Abstract: Knowledge of the efficiency of sieve tray columns as most common distillation equipments is necessary for the interpretation of separation and purification processes performance. In this study a new method based on Artificial Neural Network (ANN) for estimation of sieve tray efficiency has been proposed. In this case to develop data base several experimental data were collected from literatures. The network inputs are liquid and vapor density, liquid and vapor viscosity, liquid and vapor diffusivity, surface tension, slope of the equilibrium curve, hole diameter, weir height, weir length, liquid and gas flux, ratio of hole area to active area of the tray while the output is point efficiency. In order to find the best efficiency estimator of sieve tray, different training schemes for the back-propagation learning algorithm, such as; Scaled Conjugate Gradient (SCG), Levenberg-Marquardt (LM), Gradient Descent with Momentum (GDM), variable learning rate BP (GDA) and Resilient BP (RP) methods were examined. Finally among those trained networks, the SCG algorithm with ten neurons in the hidden layer shows the best suitable algorithm with the minimum average absolute relative error 0.029817. Finally, the capability of ANN and two recently published empirical models were compared. This ANN model reduced the prediction error by 64.03 and 92.64% relative to Garcia and Fair and Chan and Fair models, respectively. This is further proof that the proposed procedure can build a useful and robust model.
INTRODUCTION
Efficient and economical performance of distillation towers is vital for separation processes in petroleum, petrochemical, chemical and related industries. The efficiency of a sieve tray is a crucial factor in the analysis of sieve tray columns because it relates theoretical stages to real plates (Garcia and Fair, 2000b). This factor depends on the geometry of the separation device, the physical properties of the chemical system and the operating conditions of the column (Olivier and Eldridge, 2002).
Several empirical and semi empirical methods have been proposed for the determination of point efficiency from commercial equipment and some laboratory data and serve the majority of design problems for the average hydrocarbon and chemical systems (Prado and Fair, 1990). Also few models have been developed for prediction of sieve tray efficiency based on a fundamental understanding of the transport between phases in the context of the turbulent and two-phase dispersion (Garcia and Fair, 2000a; Rahimi et al., 2006). These models are empirical correlations and their application in new system is unpredictable.
The ANN as a new method and good predictor has been recently applied in many fields especially in chemical engineering processes (Hajir et al., 2010). The ANN is a model that attempts to mimic simple biological learning processes and simulate specific functions of human nervous system (Moghadassi et al., 2009). This model creates a connection between input and output variables and keeps the underlying complexity of the process inside the system. The ability to learn the behavior of the data generated by a system is the neural network's versatility and privilege. Fast response, simplicity and capacity to learn are the advantages of ANN compared to classical methods.
MacMurry and Himmelblau (1995) and You and Yang (2003) showed using of ANN to model and control a packed distillation column, where the column represents a change in the sign of the gain under various operating conditions. They showed the performance of the column could be modeled well by applying ANN. Zamprogna et al. (2001) developed a virtual sensor based on a recurrent artificial neural network to estimate the composition in a middle-vessel batch distillation column. Olivier and Eldridge (2002) have presented a neural network model for prediction of the trayed distillation column mass-transfer performance. The absolute average errors averaged on the 10 training sets was equal to 7.79% (18.22% for Garcia and Fair model), where they believed that is a satisfying approximation for the tray efficiency. You and Yang (2003) presented an estimation of the relative tray efficiency of sieve distillation trays by applying artificial neural networks, but they used six quantities (i.e., liquid flow rate, outlet weir height, superficial gas velocity, free area rate of sieve tray, surface tension, Raoult’s law constant) to determine the relative tray efficiency.
In this study, an expert model based on ANN by using some experimental results is proposed to predict the sieve tray point efficiency. In the following sections after definition of tray efficiency and an introduction to ANN, also discuss the experimental data, the best ANN structure is chosen and trained. Finally results of the ANN model is evaluated against with the unseen data and then compared with the experimental data and two fundamental models (Garcia and Fair, 2000b). The novelty of this study is applying an expert model for direct estimation of sieve tray efficiency instead of using the theoretical methods and doing indirect and complex calculations.
DEFINITION OF TRAY EFFICIENCY
Determination of theoretical trays necessary for given separation is the first step of a distillation column design. Therefore the actual number of trays is determined by using column efficiency which is computed from the overall tray efficiency. The overall tray efficiency is calculated from the point efficiency. Consequently, the point efficiency is a key parameter for the design of sieve-tray distillation columns. The point efficiency deals with the approach to equilibrium between the vapor and liquid on specific point on the tray. The point efficiency of tray n is expressed as follows:
| (1) |
where, yn is the mole fraction in the gas at a given point on tray n, yn+1 is the mole fraction in vapor coming from the tray below and yn* is the vapor phase mole fraction which is in equilibrium with the liquid at the same point on tray n (Fig. 1).
| Fig. 1: | Typical trays in definition of point efficiency |
The point efficiency is thus defined as the ratio of the actual change in the vapor concentration of a component as it passes through a tray at a given point to the change that would occur if the vapor reaches equilibrium with the liquid.
Sieve tray efficiency dependents on three types of variables: (1) Parameters which describe the physical properties of the system (ρL, ρV, μL, μV, DL, DV, σ and m), (2) parameters that describe the geometry of the tray (DH, Hw, Lw and HA/AA) and (3) parameters which describe the flow regime (Lf and Gf).
ARTIFICIAL NEURAL NETWORK
In order to find relationship between the input and output data derived from experimental work, a more powerful method than the traditional methods are necessary. ANN is an efficient algorithm to approximate any function with finite number of discontinuities by learning the relationships between input and output vectors (Bozorgmehry et al., 2005; Hagan et al., 1996).
These algorithms can learn from the experiments and also are fault tolerant in the sense that they are able to handle noisy and incomplete data. The ANNs are able to deal with non-linear problems and once trained can perform prediction and generalization rapidly. They have been used to solve complex problems in control, optimization, pattern recognition and classification (Moghadassi et al., 2009). The ANNs are biological inspirations based on the various brain functionality characteristics. They are composed of many simple elements called neurons that are interconnected by links and act like axons to determine an empirical relationship between the inputs and outputs of a given system. Multiple layers arrangement of a typical interconnected neural network is shown in Fig. 2. It consists of an input layer, an output layer and one hidden layer with different roles. Each connecting line has an associated weight. An ANN is trained by adjusting these input weights (connection weights), so that the calculated outputs may be approximated by the desired values.
| Fig. 2: | Schematic of typical multi-layer neural network model |
The output from a given neuron is calculated by applying a transfer function to a weighted summation of its input to a give output, which can serve as input to other neurons, as follow (Moghadassi et al., 2009):
| (2) |
where, αjk is neuron j’s output from k’s layer βjk is the bias weight for neuron j in layer k. The neurons in the hidden layer perform two tasks: summing the weighted inputs connected to them and passing the result through a non linear activation function to the output or adjacent neurons of the corresponding hidden layer.
The model fitting parameters wijk are the connection weights. The nonlinear activation transfer functions Fk may have many different forms (Lang, 2000; Bulsari, 1995).
The training process requires a proper set of data i.e., input (Ii) and target output (ti). During training the weights and biases of the network are iteratively adjusted to minimize the network error function (Demuth and Beale, 2002). The typical error function that is used is the Average of Absolute Relative Errors (AARE) Eq. 3.
| (3) |
There are many different types of neural networks, differing by their network topology and/or learning algorithm.
In this study the Back Propagation (BP) learning algorithm, which is one of the most commonly used algorithms is applied to predict the sieve tray efficiency. BP is a multilayer feed-forward network with hidden layers between the input and output (Moghadassi et al., 2009). The simplest implementation of BP learning is the network weights and biases updates in the direction of the negative gradient that the performance function decreases most rapidly. An iteration of this algorithm can be written as follows (Moghadassi et al., 2009):
| Table 1: | Minimum and maximum range of data |
| (4) |
There are various BP algorithms such as Scaled Conjugate Gradient (SCG), Levenberg-Marquardt (LM), Gradient Descent with Momentum (GDM), variable learning rate BP(GDA) and Resilient BP(RP). The LM is the fastest training algorithm for networks of moderate size and it has the memory reduction feature to be used when the training set is large. One of the most important general purpose BP training algorithms is SCG (Lang, 2000; Moghadassi et al., 2009). Performance of neural networks is at least as good as classical statistical modeling and even better in most cases (Demuth and Beale, 2002). The neural networks built models are more reflective of the data structure and are significantly faster.
EXPERIMENTAL DATA
A large database for trays with single cross-flow of the liquid is collected from (Jones and Pyle, 1955; Rush and Striba, 1957; Sakata and Yanagi, 1979; Nutter and Perry, 1995; Kastanek and Standart, 1967; Chen et al., 2002; Biddulph et al., 1991; Anderson et al., 1976; Chen and Chuang, 1993; Kaeser and Pritchard, 2006) which contained 300 data of parameters which are needed for training and testing the Neural Network.
Most of data are at total reflux condition and some at finite reflux condition. Conversion of overall efficiency to point efficiency was done according to correlations given by Lewis and Plate (1936). For converting Murphree efficiency to point efficiency, a relationship of Bennett and Grimm (1991) was used. Table 1 shows the range of data that are used to predict the point efficiency of sieve tray. The network inputs are liquid and vapor density, liquid and vapor viscosity, liquid and vapor diffusivity, surface tension, slope of the equilibrium curve, hole diameter, weir height, weir length, liquid and gas flux, ratio of hole area to active area of the tray while the output is point efficiency.
MODEL DEVELOPMENT
Developing the neural network model to accurately predict, point efficiency requires its exposure to a large data set during the training phase.
| Fig. 3: | Determining the optimum number of hidden layer neurons |
| Table 2: | AARE comparison between different algorithms to train ANN |
The BP method with SCG, LM, RP and GDA learning algorithms has been used in feed forward, single hidden layer network (such as Fig. 2). Input layer neurons have no transfer functions. For programming MATLAB software was employed. Two thirds of data set was used in ANN training and the remaining data were employed to evaluate the best obtained network generalization capacity. The number of the hidden layer neurons is systematically varied to obtain a good estimate of the trained data. The selection criterion is the error function. The AAREs of various number of hidden layer neurons are shown in Fig. 3. According to the Fig. 3 the optimum number of hidden layer neurons is ten.
Similarly the AARE of various training algorithms were calculated and shown in Table 2 for the obtained ten hidden layer neurons. As Table 2 shows the LM and SCG algorithms have the minimum AARE. In this step of study, the trained ANN models are ready to be tested and evaluated against the new data. Table 3 show the various AARE of the network testing.
According to Table 3 the SCG algorithm is the most suitable algorithm with the minimum AARE in test (generalization part).
Figure 4 shows the scatter diagrams that compare the experimental data versus the computed neural network data over the full range of operating conditions for both training and testing networks. Figure 4 shows an excellent agreement between the experimental and the calculated data.
| Fig. 4: | Evaluation of ANN performance; a scatter plot of typically measured experimental data against the ANN model for (a) training and (b) testing |
| Table 3: | AARE comparison of different algorithms to test ANN |
RESULTS AND DISCUSSION
The results show that the ANN predicts point efficiency very close to the experimentally measurements. Figure 5-7 confirm the capability and accuracy of the ANN model to predict the point efficiency of sieve tray in comparison with two fundamental models which were developed by Chan and Fair (1984) and Garcia and Fair (2000b).
In these figures the point efficiency of five systems were plotted against Fs in order to have a meaningful comparison against experimental data and with values predicted by fundamental models. Fs is F-factor based on total (superficial) cross section of column and is defined as following:
| (5) |
| Fig. 5: | Comparison of ANN prediction with experimental data and fundamental
models for point efficiency of cyclohexane/n-heptane system; operating pressure
= 165 kPa, column diameter = 1.22 m |
| Fig. 6: | Comparison of ANN prediction with experimental data and fundamental
models for point efficiency of i-butane/n-butane system; operating pressure
= 2758 kPa, column diameter = 1.22 m |
where, UA is the superficial vapor velocity over the active area, with the volumetric flow determined below the tray bundle and ρv is the vapor density.
Table 4 shows the relative errors of typical ANN simulations, Garcia and Fair and Chan and Fair models and compares the amounts of average absolute relative error for these cases.
The results show artificial neural network has the best performance with minimum error that can be used for prediction of the sieve tray efficiency.
For better comparison of ANN and fundamental models based on average error criteria we defined advantage factor (AF) of ANN as follow.
| Fig. 7: | Comparison of ANN prediction with experimental data and fundamental
models for point efficiency of octanol/decanol system; operating pressure
= 1.3 kPa, column diameter = 1.22 m |
| Table 4: | Comparison between relative error of ANN and two fundamental models |
| *Relative error = (Exp.-cal.)/Exp. | |
| (6) |
The analysis of results showed that the AARE corresponding to ANN, Garcia and Chan model.
CONCLUSION
The ability of ANN to model and predict sieve tray efficiency has been investigated in this study. Predicted tray efficiencies have been compared with estimated tray efficiencies based on available methods. The results show a good agreement between experimental data and those predicted by ANN. An important feature of the model is that it doesn't require any theoretical knowledge or human experience during the training process. It has been clearly shown that of the ANN calculates the point efficiency based on the experimental data only, instead of using empirical models. In comparison with Garcia and Fair and Chan and Fair models, this ANN model reduced the prediction error for point efficiency by 64.03 and 92.64%, respectively. Therefore the proposed procedure can build a useful and robust model to predict point efficiency.
ACKNOWLEDGMENTS
Financial support of Universiti Teknologi Malaysia under grant No. 77537 is gratefully acknowledged.
NOMENCLATURE
| AA | = | Active (bubbling) area of the tray (m2) |
| AH | = | Hole area of the tray (m2) |
| Dh | = | Hole diameter (m) |
| DL | = | Liquid diffusivity (m2 sec-1) |
| DV | = | Vapor diffusivity (m2 sec-1) |
| EOG | = | Point efficiency |
| F | = | Transfer function |
| g | = | Gradient |
| Gf | = | Gas flux (m3 h-1) |
| Hw | = | Weir height (mm) |
| I | = | Input data |
| Lw | = | weir length (mm) |
| Lf | = | Liquid flux (m3 h-1) |
| l | = | Learning rate |
| m | = | Slope of the equilibrium curve dy/dx |
| N | = | No. of data |
| t | = | Target data |
| UA | = | Superficial vapor velocity over the active area (m sec-1) |
| w | = | Connection weights |
| x | = | Vector of weights |
| y | = | Local gas-phase concentration mole fraction |
| y* | = | Gas mole fraction in equilibrium with the liquid |
| w | = | Connection weights |
| α | = | Output of neuron |
| β | = | Bias weight |
| ρl | = | Liquid density (kg m-3) |
| ρv | = | Vapor density (kg m-3) |
| μl | = | Liquid viscosity (mpa.s) |
| μv | = | Vapor viscosity (mpa.s) |
| σ | = | Surface tension (mN m-1) |