Abstract: A new method, based on Artificial Neural Networks (ANN) of Multi-Layer Perceptron (MLP) type, has been developed to estimate the viscosity at moderate pressure for pure nonpolar gases over a wide range of temperatures. An ANN was trained, using four physicochemical properties: Molecular weight (M), boiling point (Tb), critical Temperature (Tc) and critical Pressure (Pc) combined with absolute Temperature (T) as its inputs, to correlate and predict viscosity. A group of 52 nonpolar gases were used to train and test the performance of the ANN. The viscosity and input data for each individual gas was compiled on average at fifty different temperatures, ranging from the boiling points for each of the chosen gases to 1100 K. The maximum absolute error in viscosity, predicted by the ANN, was approximately 15%.
INTRODUCTION
Viscosity is indicated as being one of the most significant transport properties because it’s related to the movement of molecular agitation. That means, the molecular transport of momentum is the corollary of the fluid forces of cohesion. Viscosity is required by chemical engineers involved in reactor applications, heat and mass transfer.
Accurate experimental measurements of viscosity, particularly at very high and/or very low temperature, are laborious and complex task. On the other hand, kinetic theory of gases made it possible to establish formulas for the calculation of gases viscosity of which have recently gained a wider acceptance, but very difficult to use because it comprises several parameters, which are often not easy to acquire.
After bibliographical synthesis, some empirical models were recapitulated among the most used in the calculation of nonpolar gases viscosity (Table 1).
At present, there is a considerable empirical models for estimating gases viscosity which have some limited success (Reid et al., 1977; Zhao, 1997; Adel Elsharkawy, 2004; Scalabrin et al., 2002; Maloka, 2005).
| Table 1: | Theoretical models for nonpolar gases viscosity calculation (Reid et al., 1977) |
These approaches are typically limited to narrow ranges of compounds across narrow ranges of temperature.
The advances in Artificial Neural Networks (ANN) have provided a tool that may be used to avoid the shortcomings involved in empirical methods. Indeed, the ANN can approach uniformly any sufficiently regular bounded function, with an arbitrary precision, in a limited domain of its variables space, with faster speed of information processing, learning ability, fault tolerance and multi-output ability.
Although there are a few reports (Homer et al., 1999; Lee et al., 1994; Chang et al., 1995; Adel Elsharkwy and Gharbi, 2001) of using ANNs in the prediction of physicochemical properties, these reports have generally been restricted to liquid rather than gases. The present research presents the findings of a programme of work devoted to the application of ANNs gases viscosity.
PROCEDURE
Data base: The pool of compounds in this study consisted of 52 nonpolar gases (Table 2), because of experimental data deficiency over wide range of temperature, the viscosity were deduced by the first correlation shown in Table 1 and corrected by some experimental viscosity data (Reid et al., 1977; Division Scientifique de L’AIR LIQUIDE, 1976; Gosse et al., 1991) every 20 K ranging from the boiling point for all the compounds to 1100 K.
| Table 2: | List of gases used to provide training and test data of ANN (Reid et al., 1977; Division Scientifique de L’AIR LIQUIDE, 1976; Gosse, Déroulède et al., 1991) |
This resulted in each gas being studied at approximately 50 different temperatures. The viscosity data obtained consist of 2652 vectors which were divided into two sets and used separately. One set of 1989 randomly selected vectors was used to train the ANN. The remaining 664 vectors, which contained approximately a third of the data base, were used as test set for checking the predictive performance of the ANN.
The inputs to the ANN consisted of absolute temperature and four physical properties (M, Tc, Tb, Pc). The choice of the nature and the number of ANN inputs has been done after bibliographical synthesis (Reid, Prausnitz and Sherwood, 1977; Gosse, Déroulède et al., 1991), particularly the model of Chapman-Enskog (Table 1).
Neural network design: Within the literature, ANNs which have been used for the estimation and the prediction of physicochemical properties have generally been multi-layered feedforward non-linear ANNs trained via the back-propagation rule to perform a function approximation. It has been shown that non-linear feed forward neural networks are capable of universal functional approximation and that a single hidden layer with sigmoid transfer function and one neuron in the output layer with linear transfer function is sufficient to uniformly approximate any continuous bounded function (Dreyfus et al., 2002).
ANNs are also sensitive to the number of neurons in their hidden layers. Too few neurons can lead to underfitting. Too many neurons can contribute to overfitting. In the first case, the training points and the fitting curve points are all inaccurately estimated but in the second case, the training points are accurately estimated, however the fitting curve tasks wild oscillations between these points and this leads to poor generalization.
The choice of the number of neurons in the hidden layers is, therefore, a delicate compromise between providing sufficient neurons to adequately determine an approximate functional relationship and avoiding the use of too many neurons which can lead to overfitting.
After the evaluation of a considerable number of differently structured neural networks, the best ANN selected in this investigation had a single hidden layer with 30 neurons and an output layer with one neuron. The hidden layer had a tansig transfer function and the output layer had a purelin transfer function (Fig. 1).
The output viscosity of the designed ANN is given by this expression:
| (1) |
where Xj represents the inputs variables (M, Tb, Tc, Pc and T) and Wij being the weights from input (j) to neuron (i) with bi and b31 representing bias of the neurons in hidden layer and bias of the neuron in output layer, respectively.
Normalization: As the values of the physical input properties to the ANN differed by several orders of magnitude, which may not reflect the relative importance of the properties in determining viscosity, all of the inputs matrix variables (Xi) were normalized by using
| Fig. 1: | Schematic operation of the ANN |
| (2) |
Where
RESULTS AND ANALYSIS
We have opted to use in this investigation the commercially available neural network toolbox supplied for the Matlab package due to its versatility. The ANN was trained using the Lavenberg-Marquadt back propagation algorithm.
The training algorithm used in the Matlab neural network toolbox was therefore trainlm which encompasses Lavenberg-Marquadt back propagation. To prevent over training, we have chosen to train the ANN until the minimum of the Mean of Squared Errors (MSE) performance function.
The weights and bias of the final trained ANN are summarized in the Table 3. Figure 2 shows a plot of target viscosity vs the correlated viscosity and Fig. 3 shows a plot of target viscosity vs. predicted viscosity by the ANN.
The statistical quality of the ANN for both training and test sets was then evaluated using following parameters: Squared correlation coefficient R,
| Fig. 2: | Training or correlated results |
| Table 3: | Weights and bias of the designed ANN |
| Fig. 3: | Predicted results |
| (3) |
where
| (4) |
and the root-mean-square error, RMSE, is
| (5) |
In order to compare the results of the ANN with the data base viscosity, we also evaluated the following parameters: Absolute Error (AE)
| (6) |
Average Absolute Error (AAE),
| (7) |
and Standard Deviation (STDEV),
| Table 4: | Statistical performance of the trained ANN for gas viscosity |
| Table 5: | Comparison between ANN and theoretical models according to the experimental values of low-pressure gas viscosity |
| 110 poise (P) = 1Pa.s., 2Values were obtained from Reid, Prausnitz and Sherwood (1977) | |
| (8) |
In these formulas, yi represents either the ith trained or test viscosity value and representing the corresponding target viscosity value, with n being the number of input vectors (1989 and 664 for the training and prediction sets, respectively). The results are summarized in Table 4.
Table 5 represents the results obtained by the designed ANN, these were compared with various theoretical models. The average absolute error for the estimated viscosity by the designed ANN is 1.38%, according to the experimental viscosity, for 13 different gases, at various temperatures. However, the AAE of other models, except the first one (1.29%), are all greater than the AAE value of the ANN.
CONCLUSIONS
The use of the designed ANN has been shown to accurately correlate and predict the nonpolar gases viscosity at moderate pressure (about 1 bar), over wide range of temperature (Temperature from boiling points of the chosen gases to T ≈ 2000K), for substantial number of variously gases (both organic and nonorganic compounds). The AAE in the predicted set of compounds was less than 1.39% for the gases in the Table 5. When the correlated (Fig. 2) and predicted values (Fig. 3 and Table 5) are considered jointly the AAE is approximately 0.93%, which is a serious competitor of commonly used method summarized in the Table 1. On the other hand, this method can be applied without depending on many complicated factors like σ, ω, ε/k and ΩV, that are used in the almost other methods.
Nomenclature
| b | Bias |
| Ci | Group contribution |
| k | Boltzmann’s constant |
| M | Molecular weight, g/mol |
| ni | Number of atomic groups of ith type |
| Pc | Critical pressure, bar |
| T | Temperature, K |
| Tc | Critical temperature, K |
| Tr | Reduced temperature, T/Tc |
| W | Weights |
Greek
| ε | Energy-potential parameter |
| η | Viscosity in Pa.s |
| σ | Molecular diameter, Å |
| ω | Acentric factor |
| Ωv | Collision integral for viscosity |