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 multioutput 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 ChapmanEnskog
(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 multilayered feedforward nonlinear ANNs trained via the backpropagation rule to perform a function approximation. It has been shown that nonlinear 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:
where X_{j} represents the inputs variables (M, T_{b}, T_{c},
P_{c} and T) and W_{ij} being the weights from input (j) to
neuron (i) with b_{i }and b_{31} 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 (X_{i}) were normalized by using

Fig. 1: 
Schematic operation of the ANN 
Where are the rescaled input values and n=1,…,5 labels the input patterns. However, the target viscosity values weren’t different by important orders
of magnitude so there wasn’t a need to be normalized.
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 LavenbergMarquadt back propagation algorithm.
The training algorithm used in the Matlab neural network toolbox was therefore trainlm which encompasses LavenbergMarquadt 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 
where
and the rootmeansquare error, RMSE, is
In order to compare the results of the ANN with the data base viscosity, we
also evaluated the following parameters: Absolute Error (AE)
Average Absolute Error (AAE),
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 lowpressure gas viscosity 

^{1}10 poise (P) = 1Pa.s., ^{2}Values were
obtained from Reid, Prausnitz and Sherwood (1977) 
In these formulas, y_{i} 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 Ω_{V}, that are used
in the almost other methods.
Nomenclature
b 
Bias 
C_{i} 
Group contribution 
k 
Boltzmann’s constant 
M 
Molecular weight, g/mol 
n_{i} 
Number of atomic groups of ith type 
P_{c} 
Critical pressure, bar 
T 
Temperature, K 
T_{c} 
Critical temperature, K 
T_{r} 
Reduced temperature, T/Tc 
W 
Weights 
Greek
ε 
Energypotential parameter 
η 
Viscosity in Pa.s 
σ 
Molecular diameter, Å 
ω 
Acentric factor 
Ω_{v} 
Collision integral for viscosity 