2-prediction of liquid and vapor enthalpies of ammonia-water mixture
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Prediction of Liquid and VaporEnthalpies of Ammonia-water MixtureA. encan a , S. Gk a & E. Dikmen aa Department of Mechanical Education, Technical EducationFaculty , Sleyman Demirel University , Isparta, TurkeyPublished online: 19 May 2011.
To cite this article: A. encan , S. Gk & E. Dikmen (2011) Prediction of Liquid and Vapor Enthalpiesof Ammonia-water Mixture, Energy Sources, Part A: Recovery, Utilization, and Environmental Effects,33:15, 1463-1473, DOI: 10.1080/15567030903397891
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Energy Sources, Part A, 33:14631473, 2011
Copyright Taylor & Francis Group, LLC
ISSN: 1556-7036 print/1556-7230 online
DOI: 10.1080/15567030903397891
Prediction of Liquid and Vapor Enthalpies
of Ammonia-water Mixture
A. SENCAN,1 S. GK,1 and E. DIKMEN1
1Department of Mechanical Education, Technical Education Faculty,
Sleyman Demirel University, Isparta, Turkey
Abstract The ammonia-water mixture may be commonly employed as a work-ing fluid in the absorption chiller, especially because both ammonia and water are
natural substances and are harmless. In addition, these substances have excellentthermodynamic properties. In this study, an alternative method using the artificial
neural network (ANN) to determine liquid and vapor enthalpies of ammonia-watermixture is presented. The training and validation was performed with good accuracy.
The correlation coefficient obtained when unknown data were used to the networkswas 0.975 for the liquid enthalpy and 0.887 for the vapor enthalpy. Using the
weights obtained from the trained network, a new formulation is presented for thedetermination of the vapor and liquid enthalpies of ammonia-water mixture. The
results of the study show that the ANN is a perfect alternative method for thecalculation of thermodynamic properties of ammonia-water mixture. The faster and
simpler solutions with equations derived from the ANN can be carried out.
Keywords ammonia-water, liquid enthalpy, neural network, thermodynamic proper-ties, vapor enthalpy
1. Introduction
The ammonia-water mixture can be used as a working fluid in the absorption chillers. In
the absorption chillers operating with ammonia-water solution, water is the absorbent and
ammonia is the refrigerant. Since the 1970s, they are under consideration for residential
and commercial heating and cooling (Herold et al., 1996; ASHRAE, 1997; Darwish et al.,
2008).
Thermodynamic properties of fluid couples are very important parameters affecting
the performance of absorption systems. The engineering calculation and simulation of
absorption systems require the availability of simple and efficient mathematical formu-
lations for the determination of thermodynamic properties of fluid couples. Vapor and
liquid enthalpies of ammonia-water mixture were presented in the literature as limited
data (Yamankaradeniz et al., 2002). In this study, in order to determine liquid and vapor
enthalpies of this mixture, artificial neural networks (ANNs) were used. Vapor and liquid
enthalpies of ammonia-water mixture with new formulations obtained from an ANN
can be easily estimated. The method proposed offers more flexibility and, therefore,
thermodynamic simulation of absorption systems is fairly simplified.
Address correspondence to Dr. Arzu Sencan, Department of Mechanial Education, TechnicalEducation Faculty, Sleyman Demirel University, Isparta 32260, Turkey. E-mail: [email protected]
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1464 A. Sencan et al.
Figure 1. Neural network process.
2. ANNs
ANN is an information processing paradigm that is inspired by the way biological nervous
systems, such as the brain, process information. The key element of this paradigm is the
novel structure of the information processing system. It is composed of a large number of
highly interconnected processing elements (neurons) working in unison to solve specific
problems. ANNs, like people, learn by example. An ANN is configured for a specific
application, such as pattern recognition or data classification, through a learning process.
Learning in biological systems involves adjustments to the synaptic connections that
exist between the neurons (Haykin, 1999; Fu, 1994; Tsoukalas and Uhrig, 1997; Lin and
Lee, 1996). The neural network process is described in Figure 1. Neural networks have
been used in the estimate of thermodynamic properties and analysis of energy systems
(Kalogirou, 2000a, b; Chouai et al., 2002; Pacheco-Vega et al., 2001; Bechtler et al.,
2001; Szen et al., 2004a, b; Szen and Akayol, 2004; Lazzs, 2009; Eslamloueyan
and Khademi, 2009).
ANN with a back-propagation algorithm learns by changing the connection weights,
and these changes are stored as knowledge. Some statistical methods, such as the root-
mean-squared (RMS), the coefficient of multiple determination (R2), and the coefficient
of variation (cov) may be used to compare predicted and actual values. These formulations
have been given in Bechtler et al. (2001).
3. Modeling of the Thermodynamic Properties Using ANN
In order to analyze and evaluate the performance of absorption systems, reliable thermo-
dynamic property models to predict enthalpy values depending on temperature, pressure,
and concentration values are required. These relationships have been provided using
ANN. In order to train the network, limited data reported by Yamankaradeniz et al.
(2002) were used. The inputs of the network are temperature, pressure, and concentration
of NH3-water mixture, whereas output is the liquid and vapor enthalpies. For this purpose,
neural networks were trained. There are different algorithms that can be applied to train a
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Enthalpies of Ammonia-water Mixture 1465
neural network. The most popular of them is the back propagation algorithm, which has
different variants. Standard back propagation is a gradient descent algorithm. It is very
difficult to know which training algorithm will be the fastest for a given problem, and the
best one is usually chosen by trial and error. In this study, LevenbergMarquardt (LM)
back-propagation training in a feed forward, single hidden layer network was repeatedly
applied until satisfactory training was achieved. Trainlm is a network training function
that updates weight and bias values according to LevenbergMarquardt optimization.
Inputs and outputs are normalized. Tan-sig activation function has been used for both the
hidden layer and the output layer. The function used is given by:
F.z/ D2
1C e2z 1; (1)
where z is the weighted sum of the input. The computer program was performed under
MATLAB environment using the neural network toolbox. In the training, we used a
variable number of neurons (from 3 to 12) in the hidden layer to define the output
accurately. The dataset for the liquid and vapor enthalpies of NH3-water mixture available
included 1,048 data patterns. Data patterns were collected from Yamankaradeniz et al.
(2002). From these, 838 data patterns were used for the training of the network and the
remaining 210 patterns were randomly selected and used as a test dataset. Figure 2 shows
the architecture of the ANN used for predicting the liquid and vapor enthalpies of NH3-
water mixture. In this figure, the temperature, pressure, liquid, and vapor concentration are
the input data and liquid enthalpy of the mixture is the actual output. The configuration
4-9-2 appeared to be the most optimal topology for liquid enthalpy. The configuration
4-10-2 appeared to be the most optimal topology for vapor enthalpy.
Training results based on the 4-9-2 configuration for liquid enthalpy is shown in
Figure 3. Training results based on the 4-10-2 configuration for liquid enthalpy is shown
in Figure 4.
Figure 2. ANN topology used for liquid enthalpy and vapor enthalpy prediction.
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Figure 3. Training results based on the 4-9-2 configuration.
In order to achieve the optimal result, different numbers of hidden neurons were
used. Statistical values, such as RMS, R2, and cov, are given in Tables 1 and 2 for liquid
and vapor enthalpy for LM algorithm and 312 neurons in the hidden layer.
From the data presented in Table 1, it is shown that liquid enthalpy of NH3-water
mixture LM algorithm with nine neurons in the hidden layer (LM-9) appeared to be
the most optimal topology. From the data presented in Table 2, it is shown that vapor
enthalpy of NH3-water mixture LM algorithm with ten neurons in the hidden layer (LM-
10) appeared to be most optimal topology.
Figure 4. Training results based on the 4-10-2 configuration.
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Table 1
Statistical values of liquid enthalpy for
NH3-water mixture
Algorithm neurons RMS Cov R2
LM-3 40.218 0.580 0.968
LM-4 37.108 0.535 0.972
LM-5 36.804 0.531 0.973
LM-6 35.517 0.512 0.975
LM-7 35.538 0.513 0.975
LM-8 35.480 0.512 0.975
LM-9 35.314 0.509 0.975
LM-10 37.020 0.534 0.972
LM-11 35.901 0.518 0.974
LM-12 35.890 0.518 0.974
The regression curve of the output variable (liquid enthalpy) for the test data set is
shown in Figure 5. The correlation coefficient obtained in this case is 0.975, which is
very satisfactory.
Figure 6 shows the regression curve of the output variable (vapor enthalpy) for the
test data set. The correlation coefficient obtained in this case is 0.887, which is very
satisfactory.
4. Results and Discussion
Mathematical formulations derived from the ANN model are presented here. The best
approach, which has minimum errors, is performed by the LM algorithm with 9 neurons
for liquid enthalpy and the LM algorithm with 10 neurons for vapor enthalpy. In order
to calculate the liquid enthalpy and vapor enthalpy values of NH3-water mixture, the
Table 2
Statistical values of vapor enthalpy for
NH3-water mixture
Algorithm neurons RMS Cov R2
LM-3 88.846 0.061 0.870
LM-4 86.130 0.059 0.878
LM-5 84.435 0.058 0.883
LM-6 83.958 0.057 0.884
LM-7 83.795 0.057 0.884
LM-8 84.421 0.058 0.883
LM-9 84.057 0.058 0.884
LM-10 82.935 0.057 0.887
LM-11 89.085 0.061 0.870
LM-12 83.033 0.057 0.887
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Figure 5. Comparison of actual and ANN-predicted values of NH3-water mixture liquid enthalpyfor the test data set.
following equations are derived:
Ei D
4XnD1
Inwni C bn; (2)
Fi D2
1C e2Ei 1: (3)
In the above equations, for Ei the first two values are the multiplication of the input
parameters (In) with their weights at location n, and the last constant value (bn) represents
Figure 6. Comparison of actual and ANN-predicted values of NH3-water mixture vapor enthalpyfor the test data set.
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the bias term. The subscript i represents the number of hidden neuron. The four input
parameters are:
I1 D Pressure .P /; (4)
I2 D Temperature .T /; (5)
I3 D Liquid concentration .Xf /; (6)
I4 D Vapor concentration .Xv/: (7)
In the ANN, nine hidden neurons are used for liquid enthalpy; thus, nine pairs of
equations, i.e., E1 to E9 and F1 to F9 are required, which represent the summation and
activation functions of each neuron of the hidden layer, respectively. The coefficients of
Eq. (2) are given in Table 3.
In the ANN, ten hidden neurons are used for vapor enthalpy; thus, ten pairs of
equations, i.e., E1 to E10 and F1 to F10 are required, which represent the summation and
activation functions of each neuron of the hidden layer, respectively. The coefficients of
Eq. (2) are given in Table 4.
Additionally, the actual input data of the various parameters need to be normalized.
For this purpose, the actual values of each parameter are divided with the coefficients
shown in Table 5.
Finally, the liquid enthalpy .hf / of NH3-water mixture depending on temperature,
pressure, and concentration values can be computed from:
E10 D F1 .44:4563/C F2 .0:095365/C F3 .0:76719/C F4 .0:0055292/
C F5 .0:64566/C F6 .0:04226/C F7 .40:0133/C F8 .0:23058/ (8)
C F9 .0:75846/C 5:4231;
hf D
2
1C e2E10 1
:991: (9)
Table 3
Weight coefficients and bias values used for the determination of liquid enthalpy
Neuron
position (wni) I1 (P ) I2 (T ) I3 (Xf ) I4 (Xv) bn
1 0.113 0.206 1.573 0.424 2.432
2 10.791 8.848 3.333 20.526 0.361
3 0.423 0.015 1.125 0.523 0.526
4 49.229 4.814 2.330 16.648 26.587
5 0.016 0.017 2.606 0.476 0.231
6 16.804 0.028 3.527 14.241 1.401
7 16.519 0.920 0.577 0.035 2.690
8 4.510 0.467 0.370 0.912 1.121
9 16.825 2.030 15.251 5.760 24.384
Note: In weights, n represents the input number and i represents the hidden neuron number.
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Table 4
Weight coefficients and bias values used for the determination of vapor enthalpy
Neuron
position (wni ) I1 (P ) I2 (T ) I3 (Xf ) I4 (Xv) bn
1 3.831 2.016 2.918 7.828 15.779
2 5.275 0.88 16.090 17.312 1.196
3 3.446 1.270 26.761 24.422 7.158
4 0.105 0.803 1.703 2.580 1.084
5 18.061 19.350 14.800 14.410 29.720
6 0.093 0.034 0.366 0.225 0.973
7 5.229 0.592 0.209 0.568 1.743
8 4.239 8.328 26.906 25.099 5.319
9 0.252 6.855 3.450 15.297 1.594
10 0.117 0.844 1.667 2.656 1.198
Note: In weights, n represents the input number and i represents the hidden neuron number.
The coefficient shown in Eq. (5) is used to convert the normalized output to actual output
.hf / of NH3-water mixture.
Similarly, vapor enthalpy .hv/ of NH3-water depending on temperature, pressure,
and concentration values can be computed from:
E11 D F1 .2:3056/C F2 .0:76346/C F3 .0:095678/C F4 .0:99316/
C F5 .10:1107/C F6 .5:7407/C F7 .8:3/C F8 .0:67651/ (10)
C F9 .15:2952/C F10 .1:1989/C 5:0796;
hv D
2
1C e2E11 1
:2802: (11)
Table 5
Normalization coefficients for the input
and output parameters
Coefficient
Input parameter
Pressure (TG) 3,000
Temperature (T ) 232
Liquid concentration (Xf ) 102
Vapor concentration (Xv) 102
Output parameter
Liquid enthalpy (hf ) 991
Vapor enthalpy (hv) 2,802
Note: The actual values are divided with the abovecoefficients to obtain the normalized values.
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Table 6
Comparison between actual liquid enthalpy and liquid enthalpy obtained
with equations derived from ANN for NH3-water mixture
P ,
kPa T , C Xf , % Xv , %
Actual
hf , kJ/kg
Predicted
hf , kJ/kg Error
Percentage
difference,
%a
60 55.4 10 74.9 155.4 155.32 0.08 0.050
140 56 20 90.47 87.3 86.95 0.35 0.406
480 78.2 28 93.77 136.6 136.99 0.39 0.284
520 76.8 30 94.4 125 124.70 0.30 0.241
560 79.4 30 94.14 138.4 138.32 0.08 0.057
600 84.2 28 92.44 177 178.33 1.33 0.752
640 97.5 24 88.41 247.2 247.22 0.02 0.01
680 99.7 24 88.05 261.1 259.89 1.21 0.463
720 80.7 34 95.33 131 130.97 0.03 0.024
800 92.7 30 92.59 202.7 202.26 0.44 0.216
840 172.4 0 0 729.3 729.68 0.38 0.051
920 107.1 26 88.42 284.9 286.18 1.28 0.449
1,000 89.2 36 95.08 164.7 165.43 0.73 0.443
1,200 70 50 98.36 66.3 66.41 0.11 0.161
1,400 68.3 55 98.74 68.7 68.60 0.10 0.148
1,600 112.7 34 92.04 281.8 281.58 0.22 0.076
1,800 47.5 94 99.95 182.8 183.556 0.756 0.413
2,000 59.4 80 99.79 142 143.180 1.180 0.831
2,400 58 96 99.99 250.2 251.224 1.024 0.409
aPercentage difference (%) D (error/actual vapor pressure) 100.
The coefficient shown in Eq. (7) is used to convert the normalized output to actual output
.hv/ of NH3-water mixture.
In Table 6, a comparison is presented between the actual liquid enthalpy and liquid
enthalpy predicted with the equations derived from ANN for NH3-water mixture. In
Table 7, a comparison is presented between the actual vapor enthalpy and vapor enthalpy
predicted with the equations derived from ANN for NH3-water mixture. As can be seen,
the error in both cases is very small.
5. Conclusions
A new methodology for forecasting NH3-water mixture enthalpies is presented. This
methodology, based on ANN, is successfully applied to determine NH3-water mixture
enthalpies. In order to calculate NH3-water mixture enthalpies, mathematical formulations
were derived from the ANN model. Mathematical formulations have been obtained from
formulations of the summation and activation functions used in the ANN model and
weights of neurons. This approach is valid for estimating liquid and vapor enthalpies of
NH3-water mixture at any temperature, pressure, and concentration. This formulation
can especially help manufacturers and engineers with thermodynamic simulation of
absorption systems.
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Table 7
Comparison between actual vapor enthalpy and vapor enthalpy obtained with equations
derived from ANN for NH3-water mixture
P ,
kPa T , C Xf , % Xv , %
Actual
hf , kJ/kg
Predicted
hf , kJ/kg Error
Percentage
difference,
%a
140 56 20 90.47 1,527.6 1,531.10 3.50 0.229
320 17.7 55 99.87 1,318.2 1,326.80 8.60 0.652
480 78.2 28 93.77 1,535.9 1,544.20 8.30 0.540
520 153.3 0 0 2,747.4 2,752.90 5.50 0.200
560 79.4 30 94.14 1,523.5 1,532.00 8.50 0.557
600 84.2 28 92.44 1,558.2 1,572.50 14.30 0.917
680 99.7 24 88.05 1,639.8 1,646.30 6.50 0.396
720 106.1 22 85.23 1,686.3 1,693.90 7.60 0.450
760 82.7 34 95.13 1,508.8 1,516.40 7.60 0.503
800 92.7 30 92.59 1,563.3 1,576.60 13.30 0.850
880 20.8 100 100 1,280.9 1,292.00 11.10 0.866
920 111.7 24 86.16 1,683 1,685.30 2.30 0.136
1,000 89.2 36 95.08 1,520.1 1,520.00 0.10 0.006
1,100 85.2 40 96.28 1,491.6 1,491.50 0.10 0.006
1,400 120.8 28 88.1 1,669.2 1,666.60 2.60 0.155
1,600 50.3 80 99.82 1,324.5 1,325 0.5 0.037
1,800 55 80 99.81 1,328 1,325.7 2.3 0.173
2,000 59.4 80 99.79 1,331.6 1,328.8 2.8 0.210
2,400 58 96 99.99 1,300.3 1,306.7 6.4 0.492
aPercentage difference (%) D (error/actual vapor pressure) 100.
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