Computers chern. Engng, Vol. 18, Suppl., pp. S63-S67, 1994 Printed in Great Britain. All rights reserved
0098-1354/94 $6.00+0.00 Copyright © 1993 Pergamon Press Ltd
Artificial Neural Networks as a Predictive Tool for Vapor-Liquid Equilibrium
Rene Petersen, Aage Fredenslund and Peter Rasmussen Engineering Research Center IVC-SEP, Department of Chemical Engineering, The Technical University of Denmark, DK 2800 Lyngby, Denmark
ABSTRACT A new type of group-contribution method for calculation of liquid phase activity coefficients is presented. The method is implemented by using an artificial neural network. Calculated results are compared with the UNIF AC method and experimental data.
KEYWORDS Group-contribution method, activity coefficient model, neural networks.
INTRODUCTION In process design and optimization up to 80% of the total computation time can be spent on evaluation of thermodynamic properties. New methods for rapid and cost effective evaluations are needed to minimize computation time. One important problem is a correct presentation of vapor-liquid equilibria (VLE) since most simulators rely heavily on such calculations. Liquid phase nonideality can often be described using a group-contribution method such as the UNIF AC method (Fredenslund et aI., 1977). This work will introduce a new group-contribution model for estimation of liquid phase activity coefficients. The model will be implemented by using an artificial neural network (ANN) and will be compared with the UNIF AC method.
A NEW TYPE OF GROUP-CONTRIBUTION METHOD The activity of a component in a mixture can be considered to be a correction to the concentration due to interactions and size differences between the molecules in the mixture. Information about the different kinds of interactions must be available in some form in order to calculate the activity. A molecule can be considered to consist of different structural groups, . which wiIl influence the physical and chemical behavior of a given molecule, Molecules belonging to the same class of compounds, ego ketones, retain some common properties. This is the basis of group-contribution methods and it provides an easy way to classify molecules by their structural groups. Assuming that all information about the interactions between the CACE 18 Suppl-O
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molecules in the mixture can be related to the distribution of well defined structural groups in the molecule and in the mixture, the activity and thereby also the activity coefficient can be calculated as a function of these group distributions and temperature only:
(1)
where 'Iji
=
(2)
and NC
8j =
li !; li -1
Vj0k
NG
NC
=1
-1
(3) V10k
Yi is the activity coefficient, a, is the activity and Xi is the mole fraction for component i. vij is the number of groups j in component i. NC is the number of components and NG the number of structural groups. If an analytical expression for M is known, model parameters can be fitted using experimental data. In this case there is no analytical expression for M and a new and different approach is used.
DETERMINATION OF M BY USING AN ARTIFICIAL NEURAL NET A neural net consists of a number of local information processing units, called neurons. Artificial neural networks are (or can be) mathematical models of adaptive systems, originally inspired by studies on the human brain. A detailed description on artificial neural networks can be found in: (Rumelhart and McClelland, 1986) and (Hecht-Nielsen, 1990). This work will use the backpropagation algorithm for training a feedforward net. All neurons in a given layer will be connected to all neurons in the next layer, but no connections between neurons on the same layer will be allowed. The neuron will receive a number of inputs. These will be amplified or damped and the overall sum will be transformed by an activation function. The basic model of a neuron is: (4)
where Yj is the output from neuron j. Yj is a function f of the added inputs U i from neuron i on the previous layer multiplied by the weight Wi' As activation function a sigmoidal function is used: f (u)
1 = -:---=:----;--, 1 + Exp(-u)
(5)
The network performance is a function of the topology, that is the number of neurons and the number of layers and also of the weights. The network can be operated in two ways: Training
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and running. In the training process known values of both input and output are presented to the net. Based on the error in the calculated output, corrections to the weights are made. When the performance is satisfactory, the network can be used to calculate new output values based on new input values. For the purpose of this work the artificial neural network can be regarded as a black-box capable of learning a mathematical function by running a large number of examples.
TRAINING THE NETWORK To demonstrate the method the same groups as in UNIF AC were used as shown in Table 1: Table 1. A sub-set of UNIF AC groups. Group 1
CH3
Group 2
CH2
Group 3
CH3CO
Group 4
CH2CO
With these groups systems containing alkanes and ketones can be described. In order to make the comparison between this method and UNIF AC as accurate as possible the same information which originally was used in UNIF AC to estimate the interaction parameters between the CH2 and the CH2 CO groups have been used: Pentane - Acetone (Lo et al. 1962), Cyclohexane Acetone (Puri et al. 1974), Hexane - Acetone (Schaefer and Rall, 1958), Heptane- 2-Butanone (Steinhauser and White, 1949), Heptane- 3-Pentanone (Geiseler and Koehler, 1968) and Octane2-Butanone (Maripuri and Ratcliff, 1972). Four UNIF AC sub-groups are needed for describing the components in the training data set. By using the structural groups from table 1, the activity coefficient is given by: Yl
=
JlI (~CH311
~CH211 ~CH3COli ~CH2COli 8CH3 1 8CH2 '
8 CH3C01 8 CH2C01
'1')
(6)
, The artificial neural network is set up to generate the function M': 9 neurons in the inputlayer, 1 neuron in the outputlayer. Two network topologies have been evaluated: A net with one hidden layer with four neurons (ANN,9,4,1) using 45 parameters/weights and a net with no hidden layer (ANN,9,1) using 10 parameters/weights. The training data was repeatedly presented to the net until the values of the weights converged.
RESULTS The performance of the network has been tested by calculating activity coefficients for 3 binary alkane - ketone VLE-systems not included in the training data: Cyclohexane - 2-Butanone (Kurmanadharao and Rao, 1957), Cyclohexane - Cyclohexanone (Boublik and Lu, 1977) and nHexane - 2-Butanone (Hanson and van Winkle, 1967). The sum of squared errors and the standard deviation on the error are reported in table 2. The error is calculated as:
Yi,Calculated- Yi,Experimcntal'
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Table 2. Results comparing UNIF AC and ANN. Method
Sum of Squared Errors
Standard Deviation
Training Data
UNIFAC
64.3764
0.2953
Cyclohexane - 2-Butanone
UNIFAC
4.8264
0.1142
Cyclohexane - Cyclohexanone
UNIFAC
1.3680
0.0484
Hexane - 2-Butanone
UNIFAC
0.8877
0.0364
Training Data
ANN,9,4,1
81.6618
0.3745
Cyclohexane - 2-Butanone
ANN,9,4,1
14.7181
0.3470
Cyclohexane - Cyclohexanone
ANN,9,4,1
9.8623
0.3432
Hexane - 2-Butanone
ANN,9,4,1
2.6078
0.1087
Training Data
ANN,9,1
126.7279
0.5813
Cyclohexane - 2-Butanone
ANN,9,1
68.2063
1.6239
Cyclohexane - Cyclohexanone
ANN,9,1
10.4676
0.3679
Hexane - 2-Butanone
ANN,9,1
3.1356
0.1303
System:
DISCUSSION UNIF AC gives very good predictions as compared to the artificial neural network. Since no theory and no physical constants are used for predictions by the artificial neural network, one can not expect the same quality of predicions as with UNIF AC. The network will try to minimize the error for the calculated activity coefficients based on all available data points since all weights are adjusted during training. This is not the case with UNlFAC where groups of binary interactions are optimized individually. The performance of the network will improve with the number of known datapoints available. The results shown for a limited number of data points indicate a possible use of artificial neural networks as a tool for prediction of VLE data, but further work is still required.
CONCLUSION An alternative group-contribution method has been suggested and implemented by using an artificial neural network. Predictions for 3 different binary system have been reported. A comparison with UNIF AC and the artificial neural net against experimental data points shows that UNIF AC is superior to the artificial neural network. It is however shown that with no analytical expression for the activity coefficient an artificial neural networks can be used to predict activity coefficients with an average error of +/- 5%.
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