Artificial Neural Approach for Modeling the Heat and Mass Transfer

Chemical Engineering Department, UniVersity of Technology, Baghdad, Iraq. The study of reactor design and modeling is conducted frequently both at the...
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CORRELATIONS Artificial Neural Approach for Modeling the Heat and Mass Transfer Characteristics in Three-Phase Fluidized Beds Farouq S. Mjalli*,† and A. Al-Mfargi‡ Chemical Engineering Department, UniVersity of Malaya, Kuala Lumpur, Malaysia, and Chemical Engineering Department, UniVersity of Technology, Baghdad, Iraq

The study of reactor design and modeling is conducted frequently both at the initial stage of equipment design as well as during further stages of equipment operation. Fluidized bed three-phase reactors have very complex behavior which relies to a high extent on the mass and heat transfer characteristics of the reaction constituents. Numerous previous experimental and theoretical based studies for modeling heat and mass transfer coefficients have the common shortcoming of low prediction efficiency compared to experimental data. In this work, an artificial neural network approach is used to capture the reactor characteristics in terms of heat and mass transfer based on published experimental data. The newly developed heat and mass transfer coefficients models proved to be of high prediction quality compared to experimental data and previous correlations. The new correlations will be used in a further study for the hybrid steady state and dynamic modeling of fluidized bed catalytic reactors. 1. Introduction System designers are always confronted with the task of modeling the complex behavior of fluidized catalytic bed continuous reactors (FCR). Such a task is encountered in many situations. Examples of such situations involve the early stage of equipment design, the prediction of reactor conversion and yield of reacting species, performing optimization and process parameters sensitivity analysis, and using the process model as part of a model-based control system. These models rely solely on the heat and mass characteristics of the reacting medium and the nature of the process involved. In these situations the model developer, unless equipped with specific experimentally based empirical correlations for the reactor system under consideration (which is very seldom), is required to use the correlations for mass and heat transfer coefficients published in the open literature. In a typical FCR, three-phase fluidized beds are generated in the form of gas-liquid reactions in which a catalyst is required to enhance the reaction conversion. This process is very common in the industry; examples are polymerization reactions, biochemical processes, natural gas processing, coal liquefaction,1 etc. The reactors in these processes entail very complex flow behavior.2 Moreover, the presence of three phases in the reaction medium complicates the transport phenomena involved which dictates the need for a thorough understanding of their heat and mass transfer characteristics. The currently available correlations for mass transfer and heat transfer coefficient differ by up to 7 and 5 orders of magnitude, respectively.3 The main aim of this work is to explore an alternative method for estimating heat and mass transfer coefficients used in the FCR reactor model equations. The alternative method has the following properties: (i) it attains high degree of prediction of * To whom correspondence should be addressed. Email: farouqsm@ yahoo.com. † University of Malaya. ‡ University of Technology, Baghdad.

the two parameters as compared to experimentally available data, (ii) it applicable to a wide range of reactor geometries and operating conditions, (iii) it does not include any assumption about the hydrodynamic and flow patterns involved in the process, (iv) it does not need any additional parameter other than the operating conditions, and (v) it is usable as a part of numerical solution for complex design and simulation studies. Consequently, artificial neural networks (ANN) were selected for the estimation of heat and mass transfer coefficients instead of a correlation model. Numerous function approximation techniques are available; examples include traditional regression methods, fuzzy logic, and neural networks-based methods. The selection of best function approximator technique is somehow controversial, as any of these techniques or a combination of them can be used (with proper design methodology) almost equivalently for the purpose of system identification. However, each of these function approximation methods has favorable features that make it more attractive (than the rest) for application in certain system identification problems. The use of ANNs as function approximators has many favorable features. However, there are some disadvantages that need to be considered and treated appropriately. ANNs require large training data sets; they are regarded as black-box models (i.e., the individual relations between the input variables and the output variables are not developed by engineering judgment) and they (as well as other regression methods) may attain overfitting/underfitting. The aforementioned disadvantages necessitate special attention. The problem of large training data can be solved by the careful selection of input/output process data to truly represent the intrinsic relationships among process variables which will reduce the amount of training data needed. The fact that ANNs are black-box models is comprehensible specially for situations when mechanistic modeling techniques fail to describe the variables inter-relationships due to the high complexity of the physicochemical phenomena involved. In this case the use of black-box modeling methods may be justifiable as a feasible

10.1021/ie0715714 CCC: $40.75  2008 American Chemical Society Published on Web 05/29/2008

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alternative. The problem of overfitting usually results when the degrees of freedom in parameter selection exceed the information content of the data, this leads to arbitrariness in the final (fitted) model parameters which reduces or destroys the ability of the model to generalize beyond the fitting data. The likelihood of overfitting depends not only on the number of parameters and data but also on the conformability of the model structure with the data shape and the magnitude of model error compared to the expected level of noise or error in the data. Due to practical considerations, the designer is usually given a fixed amount of training data. Consequently, some measures should be taken into consideration to avoid overfitting in order to get good generalization. Techniques for doing this involve reducing number of model parameters, careful selection of the model design, early stopping, and Bayesian estimation. In this work, the fitted models were optimized in terms of number of parameters, and the early stopping technique was used for avoiding overfitting. In the area of mass transfer prediction, many ANN-based studies have been conducted. Garca-Ochoa and Castro,4 used a hybrid ANN approach to model oxygen mass transfer coefficient in stirred tank reactors. In their study, they constructed a ANN with inputs including several operational conditions such as stirrer rotational speed, slip velocity, reactor geometry, and physical properties. Lam et al.5 used a neural-based technique for the estimation of mass transfer coefficient for cyclohexane oxidation process for nylon manufacture. Two neural networks approaches were devised and tested. The first is a single network approach whereas the second contains multiple neural networks in a hierarchy, termed a “mixture of experts” approach. These models are assessed using cross validation, which leverages the available data for both the construction and validation of the predictive models. Fonseca et al.6 investigated the use of neural networks for predicting mass transfer coefficients from supercritical carbon dioxide/ethanol/water system. They used a technique which combines experimental mass transfer coefficient with that obtained from correlation available in the literature to generate new semiempirical data for the network training. Prediction of heat transfer coefficient from experimental data is another area that has been investigated by many researchers. Sablani7 calculated the fluid-to-particle heat transfer coefficient in fluid-particle systems of an agitation processing of cans containing liquid/particle mixtures where fluid temperature is time-dependent. Two ANN configurations were evaluated: (i) the input parameters (fluid and particle temperatures) and output parameters (Biot number, Bi) were taken initially on a linear scale, and (ii) input/output parameters were transformed using logarithmic and arctangent scales. The second configuration yielded an optimal ANN model with eight neurons in each of the three hidden layers. Islamoglu and Kurt8 used artificial neural networks for heat transfer analysis in corrugated channels. The experimentally generated data was fed to an ANN and a backpropagation algorithm was used for the training. In the field of food processing, Sreekanth et al.9 used an ANN for the evaluation of surface heat transfer coefficient at the liquid-solid interface using the temperature profile information within the solid. Although the concept is quite generic, the specific cases considered have a particular relevance to engineering applications. Larachi et al.10 used a combined neural networks and dimensional analysis technique to predict the macroscopic mass and heat transfer characteristics relevant to gas-liquid cocurrent, down-flow, and up-flow packed-bed reactor. From the preceding short review of the literature, it is evident that ANNs better serve for thermal and mass transfer analysis

Figure 1. Schematic of the catalytic fluidized bed reactor. Table 1. Published mass transfer coefficient correlations used for comparison reference

correlation

Nguyen-Tien et al.17 Patwari et al.18 Shumpe et al.19 Kim and Kim20

KLa ) 0.39[1 - εs/(1 - εg) / 0.58 ]U0.67 g -1.3 0.5 KLa ) 1.68×10-2U0.36 De g µt 0.42 -0.34 0.71 KLa/Dr0.5 ) 2988U0.44 Ut g Ul µ 1.11 0.45 0.71 KLa ) 0.73U0.87 U d g l p [1 + 0.036( Vf / Vs ) 2.09 -3 1.348×10 ( Vf / Vs ) ] bubble disintegrating regime KLa ) 2.36×105U0.686 Ul0.469d0.788 σl-1.532µl-0.34 g p bubble coalescing regime KLa ) 1.10×106U0.940 Ul0.381d0.790 σl-2.273µl-0.67 g p a5 a2 a 3 a 4 KLa ) a1Ug Ul dp (1 - εs) (1 + a6√z + a7z)

Lee et al.37

Zheng et al.21

in engineering applications. Nonetheless, ANNs modeling strategies have not been used for heat and mass transfer coefficient modeling in FCRs yet. Consequently, this work focuses on this area as compared to conventional correlation based methods. The success of this technique provides a more efficient methodology for predicting heat and mass characteristics of FCR reactors and improves the prediction quality of their mathematical models. 3. Process Description The widest use of this reactor is found in the polymerization industry. Other applications include oxidation of naphthalene and ethylene and the production of alkyl chlorides. This is a three-phase system where solid support-catalyst-polymer particles react in a bed through which a continuous flow of a gaseous stream composed of monomers and other species is passed. Fluidization of the particles is achieved by controlling the rate of the gas monomer stream. The gas leaving the top of the reactor is used as the energy carrier to convey the heat of polymerization out of the reactor. The gas is circulated through gas-liquid, shell-and-tube heat exchanger where it is cooled, to be later recompressed and recycled to the reaction vessel. As shown in Figure 1, the small catalyst particles form a complex with the monomers reactants and produce larger polymer particles that continue growing as they travel upward the reactor.

4544 Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 Table 2. Operating Conditions for the Heat Transfer Experimental Studies particle type glass beads FCC FCC limestone sand sand quartz sand

Fp dp (m) × 106 (kg/m3) 60 97 139 115 264 150 264 248

2370 1670 1670 2100 2500 2650 2620 2600

Dt (m)

Ug (m/s)

reference

0.1397 0.1397 0.1397 0.1397 0.0375 0.2 0.375 0.102

0.013-0.33 0.013-0.33 0.013-0.33 0.022-0.38 0.5-1.25 3.6-10.15 4.7-6.5 4.2-6.4

Bearg et al.31 Bearg et al.31 Bearg et al.31 Bearg et al.31 Gupta and Nag32 Fox et al.33 Li et al.34 Reddy and Nag35

Table 3. Operating Conditions for the Mass Transfer Experimental Studies particle type dp (m) × 103 Fp (kg/m3) εs (%) glass beads glass beads glass beads

0.530 0.670 0.755

2292.4 2361.8 2360.0

Ug (m/s)

Ul (m/s)

2-10 0.7-10.67 0.44-4.80 1-7.7 0.7-10.67 0.44-4.80 2-16.5 0.7-10.67 0.44-4.80

The reactant gas enters the bottom of the bed and flows up the reactor in the form of bubbles. As these bubbles rise, mass transfer of the reactant gases takes place between the bubbles and the clouds and between the clouds and the emulsion without chemical reaction. The mass transfer between emulsion and solid with chemical reaction occurs on the surface of the catalyst particles. The catalyst is fed at a point about 25% of the overall reactor height measured from the gas distribution level. An inert gas is fed slightly above the gas distribution plate at the bottom of the reactor. This inert gas facilitates the carryover of catalyst particles. The catalyst particles will grow in size as they climb up the reactor due to polymerization; however, their terminal velocity will keep these particles flowing within the fluidized bed away from the area exposed to drag. Moreover, high recycle ratios, typically of the order of 50, are usually used in order to make sure that there will be no drag of the smallest particles with the exiting stream. The product is discharged using a double hatch air-lock system.11 4. Heat and Mass Modeling Studies 4.1. Heat Transfer Studies. The attempts for describing heat transfer behavior of fluidized beds basically distinguish one of two approaches, the particle-based approach (PBA) which studies the heat transfer process at the particle level and the cluster-based approach (CBA) which studies unsteady state heat transfer between the pockets of particles as they are brought into contact with the heat transfer surface. Experimental investigations on the hydrodynamics of fluidized beds showed formation of clusters from individual solid particles is a major characteristic of fluidized beds. Hence, in this analysis, only CBA-based models will be considered. The pioneering work in this context is that of Mickley and Fairbanks12 and the modifications of Baskakov,13 who introduced additional thermal resistance of the gas film on the heat transfer surface. In these models heat transfer by the particle convection is modeled as the process of unstationary conduction of the particle clusters which are in contact with heat exchange surface for a definite period of time. Nearby the heat exchange surface there is a gas film which transfers heat by gas conduction. Particle convection heat transfer coefficient is defined as: hpc ) (1/hw + 1/hs)-1

gas boundary layer on the heat exchange surface is due to a thin gas layer between the packet and the wall and is expressed as hw ) mkg/dp where m is a constant. It has been found that the thickness of gas film has values of 10-40% of particle diameter. The component of particle convection hs, which comes from suspension or clusters, is modeled depending on suspension hydrodynamics on the heat exchange surface. Lints and Glicksman14 described the convective thermal exchange at the wall of a circulating fluidized bed as two processes in series leading to an overall heat transfer coefficient h defined as h ) fhhc + (1 - fh)hg

(2)

The first of these processes is an exchange characterized by the coefficient hc involving particle clusters covering the fraction fh of the wall surface. The second has coefficient hg and involves an emulsion around the clusters that is largely devoid of particles. Lints and Glicksman14 then modeled the heat transfer to clusters as two parallel processes, namely a conduction through the thin gas film of thickness δ followed by a convective exchange to the clusters with coefficient hH 1/hc ) δ/k + 1/hH

(3)

14

To capture hH, Lints and Glicksman borrowed from the model of Mickley and Fairbanks12 for bubbling beds. In that model, the authors treated the emulsion as a semi-infinite homogeneous medium and adopted the classical heat flux expression for transient conduction into a semi-infinite slab. Then, after invoking effective thermal properties for the emulsion phase and time averaging, they derived the form of the wall heat transfer coefficient as hH ≈



keFscpνc τc

(4)

where ke is the effective conductivity of the emulsion phase, τc is its average contact time with the wall, νc is its solid volume fraction, and Fs and cp are respectively the material density and specific heat of the solids. Elizabeth and Louge15 simplified the previous model by assuming that the effective cluster conductivity is governed by the gas conductivity k, neglecting the second term in eq 3, and that the convective transfer to the clusters dominates the conduction through the thin gas layer. They used and verified experimentally an equation of the form: Nud )

hds ≈ fh k



τpνc τc

(5)

(1)

Heat transfer coefficient of the gas conduction through the

Figure 2. Structure of heat and mass transfer coefficients prediction ANNs.

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Figure 3. Training propagation of the heat transfer coefficient ANN (a) and the mass transfer coefficient ANN (b). Table 4. Parameters Values of the Trained Heat and Mass Transfer Coefficient ANNs

W1

[

heat transfer parameters - 1 . 493 0 . 744 0 . 027 -9 . 428 3 . 782 6 . 941 -0 . 493 -0 . 375 -3 . 549 0 . 403 -0 . 102 -25 . 824

-0 . 383 -1 . 303 -0 . 584 32 . 324

]

[

mass transfer parameters 2 . 173 -0 . 087 -1 . 481 0 . 937 -2 . 315 1 . 186 1 . 042 1 . 215 -3 . 180 1 . 968

0 . 369 -0 . 353 -0 . 162 -1 . 244 -0 . 837 0 . 620 -1 . 050 0 . 558 -1 . 054 1 . 470

-2 . 287 2 . 948 -0 . 083 -4 . 776 -0 . 721

W2

[-3 . 488 2 . 653 12 . 414 2 . 238 ]

[0 . 831 1 . 314 -1 . 527 0 . 845 0 . 856 ]

B1

[0 . 047 0 . 614 -4 . 519 14 . 174 ]-1

[-2 . 785 1 . 203 -0 . 217 -2 . 255 0 . 248 ]-1

B2

9.367

-0.4126

where τp ) Fs cp ds2/k is a characteristic time for the heating of a particle of diameter ds. Recently, Karimipour et al.16 proposed to split the process of heat transfer through clusters into two periods with respect to time. The initial period is the time needed for the heat effect to completely penetrate (from the wall side to the bulk side) into the cluster. This is the period when the heat flux diffuses into but has not reached the other side of the cluster. The second time period is when the heat flux has crossed the cluster completely and lasts until the end of the contact time of the cluster with the wall. Using the concept of penetration depth of temperature to model the heat transfer coefficient in the first period, they solved a heat balance equation and derived the following relation for the cluster heat transfer coefficient which is similar in form to that of Mickley and Fairbanks:12 hcc )

( ) 1.5t kcFccc

hcc )

( )(

){

[

m(t - t0) kc 1 1 1 ⁄ + + 1 - exp dc 2 2 Bi 1 + Bi/3

(

]

]})

(7)

where Bi is the Biot number ) (hgdc/kc); m ) hg /(Fc cpdc); t is

-0.5

(6)

For the second period where the wall-heating effect has reached the other side of the cluster, the penetration depth is equal to the width of the cluster. Their analysis revealed the following equation:

Figure 4. Experimental versus ANN prediction of the heat transfer coefficient.

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time; t0 is the enetration time of the applied heat flux to the other side of the cluster(s); hg is the heat transfer coefficient of the gas in turbulent flow; and kc, dc, and Fc refer to the particles cluster properties. 4.2. Mass Transfer Studies. Mass transfer characteristics in three-phase fluidized-bed reactors have been investigated by many researchers during the past five decades. Further similar studies are still required to reach a better understanding of the three-phase fluidized bed reactors. The main target here is to have efficient predicting models that can be used to optimize the reactor mechanistic model with respect to geometry and operating conditions.

Figure 5. Experimental versus ANN prediction of the mass transfer coefficient.

In the current study of the mass transfer in the three phases for fluidized beds, six literature reported models are considered for comparison purposes. These references as well as the corresponding expression for the mass transfer correlation are listed in Table 1. The prediction performance of the newly developed neural network-based model for the mass transfer coefficient will be compared to these correlations. The correlations considered in this work are all empirical and are functions of operational and geometrical properties of the system. Nguyen-Tien et al.17 developed a correlation that depends on liquid velocity and axial position; however, they ignored the effects of liquid velocity, diameter of particle, height of column, and diffusivity factors. Patwari et al.18 found that mass transfer coefficient is proportional to the terminal velocity of particles to a power of 0.36, and diffusivity of the gases, but nevertheless ignored the effect of liquid velocity, height of the column, emissivity, and diameter of the particles. The effect of wettability of particles on sectional volumetric mass transfer coefficient (KLa) is considerable, especially in three-phase beds of viscous liquid medium. It has been reported by Schumpe et al.19 that poor wettability of the suspended particles such as polypropylene results in very low mass transfer coefficient values in CMC solutions due to the reduction of available gas-liquid mass transfer area by the particles at the gas-liquid interface. However, they ignored the effects of fluidized bed’s height, porosity, and diameter of particle. Kim and Kim20 proved that the mass transfer coefficient is proportional to the particles’ size to power 0.71, liquid velocity to power 0.45, and superficial gas velocity to power 0.87. This

Figure 6. Heat transfer coefficient predictions as a function of superficial gas velocity in the bubbling regime.

Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 4547

Figure 7. Heat transfer coefficient predictions as a function of superficial gas velocity in the turbulent and fast fluidization regime.

relationship is widely used because it involves most important variables excluding the effect of column’s height, emissivity, and diffusivity. The mass transfer coefficient correlation of Zheng et al.21 depends on the height of fluidized bed, superficial gas velocity, liquid velocity, porosity, and diameter of particle. Their correlation considers the effect of the different flow regimes and identifies the boundaries of these flow regimes. 5. Neural Network Modeling Methodology The formulation of ANNs theory goes back to the early forties of the last century. They evolved steadily and were adopted in many areas of science. Basically, the ANNs are numerical structures inspired by the learning process in the human brain. They are constructed and used as alternative mathematical tools to solve a diversity of problems in the fields of system identification, forecasting, pattern recognition, classification, financial systems, and many others.22–26 ANN differs from traditional computing methods in many ways. The processing style of ANN is parallel, and there is not a strict rule or algorithm to follow, whereas traditional methods are sequential and logical. Moreover, traditional methods can learn by rules while ANN learns by examples and therefore artificial neural networks are known as universal function approximators.27 This makes ANNs the best modeling techniques for situations where experimental data are available. This is exactly the case for the situation under investigation where there are abundant published experimental data for the measurement of mass and heat transfer data of FCR.

The interest in ANN as a mathematical modeling tool resulted in the consolidation of its theoretical background and the development of its underlying learning and optimization algorithms. Modeling and simulation of chemical processes is one of the research areas of interest that made use of ANN modeling techniques. The implementation of mechanistic models that rely on fundamental material and energy balances as well as empirical correlations involves a great deal of mathematical difficulties and in many instances lacks accuracy. Neuron-based modeling can be used confidently as a substitute for such situations. This is due to the favorable features entailed in their use. Among these features are simplicity, fault and noise tolerance, plasticity property28 (can retain its prediction efficiency even after the removal or damage of some of its neurons), black box modeling methodology, and capability to adapt to process changes. Details of ANNs and their applications to process modeling can be found in Fine29 and Farrell and Polycarpou.30 The trained ANNs represent the general relations linking ANN inputs (physical properties and operating process parameters) to its output (the heat and mass transfer coefficients). These relations can be written as follows: h ) f2(Wh2 f1(Wh1x + bh1)) + bh2

(8)

KLa ) f2(Wk2 f1(Wk1x + bk1)) + bk2

(9)

where x represents the network input vector, Wk1,Wk2 and Wh1,Wh2 are the first and second weight matrices for mass and heat transfer, respectively, bk1,bk2 and bh1,bh2 are the respective first and second bias matrices, and f1 and f2 are the activation

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functions for the first and the hidden layers, respectively. The two ANN-based correlations can be used to predict the heat and mass transfer coefficients within the operating conditions bounds used for training the ANNs. 6. Results and Discussion Similar analysis was adopted for the neural-based modeling of heat and mass transfer coefficients of a FCR. Experimental published data of the two coefficients were used as the basis database needed for training the ANNs. The performance of the new ANN-based prediction is tested against published correlations for different cases. In the following two sections,

the details of this analysis and the results obtained are highlighted and discussed. The operation of the FCR is characterized by three different flow regimes, namely the bubble, turbulent, and fast flow regimes. Due to different hydrodynamic characteristics involved in each one of these regimes, heat transfer differs considerably among these regimes. For the bubbling regime, the data of Bearg et al.31 was used and for the turbulent regime the data of Gupta and Nag32 was used while the data of (Fox et al.,33 Li et al.,34 and Reddy and Nag35) were used for the fast flow regime. Table 2 shows the operational ranges of experimental heat transfer studies used.

Figure 8. Mass transfer coefficient predictions of the different correlations as compared to ANN and experimental values.

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Figure 9. Mass transfer coefficient predictions of the different correlations as compared to ANN and experimental values.

The data sets were combined in one set consisting of an input vector of four components namely; particle diameter dp, particle density Fp, hydraulic diameter of the bed Dt, and the superficial gas velocity Ug and an output vector containing the experimental heat transfer coefficient. The total size of the data set is 54 data records. In the case of mass transfer coefficient data, the experimental data of Zheng et al.21 was used. This investigation was done using glass beads and involved the effect of particle diameter dp, solid phase fractional holdup εs, liquid velocity Ul, gas velocity Ug, and column height h. Table 3 shows the operational ranges of experimental mass transfer studies used. The total size of the data set is 73 data record. A feed-forward type neural network (FFNN) was chosen for the modeling problem under consideration due to its simplicity, high prediction capability, and stability under experimental uncertainty. The FFNN modeling was performed using the following procedure: data preprocessing, network structure selection, network training, network validation, and testing. An early stopping technique was used as the termination methodology for network training. The inputs and the targets were normalized so that they have zero mean and unity standard deviation. This would make the neural network training more efficient. Network training was accomplished by changing its weights and biases to achieve certain performance criteria. This is a preliminary step for treating raw experimental data before starting any modeling attempt.

The first and most important step in the neural-based modeling approach is the network design step. Here, a certain structure for the network is chosen and constructed. The objective here is to have a fairly sized network that is capable of approximating the process behavior efficiently and consistently. For the problem under investigation, a small-sized network is preferred in order to facilitate less computation effort during the prediction process. Figure 2 shows the general layout of the FFNN for both heat and mass transfer coefficients cases. The input layer contains the input data vectors. Layer 1 receives the four inputs in the case of heat transfer coefficient and five inputs for the mass transfer coefficient; multiply those by the weight matrix W1 then add the result to a bias vector b1 and this is then fed to the activation function f1. In layer 2 the same steps are performed on the output of the activation function, but in this case using W2 and b2 as parameters and f2 as an activation function that sums the outputs in one vector resembling the predicted vector for the heat and mass transfer coefficients neural networks. A series of exploratory experiments were performed to select the best ANN structure. A three-layer network was finally selected with four and five sigmoidal neurons in the hidden layer for the heat and mass transfer coefficients networks, respectively, and one linear neuron in the output layer. This is the simplest possible network that can be used in this case. In order to guarantee the generalization of the trained neural network and confirm the acceptance of the network performance over a wide

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range of process operating conditions, the network needs to be trained with data which covers the entire range of possible network inputs. For the process under consideration, the input target range spans all the experimental data. The original historical data set of inputs was subdivided into three subsets for network training, validation, and testing in a ratio of 2:1:1, respectively. This is accomplished by using an optimization algorithm that searches for network parameters which minimizes the performance index. The performance index is based on the squared difference between actual plant output ai and network prediction ti for n sample points. This is expressed as performance index )

1 n

[

n

∑ (t - a )

]

2

i

i)1

i

(10)

Many optimization techniques can be utilized to accomplish the task of network training. The Levenberg-Marquardt backpropagation algorithm (LMBP)36 gives good performance with minimum optimization steps (epochs). It was adopted here for the training of the heat transfer coefficient neural network models. The evolution of the training, validation, and testing is shown in Figure 3. The number of training epochs (training steps) was 1500 and 500 for heat and mass transfer, respectively. The heat transfer coefficient ANN achieved a sum-squared errors (SSE) for the training, validation, and testing sets of 5.391, 1.346, and 2.688, respectively. The networks’ training was stopped before achieving a performance criterion of 1 × 10-3. This is because of the early stopping methodology where the training stops due to slow improvement of validation error. The number of effective heat transfer coefficient network parameters is 23.3 which indicates a fairly sized ANN. The search gradients indicate that the training goal was not achieved completely. This is acceptable in this case because we do not expect high degree of prediction due to the complexity of the correlated parameters. A certain degree of model deviation must be expected in this case. The same training behavior was followed by the mass transfer coefficient ANN with SSE values of 0.264, 0.219, and 0.044. The size of the trained mass transfer coefficient network is more than double that of the heat transfer coefficient as indicated by an effective number of network parameters of 51.9. The final trained network parameters for the two networks are given in Table 4. The trained networks predictions were compared with the experimental data as shown in Figures 4 and 5. Both networks’ predictions are very good with a squared regression coefficient of 0.98 for the heat transfer coefficient model and 0.96 for the mass transfer coefficient model. To investigate the performance of the new neural-based heat and mass transfer coefficients, they were simulated and compared to published correlations. Two heat transfer coefficients models were used in this comparison, namely that of Mickley and Fairbanks12 and the Karimipour et al.16 model. The mass transfer coefficient experimental data of Bearge et al.31 in the bubbling regime using glass beads of different sizes was compared to the prediction of these models and the ANN model as shown in Figure 6. The ANN predictions are very good within the whole spectrum of fluidization. On the other hand, the other two models show overfitting behavior near the minimum fluidization region. The Karimipour model predictions are better than those of Mickley and Fairbanks. These two models have poor prediction capability at high particle sizes. Figure 7a shows the prediction of the three models in the fast fluidization regime as compared to the experimental data of Gupta and Nag.32 Both previous models overfit the experimental data, while the ANN predictions have excellent match. The predictions for the fast fluidization regime are shown in Figure 7b-d. The three figures

indicate the improved prediction of the Karimipour correlation; however, the ANN shows the best performance and attains high degree of accuracy. In general, this analysis shows that the new ANN heat transfer coefficient model is capable of predicting efficiently the FCR heat transfer coefficient behavior under all fluidization regimes. For the mass transfer coefficient, six additional published correlations were used in the comparative study in addition to the newly developed ANN model. These are as follows: Zhen et al.,21 Nguyen et al.,17 Patwari et al.,18 Kim and Kim,20 Shumpe et al.,19 and Lee et al.37 Plots of sectional volumetric mass transfer coefficient KLa against superficial gas velocity at two different superficial liquid velocity values are given in Figure 8, a and b. All tested correlations have the same direction as that of the experimental. Zheng et al.21 correlation is relatively more accurate than the rest. However, ANN predictions are much more aligned with practical findings. The variations of KLa against superficial liquid velocity at two different values for the superficial gas velocity are given in Figure 8, c and d. Some of the tested correlations like those of Nguyen et al.17 and Patwari et al.18 are insensitive to the variation and did not show any change. The other correlations were monotonically increasing with superficial liquid velocity. On the other hand, Schumpe et al.19 correlation attained a high degree of overfit. ANN predictions were the best among all the correlations. The effect of fractional solid holdup is depicted in Figure 8, e and f. All correlations responded correctly to the variation of superficial gas velocity, with varying degree of accuracy, though. Again, Schumpe et al.19 correlation attained the highest degree of overfit. The neural predictions accurately tracked the change in experimental mass transfer coefficient data. Only Nguyen et al.17 correlation showed appreciable decrease in the mass transfer coefficient when the fractional solid holdup was increased. This is shown in Figure 9a,b for two values of superficial gas velocity. Nevertheless, its prediction accuracy was not as good as that of the ANN-based model. The influence of particle size on sectional mass transfer coefficient is plotted in Figure 9c,d for increasing values of superficial gas velocity. One can find that the effect of particle size on KLa is significant at high gas velocities, but at low gas velocities, particle size has no remarkable effect on KLa. The Schumpe et al. correlation attained highest degree of overfitting while the Zheng et al.21 correlation had the highest degree of underfitting. As noticed with previous tests, the ANN predictions were of good tracking capability and with very small residuals. 7. Conclusion Modeling the heat and mass transfer characteristics of catalytic fluidized bed reactors is a difficult task to accomplish due to the complexity of these relations and high degree of variability involved. All previous attempts at describing the heat and mass transfer behavior in FCR reactors were of limited applicability, because of the fact that these correlations were specific to certain operating conditions and reactor geometry. The many assumptions involved in their derivation hinder the generalization of these correlations. An ANN modeling approach was implemented in this study to solve this problem and to discover the interdependency between process operating conditions and variations of the heat and mass transfer coefficients. Published experimental data were used to predict the plant behavior and compare the resulting neural based models with published correlations at different fluidization regimes.

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The ANN modeling approach used in this study, namely the multi-input feed-forward network topology, gave comparable predictions of the practical values of the two coefficients. The ANN modeling technique has many favorable features such as efficiency, generalization, and simplicity, which makes it an attractive choice for modeling complex systems, such as the heat and mass transfer characteristics of the FCR reactor. These models can be embedded in a hybrid modeling approach for better explanation of the physicochemical characteristics of the FCR reactors and to optimize the reactor mechanistic model with respect to geometry and operating conditions. This will be considered in a follow-up investigation. Nomenclature a ) gas-liquid specific interfacial area (m-1) cp ) specific heat of liquid phase (J/(kg K)) dp ) particle diameter (m) h ) convection heat transfer coefficient (W/(m2 K)) k ) thermal conductivity (W/(m K)) t ) time (s) t0 ) penetration time of the applied heat flux to the other side of the cluster (s) z ) axial distance from disturbance (m) De ) molecular diffusivity (m/s2) Dr ) radial diffusivity (m/s2) KLa ) mass transfer coefficient (s-1) Nu ) Nusselt number (hwDb/kl) U ) fluid velocity (m/s) V ) volume (m3) Greek Letters  ) fractional phase holdup µ ) viscosity (Pa s) ν ) kinematic viscosity (m2/s) F ) density (kg/m3) σ ) surface tension (N/m) τ ) cluster contact time with the wall (s) Subscripts b ) bubble phase c ) convection cc ) convection through clusters e ) emulsion phase f ) floating bubble breaker g ) gas phase l ) liquid phase pc ) particle r ) radial s ) solid phase w ) wall

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ReceiVed for reView November 20, 2007 ReVised manuscript receiVed March 8, 2008 Accepted April 1, 2008 IE0715714