.01] [79.9e100.12] [0.49e15.23] [5.99e41.98] [4.64e24.04] [0.11e0.76]
Q: eva [kW] Q: ac [kW] Q gen [kW] COP :
:
Flow rates [m3/h] fixed at: meva ¼ 1:7; mac ¼ 4:8; : mgen ¼ 2:2
(1)
where a1 and a2 are the regression parameters to be fitted with experimental data and, at the same time, the constants which characterize the entropy generation of particular chiller. " in T in in Tgen Tac ac against Considering that a plot of in in $COP Tgen Tgen # in T out : Tac eva $Qeva leads to a straight line, is it possible to calculate a1 out Teva
Q eva ¼
The general thermodynamic model for absorption chillers developed by Gordon and Ng [8] is actually a combination of physical and empirical approaches. According to the authors, the dominant irreversibility of the absorption chillers is finite-rate mass transfer. The losses due to the finite-rate mass transfer can therefore be approximated as temperature independent. The original model was based on external input parameters of the four main components (generator, condenser, evaporator and absorber) assuming that manufacturers’ catalogs provide the operating conditions for each of them. However, the current manufacturers’ practice is to provide operating curves based on three circuits, i.e. treating absorber and condenser as one component. The main reason for that is the arrangement in series of the absorber and condenser in the majority of the commercial absorption chillers. Therefore, in our case the original model was modified considering
307
Q gen ¼ where:
# " # in in T out Tgen Tac eva $ in ; A ¼ out in Teva Tgen Tac # " # " in in Tgen Tac $ a1 a2 $ in B ¼ in T in Tgen Tgen ac "
(4)
2.2. Adapted characteristic equation model (DDt0 ) For the modeling of absorption chillers, Ziegler et al. [9] developed an approximate method which is able to represent both cooling capacity and driving heat input by simple algebraic equations. These equations are expressed as a function of so-called characteristic temperature difference (DDt), which depends on the average temperature of the external heat carrier fluids. One of the main assumptions is that the heat transfer processes in absorption chillers dominate their performance behavior. In this way, a complex response to all external heat carrier temperatures is reduced to a linear function of heat flow and the external temperatures. A simple linear correlation is very convenient, but it has been found that the predicted performance of the cooling capacity deviates considerably from the linear behavior, for instance, at high
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driving temperatures, due to higher internal losses. With respect to that, an adapted characteristic equation method was proposed by Kuhn and Ziegler [20]. This improved model uses a numerical fit of catalog or experimental data to improve the characteristic equation. The adapted characteristic temperature function (DDt0 ) takes the form (Eq. (5)):
DDt 0 ¼ tgen a$tac þ e$teva
(5)
And the linear characteristic equation for each component loads k (Eq. (6)): :
Q k ¼ s0 $DDt 0 þ r
(6)
Combining Eqs. (5) and (6) yields one correlation which represents the thermal performance of the components as a function of the external arithmetic mean temperatures of the generator (tgen), absorber-condenser (tac) and evaporator (teva), when the external flow rates are constant. :
Q k ¼ s0 $tgen s0 $a$tac þ s0 $e$teva þ r
(7)
The four parameters (s’, a, e and r) are estimated by using a multiple linear regression algorithm to fit the experimental data. This algorithm chooses regression coefficients to minimize the residual sum of squares. The analyses of Puig-Arnavat et al. [21] confirmed the capability of the DDt0 method to obtain good results and also better accuracy than the original method DDt. Finally, the combination of the obtained characteristic functions with the equations of the external arithmetic mean temperatures and with the external energy balances, results in a system of six equations with six unknowns which can easily be solved. The developed model requires only three temperatures (one from each of the external circuits) at constant flow rates of external heat carriers to predict 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 processes incorporated in the model structure. MPR models are a very effective tool for describing complex non-linear relationships between input and output variables without disregarding what occurs within the system. The parameters for the MPR model are calculated by fitting the experimental data minimizing the sum of squares of the residuals using a polynomial function. Due to their simple structure, MPR models have been applied in various fields such as forecasting, control, optimization, fault detection and diagnosis [17,22]. Lee et al. [17] proved that MPR model of vaporcompression water chiller can have a high prediction accuracy, with the coefficient of variation of 0.61%. Similarly, the paper of Kim et al. [22] confirmed that MPR models are acceptable for fault detection and diagnosis of residential heat pump systems. A typical polynomial regression model contains the squared and higher order terms of the estimator variable. Normally, the higher order MPR models offer better accuracy of prediction. However, high-order MPR can become impractical due to its excessive number of parameters. One of the common techniques in the case of the high order MPR models with large number of parameters is to reduce the model by retaining only those parameters that are statistically significant. Also, excessive polynomial order for a relatively small database may worsen data interpolation. These are some of the reasons why it has been decided to apply only second order polynomials to predict the absorption chiller performance. Thus, the MPR models were developed to calculate the thermal loads of the absorption chiller by using the measurements of external circuits:
generator inlet temperature, absorber/condenser inlet temperature, and evaporator outlet temperature. The generalized second order model in case of absorption chillers can be represented using Eq. (8): :
in out in in out Q k ¼ b0;k þ b1;k Tgen þ b2;k Tac þ b3;k Teva þ b4;k Tgen Tac 2 in in out in in þ b5;k Tgen Teva þ b6;k Tac Teva þ b7;k Tgen 2 out 2 in þ b8;k Tac þ b9;k Teva
(8)
2.4. Artificial neural network model (ANN) ANN models also belong to the group of black-box models. An artificial neural network is an adaptive system which can be trained to perform a particular function or behavior on the basis of input and output information that flows through the network. ANN have found their place in the fields of modeling, identification, optimization and control in steady state and dynamic systems due to their ability to model complex relationships between inputs and outputs or to find patterns in data. Various applications of neural networks in renewable energy problems such as energy prediction and optimization of energy consumption in building service systems were presented in the review of Kalogirou [23]. The review of Mohanraj et al. [24] covers the applications of ANN in energy and exergy analysis of refrigeration and air-conditioning systems, their control and in prediction of refrigerant properties. Moon et al. [25] developed an adaptive control method using the ANN model to enhance thermal comfort in buildings. Yabanova and Keçebas¸ [26] developed ANN-based PID controller for geothermal district heating system in Turkey, which increased energy efficiency and cost saving of the system by 13%. The most common ANN architecture applied in the field of absorption systems and their applications are feed-forward neural networks with back-propagation [27]. In this research, ANN models of the absorption chiller were developed by using MatLab Neural Network toolbox. Since there is no explicit rule to determine the topology of ANN (the number of neurons in the hidden layer or the number of hidden layers) the trial and error method is usually applied to find the best solution. Thus, the adopted topology for ANN models was (3e7e1), as illustrated in Fig. 1. Each model consists of one input layer with three variables, one hidden layer with seven neurons and one output layer with one output: a component load (three different ANN models are built for thermal power exchanged in the evaporator, absorber/condenser and generator). The training of the ANN was based on the error back propagation technique using the LevenbergeMarquardt algorithm of optimization. The input parameters were normalized in the [0.2, 0.8] range. A hyperbolic tangent sigmoid function (tansig) was used in the hidden layer and the linear transfer function (purelin) was used in the output layer. To test the robustness and the prediction ability of the models, the experimental dataset was split into three parts: 70% of data was used for the model training, 20% for the model validation and the remaining 10% for the model testing. The ANN absorption chiller model to calculate the thermal load in each component is given by the general equation (Eq. (9)): :
Qk ¼
j X i
" LWð1;jÞ $
2 1 P R 1 þ exp 2 1 IWðj;RÞ IR þ b1ðjÞ
!# þ b2 (9)
where I is the input, R is the number of the inputs (R ¼ 3), b1 are biases in the hidden layer, b2 are biases in the output layer, J is the
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Fig. 1. ANN topology.
number of neurons in the hidden layer (J ¼ 7), and IW and LW are the weights in the input and output hidden layer, respectively. 3. Results and discussion 3.1. Model parameters The developed models require different parameters in order to calculate the absorption chiller performance. All these parameters were estimated according to the methods explained above and they are listed in the appendix. 3.2. Evaluation of the models 3.2.1. Simple comparison Fig. 2 shows the comparison of the measured and calculated cooling capacity of the chiller with a generator and condenser/ absorber temperature of 85 C and 27 C, respectively. As can be seen from the selected dataset, GNA model prediction shows a considerable deviation when compared to the other three models. On the other hand, DDt0 , MPR and ANN models show very close agreement with experimental data. The discrepancy between the model predictions and experimental data in the worst case is less than 5%. Unfortunately, cross validation of the models with data reported by other authors was not possible due to lack of information.
Fig. 2. Comparison of the experimental data with the data obtained by simulation.
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3.2.2. Comparison through statistical indicators The goodness-of-fit of a model is usually evaluated in terms of statistical indicators. The statistical performance analysis of the evaluated models was conducted including several statistical indicators: the residual sum of squares (SSres), the coefficient of determination (R2), the root mean square error (RMSE) and the coefficient of variation (CV). The most common parameters to check how close the predicted values are to observed data are the residual sum of squares and the coefficient of determination. Residual is unexplained variation after fitting a model and is the difference between the value predicted by the model and the associated observed value. The sum of squares of these differences is called the residual sum of squares and can be understood as a measure of the discrepancy between the data and an estimation model. A smaller SSres indicates better fit to the observed data. The coefficient of determination is another parameter which quantifies the goodness of fit. The R2 can be calculated from the residual sum of squares and the total sum of squares (SStot) by the Eq. (10) and can be interpreted as a statistical measure of how well a model prediction approximates the observed data.
R2 ¼ 1
SSres SStot
(10)
An R2 of 1.0 would indicate that model prediction perfectly fits the observed data. However, the statistical analysis cannot rely only on R2, no matter how reasonable the fit is. It should be interpreted together with other indicators. RMSE is used to obtain the confidence interval (CI) which is a way to visualize the precision of each model. Narrower CI indicates better precision since RMSE is lower. Normally, CI is constructed by using the standard deviation:
CI ¼ y z$s
(11)
where y is the mean value of the measurement, s is the standard deviation of the measurement, and z is the score of the standard normal distribution. Using the RMSE instead of s and assuming a confidence level of 95% the CI can be estimated as:
CI ¼ Qk 1:96$RMSE
(12)
We also use the coefficient of variation of the root-mean-square error (CV) in order to compare the models in terms of predicting capabilities. CV is defined as RMSE divided by the dependent variable average (Eq. (13)).
CV ¼
RMSE $100% Qk
(13)
The predicted cooling capacity and driving heat input using the four models of the absorption chiller are compared in Fig. 3 applying the R2 and CI (in the form of dashed lines) indicators. The comparison was performed using the entire experimental database. The solid line represents the ideal match of the model with experiment, while the dashed lines limit the 95% confidence area. A smaller distance between the two lines indicates a more accurate prediction of the model. As it could be expected in this figure is shown that much better accuracy is obtained by pure black-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 generator heat input, as illustrated in Fig. 3(a, e). Also, the widest CI range among all the models clearly indicates that GNA has the lowest accurate prediction. The other three methods (DDt0 , MPR and ANN)
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Fig. 3. Comparison between the measured and predicted evaporator and generator loads-Pink.
had much better statistical indicators. Excellent fit with the observed data is visible through high coefficient of determination (R2 w 0.99), while the narrow CIs indicate very accurate prediction of all three modeling methods. Among them, the narrower CI ranges and highest R2 values (>0.998) were obtained with the ANN modeling method. Another statistical indicator is the coefficient of variation (CV) of the root mean square error. CV indicator is a normalized measure of dispersion of the probability distribution and is defined as a percentage of the RMSE divided by the dependant variable mean (Eq. (14)).
CV ¼
RMSE $100% Qk
according to Hydeman et al. [28], the models with CV values in the range of 3e5% are supposed to have good accuracy for performance prediction in practical applications. The best CV indicators (<2%) in this analysis were obtained by the ANN method.
(14)
The CV values for the different modeling methods are illustrated in Fig. 4. If 10% deviation of CV is assumed to be acceptable to obtain a satisfactory prediction, it is clear that the developed GNA model cannot pass this threshold. On the other hand, the calculated CV values of DDt0 , MPR and ANN are lower than 5% which is more than satisfactory. Actually,
Fig. 4. CV values.
J. Labus et al. / Applied Thermal Engineering 58 (2013) 305e313 Table 2 AICc and BIC analysis. BIC
Criterion
AICc
Model
Q eva
Q ac
Q gen
Q eva
Q ac
Q gen
GNA DDt’ MPR ANN
33.6 244.5 329.7 412.9
337.8 91.9 120.5 208.9
185.7 140.0 188.5 342.9
719.3 446.9 377.8 346.1
1023.5 596.6 587 550.1
871.4 548.5 519.0 416.0
:
:
:
:
:
:
3.2.3. Statistical tests The selection of the best model using statistical indicators could not be enough because what is meant by best is controversial. A good model selection technique should take into account not only goodness of fit but also the simplicity of the model. More complex models will be better able to adapt their shape to fit the data, but the additional parameters may not represent anything useful. The statistical analysis usually used in the case of nested models (when one model is a simpler case of the other) is the extra sum of squares test (F-test). The F-test is not appropriate for the comparison of non-nested models, which is the case here. When comparing non-nested models there are two criteria which can be used [29]: Akaike’s Information Criterion (AIC) or Schwarz-Bayesian Information Criterion (BIC). The AIC can be termed as a measure of the goodness of fit of any estimated statistical model. The BIC is a type of model selection among a class of parametric models with different numbers of parameters. The difference between AIC and BIC is that the penalty term for the number of parameters in the model is larger in BIC than in AIC. Both criteria were applied to compare the analyzed absorption chiller models. The used AICc is a second order AIC with a greater penalty for extra parameters and can be calculated by using Eq. (15):
AICc ¼ N$ln
SSres 2K$ðK þ 1Þ þ 2K þ N NK1
(15)
Similarly, BIC is defined by Eq. (16):
BIC ¼ N$lnðSSres Þ þ K$lnðNÞ
(16)
where N is the number of data points, K is the number of the model parameters and SSres is the residual sum of squares. The interpretation of the results in both tests is the same: the model with lower value is more likely to be selected. AICc and BIC analysis results of all the models developed for the Pink absorption chiller are presented in Table 2. The analysis also included the GNA model although it could be discarded after the analysis performed with the statistical indicators. The results of both AICc and BIC tests were in agreement with the goodness of fit results, recommending the most complex model ANN to be adopted. Although the ANN model, which is a complete black-box model, shows the closest fit, other more physically related models also could be applied, depending on the specific application requirements (Table 2).
4. Conclusions A comprehensive comparison of different methods for steadystate modeling of small capacity absorption chillers was presented in this paper. Four models were developed based on experimental data. The models were based only on the measurements of the external water circuits. Statistical indicators such as R2 and CV showed that GNA model has the lowest predictive capacity.
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Although the GNA model could not reproduce the chiller performance with high accuracy, it is still used in some cases due to its simplicity. This is justified by a fact that on a whole year hourly simulation, these deviations will most probably equal out to a great extent. However, the statistical analysis indicates that would be more appropriate to use one of the other three methods in order to obtain better accuracy. Excellent statistical indicators (R2 around 0.99, CV lower than 5% and narrow CI) clearly show that any of the three methods (DDt0 , MPR and ANN) is suitable for the performance prediction of absorption systems, and could be used for the chiller control and monitoring, fault detection or optimization. Nevertheless, the best prediction was obtained with the ANN method with R2 > 0.998 and CV<2%. The comparison of the models with AICc and BIC statistical tests confirmed that ANN was the most suitable method to model the selected absorption chiller. Acknowledgements The authors would like to acknowledge financial support of this work by the CITYNET project funded via the Marie Curie Research Training Network, the project TR 33049 funded by Serbian Ministry of Education and Science and the project ENE2009-14182 funded by the Spanish Ministry of Economy and Competitiveness. Appendix Two linear regression coefficients for the GNA model,
a1 ¼ 36.524 and a2 ¼ 41.527, were obtained after performing nu-
merical fit of the experimental dataset. Table A.1 shows the coefficients for characteristic functions of absorption chiller obtained through multiple linear regression fit for the DDt’ method.
Table A.1 Estimated input parameters-DDt’.
Q_ eva Q_ ac Q_
gen
s0
s0 a
s0 e
r
0.6822 1.2698 0.5662
1.0057 2.3951 1.3066
0.3801 1.0347 0.5947
1.6396 0 0
In the case of MPR models, Table A.2 shows the regression fitting parameters (bi,k) for the different heat loads (k) when a second order polynomial fit was applied on the experimental data of the absorption chiller. Table A.2 Fitting coefficients for MPR models. (k)
Q_ eva
Q_ ac
Q_ gen
b0 b1 b2 b3 b4 b5 b6 b7 b8 b9
6.818 0.3703 1.4329 1.113 0.0122 0.0014 0.0076 0.0025 0.0081 0.0217
65.3225 0.3092 4.4372 0.2036 0.0159 0.0095 0.0117 0.0001 0.0146 0.0172
28.7061 0.5642 3.037 0.5002 0.0079 0.0048 0.0184 0.0019 0.0187 0.0009
Table A.3 shows the coefficients for the ANN modeling of the absorption chiller obtained during the network training stage.
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Table A.3 ANN coefficients. Input weights IW ðQ_ eva Þ 2.6842 26.2414 27.994 37.8786 4.5898 1.0661 0.4744 2.2865 2.0043 14.9552 3.0631 3.4876 0.6453 1.4337 Output weights 0.6507 LWðQ_ eva Þ 3.5228 LWðQ_ ac Þ _ 0.4252 LWðQ gen Þ Biases in input layer b1ðQ_ ac Þ b1ðQ_ eva Þ 7.5266 2.1264 73.789 0.5695 2.7049 1.3703 3.5951 11.9184 33.3159 0.1494 1.984 9.4941 0.0916 3.7636
40.2259 151.8244 1.0163 7.902 27.8277 3.0904 0.9214
IW ðQ_ ac Þ 1.3136 1.6348 3.1738 3.85 0.6074 1.0057 0.2914
0.4582 3.8286 18.2436
1.1899 1.8483 0.4097
b1ðQ_ gen Þ 22.9476 1.0747 1.3844 2.0647 3.6732 10.9156 5.1815
b2ðQ_ eva Þ 6.993
4.368 3.5549 0.4494 1.5356 1.2485 11.3297 0.8686
1.0693 0.616 0.8719 27.8663 9.6199 0.3298 Biases in hidden layer b2ðQ_ ac Þ b2ðQ_ gen Þ 23.9447 12.8346
Nomenclature
a b 3
b1, b2 A CL COP I IW, LW J: Q R R2 RMSE SS T
2.9482 6.8414 3.5448 7.2797 0.0293 5.0261 6.4158
GNA regression coefficient MPR coefficient efficiency bias heat transfer areas [m2] confidence limits coefficient of performance [-] input variable matrix weights number of neurons in the hidden layer heat flow [kW] number of neurons in the input layer coefficient of determination root mean square error sum of squares temperature [ C]
Sub-index ac absorber/condenser eva evaporator gen generator in input k absorber/condenser, evaporator or generator thermal load out output res residual shx solution heat exchanger tot total References [1] G. Grossman, A. Zaltash, ABSIM e modular simulation of advanced absorption systems, International Journal of Refrigeration 24 (2001) 531e543. [2] R.J.R. Silverio, J.R. Figueiredo, Steady state simulation of the operation of an evaporative cooled watereammonia absorption scale ice maker with experimental basis, Journal of the Brazilian Society of Mechanical Sciences and Engineering 28 (2006) 413e421. [3] O. Kaynakli, M. Kilic, Theoretical study on the effect of operating conditions on performance of absorption refrigeration system, Energy Conversion and Management 48 (2007) 599e607.
IW ðQ_ gen Þ 4.0929 0.617 14.3784 0.8131 14.7601 3.305 0.6262
36.7107 2.0084 4.0429 2.9104 29.0476 1.8499 9.0005
0.9387 1.233 1.5636
10.0674 4.1751 1.9776
28.4331 4.4218 11.2227 7.6695 43.4843 13.8269 9.7543
[4] H. Yin, M. Qu, D.H. Archer, Model based experimental performance analysis of a microscale LiBreH2O steam-driven double-effect absorption chiller, Applied Thermal Engineering 30 (2010) 1741e1750. [5] W. Wu, X. Zhang, X. Li, W. Shi, B. Wang, Comparisons of different working pairs and cycles on the performance of absorption heat pump for heating and domestic 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 freedom in absorption chillers and heat pumps: methodology and case study, International Journal of Refrigeration 35 (2012) 1570e1582. [7] G.A. Florides, S.A. Kalogirou, S.A. Tassou, L.C. Wrobel, Design and construction of a LiBr-water absorption machine, Energy Conversion and Management 44 (2003) 2483e2508. [8] J.M. Gordon, K.C. Ng, A general thermodynamic model for absorption chillers: theory and experiment, Heat Recovery Systems and CHP 15 (1995) 73e83. [9] F. Ziegler, H.M. Hellmann, C. Schweigler, An approximative method for modeling 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 system simulation programs, ASHRAE Transactions 105 (1999). [11] J. Labus, I. Korolija, L. Marjanovic-Halburd, A. Coronas, The influence of weather conditions on energy performance of HVAC system and absorption cooling system coupling, in: IAQVEC 2010: The 7th International Conference on Indoor Air Quality, Ventilation and Energy Conservation in Buildings, 2010, Syracuse, New York, USA. [12] A. Sözen, E. Arcaklioglu, M. Özalp, A new approach to thermodynamic analysis of ejectoreabsorption cycle: artificial neural networks, Applied Thermal Engineering 23 (2003) 937e952. [13] A. Sözen, M.A. Akçayol, Modelling (using artificial neural-networks) the performance parameters of a solar-driven ejector-absorption cycle, Applied Energy 79 (2004) 309e325. [14] H.J. Manohar, R. Saravanan, S. Renganarayanan, Modelling of steam fired double effect vapour absorption chiller using neural network, Energy Conversion and Management 47 (2006) 2202e2210. [15] S. Rosiek, F.J. Batlles, Performance study of solar-assisted air-conditioning system provided with storage tanks using artificial neural networks, International Journal of Refrigeration 34 (2011) 1446e1454. [16] D.J. Swider, A comparison of empirically based steady-state models for vapor-compression liquid chillers, Applied Thermal Engineering 23 (2003) 539e556. [17] T.-S. Lee, K.-Y. Liao, W.-C. Lu, Evaluation of the suitability of empirically-based models for predicting energy performance of centrifugal water chillers with variable chilled water flow, Applied Energy 93 (2012) 583e595. [18] J. Labus, J.A. Hernández, J.C. Bruno, A. Coronas, Inverse neural network based control strategy for absorption chillers, Renewable Energy 39 (2012) 471e482. [19] J. Labus, Modelling Of Small Capacity Absorption Chillers Driven By Solar Thermal Energy Or Waste Heat. PhD thesis, Departament d’Enginyeria Mecànica, Universitat Rovira i Virgili, 2011, p. 261. [20] A. Kühn, F. Ziegler, Operational results of a 10 kW absorption chiller and adaptation of the characteristic equation, in: Proceedings of First International Conference on Solar Air Conditioning, 2005, Bad-Staffelstein, Germany.
J. Labus et al. / Applied Thermal Engineering 58 (2013) 305e313 [21] M. Puig-Arnavat, J. Lopez-Villada, J.C. Bruno, A. Coronas, Analysis and parameter identification for characteristic equations of single- and doubleeffect absorption chillers by means of multivariable regression, International Journal of Refrigeration 33 (2010) 70e78. [22] M. Kim, S.H. Yoon, W.V. Payne, P.A. Domanski, Development of the reference model for a residential heat pump system for cooling mode fault detection and diagnosis, Journal of Mechanical Science and Technology 24 (2010) 1481e1489. [23] S.A. Kalogirou, Artificial neural networks in renewable energy systems applications: a review, Renewable and Sustainable Energy Reviews 5 (2001) 373e401. [24] M. Mohanraj, S. Jayaraj, C. Muraleedharan, Applications of artificial neural networks for refrigeration, air-conditioning and heat pump systems-A review, Renewable and Sustainable Energy Reviews 16 (2012) 1340e1358.
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[25] J.W. Moon, S.-H. Yoon, S. Kim, Development of an artificial neural network model based thermal control logic for double skin envelopes in winter, Building and Environment 61 (2013) 149e159. _ Yabanova, A. Keçebas¸, Development of ANN model for geothermal district [26] I. heating system and a novel PID-based control strategy, Applied Thermal Engineering 51 (2013) 908e916. [27] J.A. Hernández, D. Colorado, O. Cortés-Aburto, Y. El Hamzaoui, V. Velazquez, B. Alonso, Inverse neural network for optimal performance in polygeneration systems, Applied Thermal Engineering 50 (2013) 1399e1406. [28] M. Hydeman, P. Sreedharan, N. Webb, S. Blanc, Development and testing of a reformulated regression-based electric chiller model, ASHRAE Transactions 108 (2002) 1118e1127. [29] G. Claeskens, N.L. Hjort, Model Selection and Model Averaging, Cambridge University Press, Cambridge, UK, 2008.
Contents lists available at SciVerse ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Performance analysis of small capacity absorption chillers by using different modeling methods Jerko Labus a, Joan Carles Bruno b, *, Alberto Coronas b a b
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia Universitat Rovira i Virgili, CREVER, Av. Països Catalans 26, 43007 Tarragona, Spain
h i g h l i g h t s Comparison of four empirically based models: GNA, DDt0 , MPR, ANN. Experimental data of 12 kW absorption chiller are used for modeling. DDt0 , MPR, ANN methods are suitable for complex simulation environments. The statistical indicators and tests show a slight advantage of the ANN method.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 February 2013 Accepted 12 April 2013 Available online 2 May 2013
This paper presents a review and comparison of simple, yet accurate steady-state models of small capacity absorption chillers using highly reliable experimental data obtained with an absorption chiller of 12 kW in a state-of-the-art test bench. These models can potentially be used in complete modeling and simulation tools or in supervisory control strategies for air-conditioning systems using absorption chillers. With respect to that, a comparative evaluation of different modeling methods for predicting the absorption 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 artificial neural networks model (ANN) were applied using the experimental data and thoroughly examined. The paper also presents statistical indicators and tests which might assist in selection of the most appropriate model. The excellent statistical indicators such as coefficient of determination (>0.99) and coefficient of variation (<5%) clearly indicate that it is possible to develop highly accurate empirical models by using only the variables of external water circuits as model input parameters. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Absorption chillers Performance analysis Modeling Statistical indicators
1. Introduction The main aim of this paper is to present a comparative evaluation of different modeling approaches for predicting the performance of small absorption chillers. The comparative evaluation can serve as a reference when there is a need for simple, but accurate models of absorption chillers, for example to integrate these models in complete energy supply and demand models included in simulation software packages. These simple chiller models, characterized by a low number of input parameters, can serve to facilitate the annual simulations of complex building systems providing at the same time an adequate level of performance prediction. Also,
* Corresponding author. Tel.: þ34 977 297068; fax: þ34 977559691. E-mail address: [email protected] (J.C. Bruno). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.04.032
this paper aims to provide a statistical approach which may help in selecting the appropriate model. With respect to absorption chiller modeling, both physical and empirical approaches were many times presented in the literature. Physical or more precise thermodynamic models were reported by many authors. Here just a brief review of the most recent or relevant will be given. Grossman and Zaltash [1] developed a modular simulation tool for absorption systems called ABSIM. With this software is possible to study various absorption cycle configurations using different working fluids. ABSIM calculates the cycle internal state points and thermal loads in each component using a cycle configuration build by the user graphically and for given working fluid specifications and operating conditions. This is enabled through the governing equations for each component of the cycle contained in the software subroutines. However, the calculation convergence is not always easy. Silverio and Figueiredo
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[2] used a thermodynamic approach for steady-state simulation of an ammonia-water absorption system. The thermodynamic state relations, the pressure drop equations and the heat transfer coefficients were solved by using an algorithm based on the Substitution Newton Rapshon method. Kaynakli and Kilic [3] performed a theoretical study on the performance of a H2OeLiBr absorption system using a thermodynamic analysis of the absorption cycle. These authors investigated the influences of the driving temperature and heat exchanger effectiveness on the thermal loads of the components and COP. Yin et al. [4] developed a detailed thermodynamic model of a 16 kW double-stage H2OeLiBr absorption chiller. The steady-state model was based on the working fluids property relations, detailed mass and energy balances, and the heat and mass transfer relationships for each chiller component. One of the most recent application of the thermodynamic approach in absorption system modeling can be found in the paper of Wu et al. [5]. The authors developed thermodynamic models of different absorption heat pump cycles to test their applicability with different heat sources, working pairs and in different cold regions. All these thermodynamic models are very demanding since they require comprehensive knowledge of the absorption cycle including some internal state points. These models need lots of input parameters such as heat transfer coefficients (U) and heat transfer areas (A) of heat exchangers, the rich solution flow rate, working fluid properties and water side flows and temperatures as well as some additional assumptions for the convenience of modeling. A more complete explanation on all these degrees of freedom in the modeling of absorption chillers can be found in Dereje et al. [6]. In practice, however, especially with commercial units, the internal parameters are not available. This is the reason why thermodynamic models are more adequate during the design stage of absorption equipment as explained in the paper of Florides et al. [7]. Also, the computation time in simulation software packages using these models is very long since they require a lot of simultaneous iterations. The annual simulation of absorption chillers under different ambient and operating conditions on an hourly time step basis is a clear example of this. Thus, there is a need for simple models which can provide sufficiently good representation of the absorption machine behavior based only on available external parameters (experimental measurements or manufacturer catalog data). Simple models can be more easily incorporated in simulation programs or used for fault detection and control. Contrary to the physical models, the empirical and semi-empirical models require less time and effort to develop and computation time is much shorter when they are built into complete energy management simulation programs. The parameters and fitting coefficients in these models are determined by using a regression method or a minimization algorithm applied to a dataset obtained performing experimental measurements or using a manufacturer catalog. The studies about development of empirically based models for absorption chillers have been reported by several authors. Gordon and Ng [8] developed a general model for predicting the absorption chillers performance. The model lays both on physical and empirical principles. The physical principles that govern the performance of the absorption chiller are fitted to the experimental or manufacturer data by using a regression method. Ziegler et al. [9,10] developed a model (Characteristic equation method) which predicts the performance of the absorption chiller by using two simple algebraic equations: one to calculate the cooling capacity and another for the driving heat input. These two previous models belong to semi-empirical (gray-box) category of models, in which the fitted parameters can be interpreted under the actual physical principles which govern the absorption chiller performance. Labus et al. [11] used a completely empirical approach to model
absorption chillers based on manufacturers curves in order to investigate the energy savings when different absorption chiller configurations were considered for their integration in a complete chiller plant. The Artificial neural networks approach has been also used for absorption chiller modeling. ANN models belong to the black-box model category, that unlike gray-box models, the estimated parameters of the model have no physical interpretation. Sözen [12] used the ANN to determine thermodynamic properties of an alternative working pair for absorption systems. The study also demonstrated that ANN can replace mathematical models in the simulation of absorption systems. In the paper of Sözen and Akçayol [13] the ANN approach was proposed for performance analysis of an absorption chiller. The ANN model used only the working temperatures in the four main components as input parameters in order to predict the performance of the chiller. Manohar et al. [14] applied ANN for the modeling of steam fired double effect absorption chiller. Later, a similar work was carried out by Rosiek and Batlles [15], who used ANN to model solar-assisted air conditioning system with hot water driven double effect absorption chiller. The last approach considered in this paper is the simple multivariable polynomial regression which also belongs to the black-box category of models. Regardless the numerous studies on the modeling of absorption equipment, literature review shows that there is a lack of information with respect to comprehensive comparative studies on different modeling techniques for predicting absorption equipment performance in a similar way as Swider [16] or Lee et al. [17] did for the case of vapor-compression chillers. The main aim of this paper is to present a comparative evaluation of different modeling approaches for predicting the performance of absorption systems. In the next section are presented the experimental data and a brief description of the evaluated types of models. Later the application of these models to the experimental data is evaluated with the help of statistical indicators and statistical tests to select the best modeling approach. 2. Experimental data and absorption chiller models Four different types of absorption chiller models were developed and examined:
Adapted GordoneNg model, Adapted characteristic equation model, Multivariate polynomial regression model and Artificial neural networks model.
The experimental data required for the models’ development were obtained in the state-of-the-art test bench of the Rovira i Virgili University in Tarragona (Spain). The test bench is fully equipped to test under controlled operation conditions a variety of units commonly used in HVAC systems. A more detailed explanation about the functionality of the test bench can be found in Labus et al. [18] and Labus [19]. For the models described in this research the data were collected in a series of experiments with a 12 kW absorption chiller Pink Chilli PSC12. The measured variables in the experiments were inlet and outlet temperatures of hot, chilled, and cooling water circuit; volumetric flow rates and pressure drop in each circuit; and electric consumption of the chiller. The raw data were processed using a comprehensive test procedure which includes several techniques: data reduction, development of steadystate detector with additional filtering and uncertainty estimation. Based on external measurable parameters only, this procedure allows the creation of the complete performance map for absorption machines based on highly accurate data. In data reduction, the
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collected data were used to calculate thermal loads and efficiency coefficients. Off-line steady-state detector for absorption chillers was developed based on analogy with steady-state detector for vapor compression chillers using the moving window average. Additional data filtering was engaged to eliminate remaining transient periods caused by time delays when changing from one steady-state to another. The evaluation of experimental uncertainty was carried out by judgment based on available information on the possible variability of input quantities. When the uncertainties of heat loads were evaluated, the following input quantities were taken into account: inlet and outlet temperatures of external circuits, volumetric flow rate, density, and specific heat capacity. Uncertainty contribution for temperature and volumetric flow rate was calculated as a combination of different sources of uncertainty: repeatability, accuracy of the instrument and resolution of the instrument. In order to be in accordance with international standards, time length for the steady-state tests was not shorter than 30 min collected in 5s intervals. The experimental database used for modeling consists of 138 steady-state points and covers the following temperatures ranges: inlet hot water temperature 80e 100 C, inlet cooling water temperature 27e35 C and outlet chilled water temperature 5e12 C, as presented in Table 1. The main criterion to select these three temperatures as input variables for the empirical models was their availability to the operating engineers in practical applications. In detail description of the test procedure as well as the experimental results can be found online in Labus’ PhD thesis [19]. Also, it is important to explain the influence of database size on modeling. Models developed with small datasets are not reliable and statistically correct, since small datasets are insufficient to form strong relationships within the models. On the other hand, the models created with large dataset which completely covers the operating range of the absorption chiller show very high level of predicting capabilities.
the absorber and condenser as a single source of heat at medium temperature. According to Gordon and Ng approximation, the finite-rate mass transfer is roughly temperature independent. With respect to that, the losses in the evaporator can be neglected, while the losses in other two heat exchangers (generator, absorber/condenser) can be viewed as a constant characteristic of each particular chiller. The general equation for the GNA model can be obtained after series of transformation, starting from the First law of thermodynamics and using the entropy balance which takes into account the dominant irreversibility [8,19]. The GNA model calculates the inverse of COP using the following equation (Eq. (1)):
# " # " # " # " in in Tgen Tgen 1 T in T out 1 þ $ ¼ ac out eva $ in : in in T in Teva COP Tgen Tac Tgen Qeva ac # " in Tac $ a1 a2 $ in Tgen
and a2 as the intercept and slope of this line using linear regression. Bearing in mind that the purpose of this analysis is to compare different modeling approaches by means of the deviations between experimental and modeled heat loads, the final equation of the GNA model (eq. (1)) was adapted to obtain the chiller capacity (Eq. (2)). The heat input can be derived from the COP (Eq. (3)). :
B 1=COP A
(2)
:
B 1 A$COP
(3)
2.1. Adapted GordoneNg model (GNA)
Table 1 Experimental operation range conditions. Variable
Range
in [ C] Teva out [ C] Tac in [ C] T:gen
[4.98e12.1] [26.95e35.01] [79.9e100.12] [0.49e15.23] [5.99e41.98] [4.64e24.04] [0.11e0.76]
Q: eva [kW] Q: ac [kW] Q gen [kW] COP :
:
Flow rates [m3/h] fixed at: meva ¼ 1:7; mac ¼ 4:8; : mgen ¼ 2:2
(1)
where a1 and a2 are the regression parameters to be fitted with experimental data and, at the same time, the constants which characterize the entropy generation of particular chiller. " in T in in Tgen Tac ac against Considering that a plot of in in $COP Tgen Tgen # in T out : Tac eva $Qeva leads to a straight line, is it possible to calculate a1 out Teva
Q eva ¼
The general thermodynamic model for absorption chillers developed by Gordon and Ng [8] is actually a combination of physical and empirical approaches. According to the authors, the dominant irreversibility of the absorption chillers is finite-rate mass transfer. The losses due to the finite-rate mass transfer can therefore be approximated as temperature independent. The original model was based on external input parameters of the four main components (generator, condenser, evaporator and absorber) assuming that manufacturers’ catalogs provide the operating conditions for each of them. However, the current manufacturers’ practice is to provide operating curves based on three circuits, i.e. treating absorber and condenser as one component. The main reason for that is the arrangement in series of the absorber and condenser in the majority of the commercial absorption chillers. Therefore, in our case the original model was modified considering
307
Q gen ¼ where:
# " # in in T out Tgen Tac eva $ in ; A ¼ out in Teva Tgen Tac # " # " in in Tgen Tac $ a1 a2 $ in B ¼ in T in Tgen Tgen ac "
(4)
2.2. Adapted characteristic equation model (DDt0 ) For the modeling of absorption chillers, Ziegler et al. [9] developed an approximate method which is able to represent both cooling capacity and driving heat input by simple algebraic equations. These equations are expressed as a function of so-called characteristic temperature difference (DDt), which depends on the average temperature of the external heat carrier fluids. One of the main assumptions is that the heat transfer processes in absorption chillers dominate their performance behavior. In this way, a complex response to all external heat carrier temperatures is reduced to a linear function of heat flow and the external temperatures. A simple linear correlation is very convenient, but it has been found that the predicted performance of the cooling capacity deviates considerably from the linear behavior, for instance, at high
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driving temperatures, due to higher internal losses. With respect to that, an adapted characteristic equation method was proposed by Kuhn and Ziegler [20]. This improved model uses a numerical fit of catalog or experimental data to improve the characteristic equation. The adapted characteristic temperature function (DDt0 ) takes the form (Eq. (5)):
DDt 0 ¼ tgen a$tac þ e$teva
(5)
And the linear characteristic equation for each component loads k (Eq. (6)): :
Q k ¼ s0 $DDt 0 þ r
(6)
Combining Eqs. (5) and (6) yields one correlation which represents the thermal performance of the components as a function of the external arithmetic mean temperatures of the generator (tgen), absorber-condenser (tac) and evaporator (teva), when the external flow rates are constant. :
Q k ¼ s0 $tgen s0 $a$tac þ s0 $e$teva þ r
(7)
The four parameters (s’, a, e and r) are estimated by using a multiple linear regression algorithm to fit the experimental data. This algorithm chooses regression coefficients to minimize the residual sum of squares. The analyses of Puig-Arnavat et al. [21] confirmed the capability of the DDt0 method to obtain good results and also better accuracy than the original method DDt. Finally, the combination of the obtained characteristic functions with the equations of the external arithmetic mean temperatures and with the external energy balances, results in a system of six equations with six unknowns which can easily be solved. The developed model requires only three temperatures (one from each of the external circuits) at constant flow rates of external heat carriers to predict 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 processes incorporated in the model structure. MPR models are a very effective tool for describing complex non-linear relationships between input and output variables without disregarding what occurs within the system. The parameters for the MPR model are calculated by fitting the experimental data minimizing the sum of squares of the residuals using a polynomial function. Due to their simple structure, MPR models have been applied in various fields such as forecasting, control, optimization, fault detection and diagnosis [17,22]. Lee et al. [17] proved that MPR model of vaporcompression water chiller can have a high prediction accuracy, with the coefficient of variation of 0.61%. Similarly, the paper of Kim et al. [22] confirmed that MPR models are acceptable for fault detection and diagnosis of residential heat pump systems. A typical polynomial regression model contains the squared and higher order terms of the estimator variable. Normally, the higher order MPR models offer better accuracy of prediction. However, high-order MPR can become impractical due to its excessive number of parameters. One of the common techniques in the case of the high order MPR models with large number of parameters is to reduce the model by retaining only those parameters that are statistically significant. Also, excessive polynomial order for a relatively small database may worsen data interpolation. These are some of the reasons why it has been decided to apply only second order polynomials to predict the absorption chiller performance. Thus, the MPR models were developed to calculate the thermal loads of the absorption chiller by using the measurements of external circuits:
generator inlet temperature, absorber/condenser inlet temperature, and evaporator outlet temperature. The generalized second order model in case of absorption chillers can be represented using Eq. (8): :
in out in in out Q k ¼ b0;k þ b1;k Tgen þ b2;k Tac þ b3;k Teva þ b4;k Tgen Tac 2 in in out in in þ b5;k Tgen Teva þ b6;k Tac Teva þ b7;k Tgen 2 out 2 in þ b8;k Tac þ b9;k Teva
(8)
2.4. Artificial neural network model (ANN) ANN models also belong to the group of black-box models. An artificial neural network is an adaptive system which can be trained to perform a particular function or behavior on the basis of input and output information that flows through the network. ANN have found their place in the fields of modeling, identification, optimization and control in steady state and dynamic systems due to their ability to model complex relationships between inputs and outputs or to find patterns in data. Various applications of neural networks in renewable energy problems such as energy prediction and optimization of energy consumption in building service systems were presented in the review of Kalogirou [23]. The review of Mohanraj et al. [24] covers the applications of ANN in energy and exergy analysis of refrigeration and air-conditioning systems, their control and in prediction of refrigerant properties. Moon et al. [25] developed an adaptive control method using the ANN model to enhance thermal comfort in buildings. Yabanova and Keçebas¸ [26] developed ANN-based PID controller for geothermal district heating system in Turkey, which increased energy efficiency and cost saving of the system by 13%. The most common ANN architecture applied in the field of absorption systems and their applications are feed-forward neural networks with back-propagation [27]. In this research, ANN models of the absorption chiller were developed by using MatLab Neural Network toolbox. Since there is no explicit rule to determine the topology of ANN (the number of neurons in the hidden layer or the number of hidden layers) the trial and error method is usually applied to find the best solution. Thus, the adopted topology for ANN models was (3e7e1), as illustrated in Fig. 1. Each model consists of one input layer with three variables, one hidden layer with seven neurons and one output layer with one output: a component load (three different ANN models are built for thermal power exchanged in the evaporator, absorber/condenser and generator). The training of the ANN was based on the error back propagation technique using the LevenbergeMarquardt algorithm of optimization. The input parameters were normalized in the [0.2, 0.8] range. A hyperbolic tangent sigmoid function (tansig) was used in the hidden layer and the linear transfer function (purelin) was used in the output layer. To test the robustness and the prediction ability of the models, the experimental dataset was split into three parts: 70% of data was used for the model training, 20% for the model validation and the remaining 10% for the model testing. The ANN absorption chiller model to calculate the thermal load in each component is given by the general equation (Eq. (9)): :
Qk ¼
j X i
" LWð1;jÞ $
2 1 P R 1 þ exp 2 1 IWðj;RÞ IR þ b1ðjÞ
!# þ b2 (9)
where I is the input, R is the number of the inputs (R ¼ 3), b1 are biases in the hidden layer, b2 are biases in the output layer, J is the
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Fig. 1. ANN topology.
number of neurons in the hidden layer (J ¼ 7), and IW and LW are the weights in the input and output hidden layer, respectively. 3. Results and discussion 3.1. Model parameters The developed models require different parameters in order to calculate the absorption chiller performance. All these parameters were estimated according to the methods explained above and they are listed in the appendix. 3.2. Evaluation of the models 3.2.1. Simple comparison Fig. 2 shows the comparison of the measured and calculated cooling capacity of the chiller with a generator and condenser/ absorber temperature of 85 C and 27 C, respectively. As can be seen from the selected dataset, GNA model prediction shows a considerable deviation when compared to the other three models. On the other hand, DDt0 , MPR and ANN models show very close agreement with experimental data. The discrepancy between the model predictions and experimental data in the worst case is less than 5%. Unfortunately, cross validation of the models with data reported by other authors was not possible due to lack of information.
Fig. 2. Comparison of the experimental data with the data obtained by simulation.
309
3.2.2. Comparison through statistical indicators The goodness-of-fit of a model is usually evaluated in terms of statistical indicators. The statistical performance analysis of the evaluated models was conducted including several statistical indicators: the residual sum of squares (SSres), the coefficient of determination (R2), the root mean square error (RMSE) and the coefficient of variation (CV). The most common parameters to check how close the predicted values are to observed data are the residual sum of squares and the coefficient of determination. Residual is unexplained variation after fitting a model and is the difference between the value predicted by the model and the associated observed value. The sum of squares of these differences is called the residual sum of squares and can be understood as a measure of the discrepancy between the data and an estimation model. A smaller SSres indicates better fit to the observed data. The coefficient of determination is another parameter which quantifies the goodness of fit. The R2 can be calculated from the residual sum of squares and the total sum of squares (SStot) by the Eq. (10) and can be interpreted as a statistical measure of how well a model prediction approximates the observed data.
R2 ¼ 1
SSres SStot
(10)
An R2 of 1.0 would indicate that model prediction perfectly fits the observed data. However, the statistical analysis cannot rely only on R2, no matter how reasonable the fit is. It should be interpreted together with other indicators. RMSE is used to obtain the confidence interval (CI) which is a way to visualize the precision of each model. Narrower CI indicates better precision since RMSE is lower. Normally, CI is constructed by using the standard deviation:
CI ¼ y z$s
(11)
where y is the mean value of the measurement, s is the standard deviation of the measurement, and z is the score of the standard normal distribution. Using the RMSE instead of s and assuming a confidence level of 95% the CI can be estimated as:
CI ¼ Qk 1:96$RMSE
(12)
We also use the coefficient of variation of the root-mean-square error (CV) in order to compare the models in terms of predicting capabilities. CV is defined as RMSE divided by the dependent variable average (Eq. (13)).
CV ¼
RMSE $100% Qk
(13)
The predicted cooling capacity and driving heat input using the four models of the absorption chiller are compared in Fig. 3 applying the R2 and CI (in the form of dashed lines) indicators. The comparison was performed using the entire experimental database. The solid line represents the ideal match of the model with experiment, while the dashed lines limit the 95% confidence area. A smaller distance between the two lines indicates a more accurate prediction of the model. As it could be expected in this figure is shown that much better accuracy is obtained by pure black-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 generator heat input, as illustrated in Fig. 3(a, e). Also, the widest CI range among all the models clearly indicates that GNA has the lowest accurate prediction. The other three methods (DDt0 , MPR and ANN)
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Fig. 3. Comparison between the measured and predicted evaporator and generator loads-Pink.
had much better statistical indicators. Excellent fit with the observed data is visible through high coefficient of determination (R2 w 0.99), while the narrow CIs indicate very accurate prediction of all three modeling methods. Among them, the narrower CI ranges and highest R2 values (>0.998) were obtained with the ANN modeling method. Another statistical indicator is the coefficient of variation (CV) of the root mean square error. CV indicator is a normalized measure of dispersion of the probability distribution and is defined as a percentage of the RMSE divided by the dependant variable mean (Eq. (14)).
CV ¼
RMSE $100% Qk
according to Hydeman et al. [28], the models with CV values in the range of 3e5% are supposed to have good accuracy for performance prediction in practical applications. The best CV indicators (<2%) in this analysis were obtained by the ANN method.
(14)
The CV values for the different modeling methods are illustrated in Fig. 4. If 10% deviation of CV is assumed to be acceptable to obtain a satisfactory prediction, it is clear that the developed GNA model cannot pass this threshold. On the other hand, the calculated CV values of DDt0 , MPR and ANN are lower than 5% which is more than satisfactory. Actually,
Fig. 4. CV values.
J. Labus et al. / Applied Thermal Engineering 58 (2013) 305e313 Table 2 AICc and BIC analysis. BIC
Criterion
AICc
Model
Q eva
Q ac
Q gen
Q eva
Q ac
Q gen
GNA DDt’ MPR ANN
33.6 244.5 329.7 412.9
337.8 91.9 120.5 208.9
185.7 140.0 188.5 342.9
719.3 446.9 377.8 346.1
1023.5 596.6 587 550.1
871.4 548.5 519.0 416.0
:
:
:
:
:
:
3.2.3. Statistical tests The selection of the best model using statistical indicators could not be enough because what is meant by best is controversial. A good model selection technique should take into account not only goodness of fit but also the simplicity of the model. More complex models will be better able to adapt their shape to fit the data, but the additional parameters may not represent anything useful. The statistical analysis usually used in the case of nested models (when one model is a simpler case of the other) is the extra sum of squares test (F-test). The F-test is not appropriate for the comparison of non-nested models, which is the case here. When comparing non-nested models there are two criteria which can be used [29]: Akaike’s Information Criterion (AIC) or Schwarz-Bayesian Information Criterion (BIC). The AIC can be termed as a measure of the goodness of fit of any estimated statistical model. The BIC is a type of model selection among a class of parametric models with different numbers of parameters. The difference between AIC and BIC is that the penalty term for the number of parameters in the model is larger in BIC than in AIC. Both criteria were applied to compare the analyzed absorption chiller models. The used AICc is a second order AIC with a greater penalty for extra parameters and can be calculated by using Eq. (15):
AICc ¼ N$ln
SSres 2K$ðK þ 1Þ þ 2K þ N NK1
(15)
Similarly, BIC is defined by Eq. (16):
BIC ¼ N$lnðSSres Þ þ K$lnðNÞ
(16)
where N is the number of data points, K is the number of the model parameters and SSres is the residual sum of squares. The interpretation of the results in both tests is the same: the model with lower value is more likely to be selected. AICc and BIC analysis results of all the models developed for the Pink absorption chiller are presented in Table 2. The analysis also included the GNA model although it could be discarded after the analysis performed with the statistical indicators. The results of both AICc and BIC tests were in agreement with the goodness of fit results, recommending the most complex model ANN to be adopted. Although the ANN model, which is a complete black-box model, shows the closest fit, other more physically related models also could be applied, depending on the specific application requirements (Table 2).
4. Conclusions A comprehensive comparison of different methods for steadystate modeling of small capacity absorption chillers was presented in this paper. Four models were developed based on experimental data. The models were based only on the measurements of the external water circuits. Statistical indicators such as R2 and CV showed that GNA model has the lowest predictive capacity.
311
Although the GNA model could not reproduce the chiller performance with high accuracy, it is still used in some cases due to its simplicity. This is justified by a fact that on a whole year hourly simulation, these deviations will most probably equal out to a great extent. However, the statistical analysis indicates that would be more appropriate to use one of the other three methods in order to obtain better accuracy. Excellent statistical indicators (R2 around 0.99, CV lower than 5% and narrow CI) clearly show that any of the three methods (DDt0 , MPR and ANN) is suitable for the performance prediction of absorption systems, and could be used for the chiller control and monitoring, fault detection or optimization. Nevertheless, the best prediction was obtained with the ANN method with R2 > 0.998 and CV<2%. The comparison of the models with AICc and BIC statistical tests confirmed that ANN was the most suitable method to model the selected absorption chiller. Acknowledgements The authors would like to acknowledge financial support of this work by the CITYNET project funded via the Marie Curie Research Training Network, the project TR 33049 funded by Serbian Ministry of Education and Science and the project ENE2009-14182 funded by the Spanish Ministry of Economy and Competitiveness. Appendix Two linear regression coefficients for the GNA model,
a1 ¼ 36.524 and a2 ¼ 41.527, were obtained after performing nu-
merical fit of the experimental dataset. Table A.1 shows the coefficients for characteristic functions of absorption chiller obtained through multiple linear regression fit for the DDt’ method.
Table A.1 Estimated input parameters-DDt’.
Q_ eva Q_ ac Q_
gen
s0
s0 a
s0 e
r
0.6822 1.2698 0.5662
1.0057 2.3951 1.3066
0.3801 1.0347 0.5947
1.6396 0 0
In the case of MPR models, Table A.2 shows the regression fitting parameters (bi,k) for the different heat loads (k) when a second order polynomial fit was applied on the experimental data of the absorption chiller. Table A.2 Fitting coefficients for MPR models. (k)
Q_ eva
Q_ ac
Q_ gen
b0 b1 b2 b3 b4 b5 b6 b7 b8 b9
6.818 0.3703 1.4329 1.113 0.0122 0.0014 0.0076 0.0025 0.0081 0.0217
65.3225 0.3092 4.4372 0.2036 0.0159 0.0095 0.0117 0.0001 0.0146 0.0172
28.7061 0.5642 3.037 0.5002 0.0079 0.0048 0.0184 0.0019 0.0187 0.0009
Table A.3 shows the coefficients for the ANN modeling of the absorption chiller obtained during the network training stage.
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Table A.3 ANN coefficients. Input weights IW ðQ_ eva Þ 2.6842 26.2414 27.994 37.8786 4.5898 1.0661 0.4744 2.2865 2.0043 14.9552 3.0631 3.4876 0.6453 1.4337 Output weights 0.6507 LWðQ_ eva Þ 3.5228 LWðQ_ ac Þ _ 0.4252 LWðQ gen Þ Biases in input layer b1ðQ_ ac Þ b1ðQ_ eva Þ 7.5266 2.1264 73.789 0.5695 2.7049 1.3703 3.5951 11.9184 33.3159 0.1494 1.984 9.4941 0.0916 3.7636
40.2259 151.8244 1.0163 7.902 27.8277 3.0904 0.9214
IW ðQ_ ac Þ 1.3136 1.6348 3.1738 3.85 0.6074 1.0057 0.2914
0.4582 3.8286 18.2436
1.1899 1.8483 0.4097
b1ðQ_ gen Þ 22.9476 1.0747 1.3844 2.0647 3.6732 10.9156 5.1815
b2ðQ_ eva Þ 6.993
4.368 3.5549 0.4494 1.5356 1.2485 11.3297 0.8686
1.0693 0.616 0.8719 27.8663 9.6199 0.3298 Biases in hidden layer b2ðQ_ ac Þ b2ðQ_ gen Þ 23.9447 12.8346
Nomenclature
a b 3
b1, b2 A CL COP I IW, LW J: Q R R2 RMSE SS T
2.9482 6.8414 3.5448 7.2797 0.0293 5.0261 6.4158
GNA regression coefficient MPR coefficient efficiency bias heat transfer areas [m2] confidence limits coefficient of performance [-] input variable matrix weights number of neurons in the hidden layer heat flow [kW] number of neurons in the input layer coefficient of determination root mean square error sum of squares temperature [ C]
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