1. Introduction
Traditional industrial processes have been affected by technologies for oxygen separation from air and partial oxidation of hydrocarbons, based on dense ceramic membranes with mixed oxygen ionic and electronic conductivity (MIECM). These provide a number of economic and environmental benefits with respect to industrial scale methods of pressure swing adsorption and cryogenic separation processes due to their possibility to integrate oxygen separation and partial oxidation into a single step and infinite theoretical perm-selectivity to oxygen, resulting from the unique transport mechanism via the crystal lattice (Kharton et al. 2002).
Oxygen transport through such membranes can occur only via hopping oxygen ions to neighboring vacant sites in the crystal lattice of mixed conductors, whereas the transport of any other species is excluded. Owing to this, gas-tight mixed-conductive membranes possess an infinite permselectivity.
The transport of oxygen in MIEC membranes is limited by surface exchange resistance, bulk diffusion limitations, or both (Gerdes et al. 2006). Chemical instability and mechanical constraints on membrane thickness limit MIECM's ability to increase the oxygen ionic flow by reducing the bulk resistance. Models of oxygen transport under mixed resistances have been proposed for monolithic disks, monolithic tubes, and disks and tubes coated with porous surface layers (Gerdes et al 2006, Van Hassel 2004). Experiments on bulk composite membranes (Yang et al. 2002, Gu et al. 2002) and disks coated with dissimilar surface layers (Ishihara et al. 2003) have been reported.
The membrane materials should satisfy numerous requirements, such as high ambipolar conductivity determined by the bulk ionic and electronic transport, fast oxygen surface exchange kinetics within a wide PO2 range, thermodynamic and dimensional stability under operation conditions, and suitable hermomechanical properties. Perovskite-related oxides represent one of most promising groups of the mixed-conducting membrane materials since their transport properties and stability in different atmospheres vary over a wide range (Kharton et al. 1999).
Gerdes et al. developed a model for radial oxygen flow through MIEC membranes (Gerdes et al. 2006). They assume that the bulk diffusion coefficient is constant and that the molar concentration of ions in the membrane is constant. However, the investigations made on MIECM have been shown the development of membrane materials and the experiments of performance evaluation with weakness of theoretical analysis and process modeling (Tan et al. 2002).
Tsai et al. developed a two-dimensional non-isothermal mathematical model to simulate the partial oxidation of methane to syngas in a mixed-conducting membrane reactor. Wang and Lin analyzed numerically the oxidative coupling of methane in an oxygen permeable membrane. Tan and Li (Tan et al. 2002) used a mathematical model to simulate La 0.6Sr0.4Co0.2Fe0.8O3-d hollow fiber membranes for air separation and investigated the module performance at various operating conditions theoretically. Nevertheless, there were not enough experimental data to verify the simulated results in these literatures. Also Yang et al. (2002) developed a mathematical model to simulate the process of air separation in the BSCF membrane permeator.
Many experiments have been carried out on the BSCFO membrane while few researchers focused on modeling the process. For instance, Wang et al (2004) focused on the investigation of the stability and oxygen permeation properties of the Ba0.5Sr0.5Co0.8Fe0.2O3-d (BSCFO) membrane. They chose one of the optimized compositions of BaxSr1-xCo0.8Fe0.2O3-d. BSCFO due to its high oxygen permeation and good structure and chemical stabilities.
Recently ANNs have been an emerging approach in modeling chemical engineering processes (Valles 2006, Zahedi et al. 2006, Biglin 2004, Sozen et al 2005, Sozen et al. 2004). Besides the high costs of the experimental work, it is often difficult if not impossible, to get a clear picture of the condition and possible problems of a process. Therefore a model based on some experimental results is proposed to predict the required data instead of doing more experiments. The major processes in the chemical engineering field are unfortunately nonlinear. ANN is a modeling approach that attempts to mimic simple biological learning processes and simulate specific functions of the human nervous system. This approach 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.
The main objective of this work is to model the performance of BSCF tubular membranes via both an ANN approach and a mechanistic approach. The predictions of both approaches will be compared to experimental measurements. The remainder of this paper is organized as follows. First a brief overview of the collected experimental measurements is outlined. These measurements represent the data sets against which both approaches will be compared to. Furthermore, the ANN approach uses part of the experimental data set in order to get trained and the other part for testing purposes. Section 3, provides an overview of ANN modeling and a description of the different algorithms used for developing these networks. Section 4 gives the actual development of the ANN models and comparisons between different training algorithms. The comparison is based on the criteria of minimum mean sum of squares of the errors. Section 5 provides a mechanistic model of the process under different sets of scenarios: co-current and cross flows with a purge model by H2 and under a vacuum mode. Section 6 gives the results of the predictions of both approaches as well as comparisons against experimental measurements. Appropriate conclusions are given in section 7.
2. Experimental Data
The experimental data sets used in this paper were provided from the work done by Wang et al. (2004) and Emrani et al. (2008). Overall, the data collected included the Oxygen partial pressure in the feed, temperature, Helium flow rate, air flow rate and Oxygen permeation flux. The data sets were collected that span a wide range of operating conditions of the process. Table (I) provides a summary of the different operating conditions and process outputs and gives the minimum and the maximum measurement for each variable.
3. Artificial neural networks
In order to find a relationship between the collected input and output patterns collected and given the inherent nonlinear nature of the process under consideration, a method more powerful than the traditional ones is necessary. ANNs represent in particular an efficient approach for approximating any function with a finite number of discontinuities by learning the relationships between input and output vectors (Bozorgmehry et al. 2005, Hagan 2006), 20]. 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. ANNs are able to deal with non-linear problems, and once trained can perform prediction and generalizations rapidly. They have been used to solve complex problems that are difficult to be solved if not impossible by conventional approaches, such as oil reservoir simulation and atmospheric pollution (Elkamel 1998, Elkamel et al. 2001). Artificial neural networks 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 Figure (1). It consists of an input layer, an output layer, and one hidden layer with different roles. Each connecting line has an associated weight. Artificial neural networks are trained by adjusting these input weights (connection weights), so that the calculated outputs may be approximated by the desired values. The output from a given neuron is calculated by applying a transfer function to a weighted summation of its input to give an output, which can serve as input to other neurons, i.e.
Where is neuron's output from 's layer βjk is the bias weight for neuronin layer. The model fitting parameters wijk are the connection weights. The nonlinear activation transfer functions may have many different forms. The classical ones are threshold, sigmoid, Gaussian and linear functions (Lang 2000). For more details of various activation functions see Bulsari (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 performance function (Demuth 2002). The typical performance function that is used for training feed forward neural networks is the network Mean Squares Errors (MSE):
There are many different types of neural networks, differing by their network topology and/or learning algorithm. In this paper the back propagation learning algorithm is employed. The simplest implementation of back propagation 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:
The process details flowchart to find the optimal model is shown in Figure (2). There are various back propagation algorithms Such as Scaled Conjugate Gradient (SCG), Levenberg-Marquardt (LM), Gradient Descent with Momentum (GDM), variable learning rate Back propagation (GDA) and Resilient back Propagation (RP). 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 back propagation training algorithms is SCG (Lang 2000, Demuth 2002).
The neural nets learn to recognize the patterns of the data sets during the training process. Neural nets teach themselves the patterns of the data set letting the analyst to perform more interesting flexible work in a changing environment. Although a neural network may take some time to learn a sudden drastic change, it is excellent in adapting to constantly changing information. However the programmed systems are constrained by the designed situation and they are not valid otherwise. Neural networks build informative models whereas the more conventional models fail to do so. Because of handling very complex interactions, the neural networks can easily model data, which are too difficult to model traditionally (inferential statistics or programming logic). Performance of neural networks is at least as good as classical statistical modeling, and even better in most cases (Bulsari 1995). The neural networks built models are more reflective of the data structure and are significantly faster.
Neural networks now operate well with modest computer hardware. Although neural networks can be computationally intensive, the routines have been optimized to the point that they can now run in reasonable time on personal computers. They do not require supercomputers as they did in the early days of neural network research.
4. Neural Network Model Development
Developing the neural network model to accurately predict oxygen permeation flux requires its exposure to a data set during the training phase.
The back propagation method with SCG, LM and RP learning algorithms has been used in a feed forward, single hidden layer network. Input layer neurons have no transfer functions. 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. A computer program has been developed under MATLAB. The number of hidden layer neurons is systematically varied to obtain a good estimate of the trained data.
The selection criterion is the net output MSE. The MSE of various hidden layer neurons are shown in Figure (3). As it can be seen the optimum number of hidden layer neurons is determined to be seven for minimum MSE.
The MSEs of various training algorithms are listed in Table (II). As can be seen the Levenberg-Marquardt (LM) and the Scaled Conjugate Gradient (SCG) algorithms have the minimum MSE.
Now the trained ANN models are ready to be tested and evaluated against a new data set. Table (III) lists the corresponding MSEs for the testing phase. Accordingly, the Scaled Conjugate Gradient (SCG) algorithm is the most suitable algorithm with the minimum MSE. Consequently, SCG provides the best minimum error average for both training and testing of the network.
5. Mathematical modeling
Several Models have been proposed in the past for BCSFO membranes based on basic principles and simplifying assumptions. We focus here on a more accurate one developed by Kim et al (1998). This model considers radial oxygen flow through the membranes, assumes that the bulk diffusion coefficient and the molar concentration of ions in the membrane are constant, and that the density of vacancies at the interfaces does not vary significantly with Po2. Based on these assumptions, the exchange fluxes i1 and i2 at the MIEC-gas interfaces at the inlet and outlet sides of the membrane are given by (Kim et al. 1998).
6. Results and discussions:
We present first the results that pertain to the ANN models. The ANN predictions are very close to experimental measurements. Figures (8-11) show crosses plots that compare the experimental data for the training and testing data sets, respectively.
As can seen, a tight cloud of points about the 45o line was obtained indicating an excellent agreement between the experimental and the calculated values. The predictions of the ANN models are in addition compared to those of the mathematical models. Figures (12-15) show such comparisons. According to the average MSEs of the ANN and the mathematical models (Table (IV)), both modeling approaches provide accurate predictions (Table (IV)). The ANN model is slightly superior.
Conclusion:
In this paper, the modeling of BSCF tubular membrane's performance was investigated via an empirical approach (ANN) as well as a mechanistic approach. The MSE analysis based results are used for verification of the different approaches. The results show a good agreement between experimental measurements and the predictions of both modeling approaches. An important feature of the ANN model is that no required theoretical knowledge or human experience is needed during the training process.
