Fluidized bed coating of metal substrates by using high performance thermoplastic powders (original) (raw)

Fluidized bed coating of metal substrates by using high performance thermoplastic powders: Statistical approach and neural network modelling

M. Barletta*, A. Gisario, S. Guarino, V. Tagliaferri
Dipartimento di Ingegneria Meccanica, Universitá degli Studi di Roma “Tor Vergata”, Via del Politecnico, 1-00133 Rome, Italy

Received 20 April 2006; received in revised form 4 June 2007; accepted 14 January 2008
Available online 18 April 2008

Abstract

This paper deals with fluidized bed coating of metal substrates with high performance thermoplastic powders (polyftalamide, PPA). Two different experimental scenarios were investigated: the conventional hot dipping fluidized bed (CHDFB) process and the electrostatic fluidized bed (EFB) coating process. The preliminary experimental plan was scheduled employing design of experiment (DOE) technique. Three experimental factors and operative ranges large enough for practical purposes were considered in both of the examined scenarios. In particular, coating time and airflow rate were chosen as experimental factors in both CHDFB and EFB. The third factor was the preheating temperature of metal substrates in CHDFB and the applied voltage in EFB.

A general linear model based upon analysis of variance (ANOVA) was used to evaluate the significance on coating processes of each experimental factor. Main effect plots (MEPs) and interaction plots (IPs) of coating thickness were drawn. Trends consistent with the settings of the operative parameters were displayed.

The experimental trends were first modelled by numerical regression of the experimental data and, subsequently, by using artificial neural network. The reliability of the neural network solution and of the built ad hoc regression models was comparatively evaluated. Multi-layer perceptron (MLP) neural network trained with back propagation (BP) algorithm was found to be the most valuable in fitting the coating thicknesses trends for both the coating processes. Examining the developed models outside the operative ranges they were designed for, the good generalization capability and high flexibility of the neural network solution was definitely stated.

© 2008 Elsevier Ltd. All rights reserved.

Keywords: Conventional hot dipping fluidized bed; Electrostatic fluidized bed; Coating; Process; Neural network

1. Introduction

The application of fluidized beds in powder coating dates back to 1960s (Richardson, 1971). In those years, scientist and technicians developed the first prototypal systems, which were based on the conventional hot dipping coating process (Richart, 1962a, b; Pettigrew, 1966a, b). With that technique, industrial and maintenance polymeric films, thicker than 1 mm and based on epoxy, polyvinyl or polyurethane powders, could be fruitfully applied. Accordingly, earliest studies were aimed at understanding the leading mechanisms of the newly developed coating process (Richart, 1962a, b)

[1]and, above all, at optimizing the choices in settings of operative parameters, that is, preheating temperature, coating time, hydrodynamic settings of the fluidized bed, part location in the bed and so on (Pettigrew, 1966a, b). In the later 1970 s , innovation in painting and coatings market was characterized by the introduction of industrial scale electrostatic fluidized bed (Strucaly, 1977). Such technique allowed to significantly improve the overall efficiency of conventional hot dipping coating process, when architectural films with thinner coating thickness (i.e. less than 150μ m150 \mu \mathrm{~m} ) and based on new painting materials like hybrid epoxy-polyester and acrylic powders had to be applied without masking the part to be coated or preheating it.

In the 1980s and 1990s, fluidized bed coating process got a limited success as coating technique (Reidenbach, 1994).

[2]

1. *Corresponding author. Tel.: +39 0672597168 ; fax: +39 062021351 . E-mail address: barletta@mail.mec.uniroma2.it (M. Barletta). ↩︎
2. 0952-1976/ \ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2008.01.004 ↩︎

The employment of powders as raw painting material, the hydrodynamic instability of fluidized bed coating process with the resulting application of uneven films, the scant aesthetic aspect of the painted goods, the need for large amount of powder to run the coating operations, the tricky colour change protocol, and, above all, the troubles in operative settings and in prediction of coating thickness as well as in automation and process control determined a progressive declining of industrial world interest towards fluidized beds, thus relegating their use to the application of some of specialty coatings (Weiss, 1997).

In recent years, the growing attention of paint sectors towards environmental problem, the concurrent increasing of powder coating market share, the availability on the market of new high performance thermoplastic and thermoset based alloy powders, the need for alternative eco-sustainable and economical powder coating techniques have been providing renovating interest in fluidized bed coating process. As a consequence, several research programs dealing with the experimentation of new painting materials specifically designed for fluidized bed applications (Leong et al., 1999a; Barletta et al., 2006; Barletta and Tagliaferri, 2006a), the definition of new plant designs and installations (Eti Literature, 2006), the best choice in settings of operative parameters (Barletta and Tagliaferri, 2006b) and, above all, with the development of support instrument in the prediction of best coating strategy (Leong et al., 1999b, 2001, 2002; Ali and Inculet, 2000a, b; Barletta et al., 2002a, 2005) are still in progress.

In particular, attention was paid by the scientific literature towards the development of new tools for prediction of coating thickness in fluidized bed process once set the leading operative parameters. Leong et al. (1999b, 2001, 2002) tried to develop numerical instruments based on finite difference to predict the growth kinetic of the polymeric film. Ali and Inculet (2000a, b) employed volume elements methods to solve the problem of coating thickness prediction in electrostatic fluidized bed. Barletta et al. (2002a) tried to operate massive experimental campaign in both conventional hot dipping and electrostatic fluidized bed coating process to develop reliable analitycal models and to calibrate sophisticated prediction models based upon finite elements method (Barletta et al., 2005). Nevertheless, the common drawbacks of the so far developed tools are related to the limited reliability of models in the film growth prediction, to the lack of generalization capability, to the low flexibility of models in suiting to different scenarios, to the troubled and often recursive computing procedure, and, above all, to the slowness of models in providing reliable responses. Therefore, at this time, no models suitable for industrial purposes are still available in the scientific and technical literature.

In this context, our paper deals with a new approach in prediction of film growth in fluidized bed coating process: the application of neural network solutions. A multi-layer perceptron (MLP) neural network based upon gradient
descent rule with momentum and adaptive learning rate back-propagation (BP) algorithm was used to predict coating thickness trends in both conventional hot dipping fluidized bed (CHDFB) and electrostatic fluidized bed (EFB) coating process with respect to the settings of the operative parameters. Regression models were developed to check both the performance and the reliability of the neural network model.

A good fitting between neural network response and available experimental data was detected. In particular, the neural network model was found able to guarantee better or, at least, comparable performance with the regression model. Besides, the generalization capability of neural network model was found to be good, being it able to guarantee very close fitting results in predicting coating thickness during both the training and testing. Finally, the convenience of the neural network model was stated by checking its capability to predict experimental data outside the operative ranges it was designed for.

2. Experimental apparatus and procedure

2.1. Fluidized bed system

The fluidized bed coating system and coating process employed in this study are thoroughly described in previous papers (Barletta et al., 2006; Barletta and Tagliaferri, 2006a). Here, it is worth remarking that our system is suitable for both CHDFB and EFB coating process. In particular, the coating system is constituted of a fluidized bed, a blower, an integrated convective furnace and an electrostatic charging media (Fig. 1). The fluidized bed is constituted of a container provided with two chambers and a porous plate distributor:

• the upper chamber that holds the powder material referred to as a fluidization chamber ( 250 mm in diameter and 500 mm in height),
• the lower chamber that allows to uniformly distribute the fluidization air from the blower all over the cross section of the bed referred to as an inlet plenum ( 250 mm in diameter and 200 mm in height),
• the fluidization chamber and the inlet plenum separated by an air distributor ( 8 mm as thickness and made from sintered brass) that is porous enough for air to pass through but not porous enough for solids to pass through.

Fluidization air is fed from the blower to the inlet plenum and up through the porous plate distributor. As the fluidization air reaches the fluidization chamber, it exerts a push on the powder particles, which are, consequently, suspended in the airflow. In this suspended state, referred to as fluidization (Richardson, 1971), the powder/air mixture assumes a liquid-like behaviour (Richardson, 1971), hence allowing the coating process to successfully take place (Reidenbach, 1994).

Fig. 1. The experimental apparatus.

Coating application in CHDFB is realized by preheating at an enough high temperature the metal substrate and dipping it into the dense phase (Fig. 1) of the fluidized bed (Barletta et al., 2006). The powder material will melt coming in touch with the hot surface, giving rise to a thick and tough continuous film on the metal surface. However, when the substrate or part of it does not have enough heat capacity to completely melt the powder, the substrate will be submitted to a short postheating cycle, typically in the range of 3−5 min3-5 \mathrm{~min} at 200−250∘C200-250^{\circ} \mathrm{C} to end the melting process and consolidate the coating.

Coating application in EFB was accomplished in the same equipment, with the only difference standing in the electrostatic charging media (direct current, 0−100kV0-100 \mathrm{kV} with negative polarity). The voltage supply feeds a set of electrodes located inside the fluidized bed. The high potential electrodes ionize the surrounding fluidization air (corona charging Castle, 2001; Guskov, 2002) so that the powder material gets charged in short order (capacitor effect Castle, 2001; Guskov, 2002) as the fluidizing air suspends it up. Afterward, electrically charged powder particles move upward, thereby forming a rarefied cloud of charged powders in equilibrium above the denser phase of fluidized powders in the fluidization chamber (Fig. 1). When the grounded metal substrate is hung or passed through the charged cloud, the particles will be attracted to its surface. In such case, a postheating cycle is compulsory to melt the just adhered powder and to consolidate it in a continuous film.

2.2. Materials

Coating tests were performed upon low carbon steel (AISI1040) rods. Six metre long steel rods 10 mm thick were cut into pieces of 60 mm in length. Before being

Table 1
Typical property of PPA 571 H

coated, the metal surfaces were submitted to a set of standard surface pretreatments (Barletta et al., 2006; Barletta and Tagliaferri, 2006a) in order to wash the surface out and to improve the adhesion of the incoming polymeric coating.

For coating purposes, a fixed bed (Richardson, 1971; Reidenbach, 1994) ( 230 mm in height) of thermoplastic polyolefin based alloy powders (polyftalamide, PPA 571 H ) was preloaded inside the fluidized bed in both CHDFB and EFB coating processes. Plascoat Systems Limited supplied PPA 571 H , specifically designed for use in fluidized bed coating, as finely divided powders (distribution peak around 120μ m120 \mu \mathrm{~m}, with 90%90 \% of distribution standing below 250μ m250 \mu \mathrm{~m} ). Table 1 summarizes the main properties of PPA 571 H . Once applied as a continuous film, PPA 571 H

guarantees good adhesion, impact and corrosion resistance as well as outstanding insulating properties.

Hydrodynamic and thermo-physical properties of PPA 571 H powders were fully depicted in previous related papers (Barletta et al., 2006; Barletta and Tagliaferri, 2006a). For the purposes of the present paper, it is worth to remark that minimum fluidization flow rate lower than 1 m3/h1 \mathrm{~m}^{3} / \mathrm{h} was found in the employed fluidized bed. Airflow rate in the range of 3−7 m3/h3-7 \mathrm{~m}^{3} / \mathrm{h} allows the establishment of the bubbling regime (Richardson, 1971; Barletta et al., 2006; Barletta and Tagliaferri, 2006a). Airflow rate higher than 10 m3/h10 \mathrm{~m}^{3} / \mathrm{h} took the bed in slug regime (Richardson, 1971; Barletta et al., 2006; Barletta and Tagliaferri, 2006a) with the first powder spouts rising from denser phase of fluidized powder to the free board zone of the fluidization chamber. Higher flow rate was found to cause massive elutriation phenomena (Richardson, 1971; Barletta et al., 2006; Barletta and Tagliaferri, 2006a) of the powders from the bed and the establishment of turbulent regime (Richardson, 1971; Barletta et al., 2006; Barletta and Tagliaferri, 2006a). As regard thermal properties, PPA 571 H powders exhibit a glass transition temperature around 30∘C30^{\circ} \mathrm{C} and two crystalline phases with peak melting points around 75 and 95∘C95^{\circ} \mathrm{C}, respectively (Barletta et al., 2006). The onset and endset temperatures of melting are close to 40 and 110∘C110^{\circ} \mathrm{C}, respectively (Barletta et al., 2006). Degradation phenomena of Plascoat PPA 571 begin nearby 300∘C300^{\circ} \mathrm{C} (Barletta et al., 2006). Thermal conductivities of 0.16−0.18 W/mK0.16-0.18 \mathrm{~W} / \mathrm{mK} in the bed and of 0.24−0.26 W/mK0.24-0.26 \mathrm{~W} / \mathrm{mK} in the melted layers on part being coated were measured (Barletta et al., 2006).

2.3. Experimental procedure and plans

Coating procedures in CHDFB and in EFB coating process are reported in previous related papers (Barletta et al., 2006; Barletta and Tagliaferri, 2006a). In CHDFB, the metal substrates were first preheated in a convective furnace (Nabertherm, model B170 30-2000 ∘C{ }^{\circ} \mathrm{C} ) and, subsequently, dipped in the denser phase of the fluidized powder in the bed (Fig. 1). In EFB, the metal substrates were commonly located in the middle of the cloud of electrostatic charged powders, that is, in the free board zone of the fluidization chamber away from the bed denser phase (Fig. 1). When coating time elapsed, the metal substrates were moved from the bed up to the convective furnace, to be postheated at 250∘C250^{\circ} \mathrm{C} for about 10 min . In both the set of experiments, all the cautions needed to guarantee the stability of environmental conditions (temperature of 20∘C20^{\circ} \mathrm{C}, moisture of 40%40 \% and ventilation control) as well as the accuracy and the reproducibility of the experimental procedure were taken.

Afterward, the amount of deposited polymer onto workpiece surface in CHDFB and in EFB coating process was measured using a magnetic thickness gauge (according to regulations ISO 2178 and ISO 2370). The coating thickness was estimated as the average of twelve measure-

Table 2
Experimental plan in CHDFB coating process
Levels Three factors

Table 3
Experimental plan in EFB coating process

ments equally spaced along the surface of each metal substrate.

Two sets of experimental tests were respectively defined for CHDFB and EFB coating process following design of experiment (DOE) approach. Tables 2 and 3 summarize the experimental factors chosen and the related ranges of experimental levels investigated. In CHDFB coating process, a full factorial experimental plan, employing three factors (i.e. coating time, preheating temperature and airflow rate) and replicated six times for a total of 324 trials was performed. In EFB coating process, a full factorial experimental plan, employing three factors (i.e. coating time, applied voltage and airflow rate) and replicated five times for a total of 300 trials was performed. These tests were scheduled for drawing the experimental trends of coating thickness according to operational parameters, for evaluating the significance of the experimental factors in the range investigated as well as for examining the simple and interacted effects of experimental factors on CHDFB and EFB coating processes.

A strict statistical approach was followed in examining and reporting experimental data. Main effect and interaction plots (IPs) were built to report the trend of coating thickness according to operative parameters both in CHDFB and EFB coating processes. The significance of operational parameters on the response coating thickness was quantitatively estimated using a general linear model ANOVA. A first approximation regression model was used to find the best fit for the experimental results as well as to asses the reliability of the neural network solution, which was to be developed.

3. Neural network model

3.1. Neural network model

A MLP neural network based upon gradient descent rule with momentum and adaptive learning rate BP algorithm (Haykin, 1994) was used to predict coating thickness trends in CHDFB and EFB coating process. As software simulator, NeuroSolutions version 5.0 developed by NeuroDimension Incorporated was employed (Principe et al., 2000).

The MLPs are layered feed-forward neural networks (Haykin, 1994). The architecture of the MLPs is characterized by non-linear PEs, with the non-linearity function generally smoothed by logistic, sigmoid or hyperbolic tangent functions. PEs are fully interconnected so that any element of the former layer feeds all the elements of the latter layer. MLPs are easy to use and they can simulate any input/output map. To the contrary, the training time of MLPs is very long and the training procedure requires many training data.

The gradient descent rule with momentum uses adaptive learning rate BP to calculate derivatives of performance cost function with respect to the weight and bias variables of the network (Haykin, 1994). Each variable is adjusted according to the gradient descent with momentum. For each step of the optimization, if performance decreases the learning rate is increased. When the BP algorithm is adopted, the operation of the neural network can be schematized into two main phases: forward computing and backward learning. In the forward phase, the synaptic weights are kept constant, and the response of the network is computed by submitting it to a prescribed set of input data. In the backward phase, the adjustments to synaptic weights are computed to minimize a cost function defined as the sum of error square.

3.2. Neural networks set-up

The three experimental factors which were found to be significant by ANOVA on the experimental response (coating thickness), that is, coating time, preheating temperature and airflow rate in CHDFB coating process and coating time, applied voltage, and airflow rate in EFB coating process were employed as input processing elements (PEs). The experimental response ‘coating thickness’ was the only output estimated by neural network model developed. The neural network response was defined as a normalized parameter to prevent the saturation of the chosen activation function.

To model CHDFB coating process, a total of 54 samples coated with PPA 571 H was considered. For simulation purposes, the averaged coating thicknesses of the six replications with each combination of experimental factors were used. Samples were randomly shared according to the following percentages: 60%60 \% as Training, 15%15 \% as Cross Validation, and 25%25 \% as Testing (Principe et al., 2000).

To model EFB coating process, a total of 60 samples coated with PPA 571 H was considered. For simulation purposes, the averaged coating thicknesses of the five replications with each combination of experimental factors were used. Samples were randomly shared according to the following percentages: 60%60 \% as Training, 15%15 \% as Cross Validation, and 25%25 \% as Testing (Principe et al., 2000).

The same topological structure of the neural network with just one hidden layer was used for both CHDFB and EFB coating processes. A sigmoid transfer function was used in the hidden and output layers to generate the output values. In specific, BP algorithm was based on the momentum with step size being set at 0.7 and a momentum coefficient for the hidden layer set at 0.5 (Principe et al., 2000). The same operational parameters were used for the output layer except for the step size, which was set at 0.1 (Principe et al., 2000). Besides, the rule of descendent gradient was adopted to calibrate the network and to identify the optimal weight values, that is, those minimizing the overall error between desired and calculated output. Afterward, the network was thoroughly trained. In order to obtain a good indicator of generalization level achieved by the network, the training was stopped when the mean square error (MSE) of the cross validation set began to increase, this being a signal that the network had come to the point of becoming over-trained. Finally, two different criteria were used to evaluate the effectiveness of the neural network model and its ability to make accurate predictions: the root MSE (RMSE) and the correlation coefficient (r)(r). Therefore, the best fit between measured and estimated values, which is unlikely to occur, would have RMSE =0=0 and r=1r=1.

4. Simulation procedure and results

4.1. The experimental results and statistical approach

A statistical approach was used to interpret coating thickness trends versus operative variables and to evaluate the significance of the experimental factors and of their interactions on the response ‘coating thickness’ in both CHDFB and EFB coating processes. This is of fundamental importance for establishing which factors or combination of them consider in the built-up of the regression and neural network model.

Tables 4 and 5 summarize the results of the analysis of variance (ANOVA) applied to CHDFB and EFB, respectively. In both cases, the calculated Fisher’s values are well above the respective Fisher’s values ( F0.05F_{0.05} ) tabulated. In fact, being very large the degree of freedom for the factor error ( 324 in Table 4 and 240 in Table 5) and being very low the degree of freedom for the other factors and interactions (from 2 to 20 in Table 4 and from 2 to 24 in Table 5), the tabulated Fisher’s (F0.05)\left(F_{0.05}\right) assume values in the range of about 1.5−31.5-3 for both coating processes (Pearson and Hartley, 1966). Therefore, even the three factors interactions seem to be apparently significant. On the other

Table 4
ANOVA for CHDFB coating process

Table 5
ANOVA for EFB coating process

hand, if the contribution percentage III I is considered, some further indications about how influential the experimental factors and their interactions can be on the experimental response. Tables 4 and 5 report the values of contribution percentage in the last column. As can be seen, the simple factors in CHDFB coating process share about the 90%90 \% of the contribution. In EFB coating process, the contribution of the simple factors comes up to 95%95 \%, hence stating the minor importance of the effect of the interactions on the response ‘coating thickness’. It is worthwhile noting how flow rate could be disregarded in CHDFB, being its contribution percentage very low (around 1.5%1.5 \% ) and very close to those of the interactions between experimental factors.

Fig. 2 reports the main effect plot (MEP) for CHDFB coating process. Increasing averaged trends of ‘coating thickness’ according to the experimental factors ‘coating time’ and ‘preheating temperature’ can be noticed. It stands to reason that if more time or pushing force (i.e. the temperature differential between substrate surface and fluidized powder) is given to coating process, thicker coating thickness can be expected in agreement with indications reported in the scientific literature (Barletta et al., 2006). On the other hand, an increase in the experimental factor ‘flow rate’ was found to produce a concurrent slight decrease in the response ‘coating thickness’. Accordingly, if more fluidizing air is provided to the bed of thermoplastic powders, less dense bed would be established and, consequently, less powder would be available to the metal substrates when dipped in the
fluidized bed for coating purposes. Therefore, thinner coating thickness can be expected in agreement with the results that one of the authors found in a previous paper (Barletta et al., 2006).

For the purposes of such paper, MEP in Fig. 2 is mostly useful to examine and compare the level means for the experimental factors involved. In specific, comparing the changes in the level means to see which factors influence the response the most, a main effect can be detected in Fig. 2 for the factors ‘coating time’ and ‘preheating temperature’. In fact, the lines representative of the trends of the response ‘coating thickness’ versus the experimental levels of the factors ‘coating time’ and ‘preheating temperature’ are not parallel to the reference line set at the overall mean. Consequently, the different levels of the considered factors affect the response differently. Besides, being the slopes of the trends of the response ‘coating thickness’ versus the experimental factors associated with the magnitude of the main effects, larger main effects can be found for the factor ‘coating time’. This results is in agreement with indications coming from ANOVA (Table 4). On the contrary, slight main effects can be noticed for the factor ‘flow rate’, where the response mean for each level remains very close to the reference line. Therefore, each level of the factor ‘flow rate’ affects the response in the same way, with this result being in good agreement with indications provided by ANOVA (Table 4).

Fig. 3 reports the IP for CHDFB coating process, with each plot showing the interaction between two different factors. As known, an interaction between factors occurs

Fig. 2. MEP for CHDFB coating process.

Fig. 3. IP for CHDFB coating process.
when the change in response from a generic level of a generic factor to another level differs from the change in response at the same two levels of a second factor, with this meaning that the effect of one factor is dependent upon a second factor. Watching to matrix plot reported in Fig. 3, all the lines representative of the averaged trends of the experimental response ‘coating thickness’ run parallel. Consequently, neither panel indicate a clear interaction among the factors examined in agreement with the values of contribution percentages reported in ANOVA Table 4.

Similar considerations could be done for EFB coating process. Fig. 4 reports the increasing trends of coating thickness versus the experimental factors ‘coating time’, ‘flow rate’ and ‘applied voltage’. This result can be attributed to the characteristic of electrostatic coating process (Guskov, 2002). If larger flow rate and voltage are

Fig. 4. MEP for EFB coating process.

selected, the cloud of powder, which is established in equilibrium above the denser phase of fluidized bed presents a higher powder concentration, with more electrical charge distributed on powder surface. Consequently, more powders surrounds and is attracted towards the grounded substrates when it is hung above the dense phase of the fluidized bed, with thicker coating thickness developing in agreement with data reported in the literature (Barletta and Tagliaferri, 2006a, b). Similarly, if larger coating time is selected, more time is allowed to the coating to grow on the metal substrate, thereby causing the built-up of thicker film as well. This was in agreement on what one of the authors found in previous studies (Barletta and Tagliaferri, 2006a, b).

Accordingly, MEP in Fig. 4 exhibits remarkable main effects for all the investigated factors. Matrix plot in Fig. 5 displays weak interactions among experimental factors.

Nevertheless, a certain interaction can be noticed between the factors ‘coating time’ and ‘applied voltage’, even if only for larger values of the experimental levels. These results are in good agreement with the values of contribution percentages summarized in ANOVA Table 5.

Figs. 6 and 7 report the residual plots for both CHDFB and EFB coating process. In both cases, the normal probability plots display residuals that roughly follow straight lines and, consequently, are normally distributed, with moderate departures from normality. The graphs of the residuals versus the fitted values display residuals that are randomly scattered about zero. The detected patterns indicate the lack of fanning or uneven spreading of residuals across fitted values, with no evidence of missing terms or outliers. The histograms of residuals show the bell-shaped distribution of the residuals, with no skewness. A couple of modest outliers can be detected in Fig. 7 for

Fig. 5. IP for EFB coating process.

Fig. 6. Residuals plots for CHDFB coating process.

Fig. 7. Residuals plots for EFB coating process.
residuals in EFB coating process. The plots of residuals in the order of the corresponding observations fluctuate in a random pattern around the centre line, with no evidence of correlation between error terms that are near each other, testified by the lack of ascending or descending trend in the residuals as well as of rapid changes in signs of adjacent residuals.

4.2. Mathematical formulation

Examining the results of ANOVA, all the experimental factors in EFB coating process were assessed to be significant. Similar considerations can be done for CHDFB coating process. Instead, the interactions among the experimental factors were found to be remarkably less significant if compared with the significance of each individual factor in CHDFB and EFB coating processes.

Concerning on what has just been mentioned, all the simple experimental factors were considered in the modelling of both coating processes for the sake of similarity, while all the interactions were disregarded without affecting too much the overall error. Consequently, a multiple regression analysis was led to find the best mathematical model for fluidized bed coating processes with the parameters coating time tt, preheating temperature TT and air flow FF being used as predictors in CHDFB and coating time tt, applied voltage VV and air flow FF in EFB.

The generic model can be so formulated:
r=y(f1,f2,f3)r=y\left(f_{1}, f_{2}, f_{3}\right)
where rr is the coating thickness, yy is the response function and f1,f2f_{1}, f_{2} and f3f_{3} the experimental factors of the two coating processes. Expressed in non-linear form Eq. (1) becomes
r=kf1αf1βf1γr=k f_{1}^{\alpha} f_{1}^{\beta} f_{1}^{\gamma}
To simplify the determination of the constant and parameters, the mathematical model was linearized per-
forming a logarithm transformation. Eq. (2) becomes
ln⁡r=ln⁡k+αln⁡f1+βln⁡f2+γln⁡f3\ln r=\ln k+\alpha \ln f_{1}+\beta \ln f_{2}+\gamma \ln f_{3}
The constant kk and parameters α,β,γ\alpha, \beta, \gamma were solved by using a multiple regression analysis with the assistance of experimental results reported in previous section. Eq. (3) can be rearranged as
r1=3.16t0.354T0.870F−0.0544r_{1}=3.16 t^{0.354} T^{0.870} F^{-0.0544}
r2=6.95t0.0935V0.753F0.632r_{2}=6.95 t^{0.0935} V^{0.753} F^{0.632}
for CHDFB and EFB coating processes, respectively.
A first validation of the regression models was operated on the experimental data available. Values of R2R^{2} around 91%91 \% were found for CHDFB coating process, stating its good capability to predict the experimental data. Furthermore, standard error of 0.04 and a correlation factor of 0.96 were calculated for CHDFB coating process. The performance of CHDFB regression model can be assessed as surprisingly good if compared with analitycal, finite difference and finite element models reported in the literature (Leong et al., 1999b, 2001, 2002; Barletta et al., 2002a, b, 2005, 2006). In particular, analytical models even being characterized by simple mathematical formulations and requiring shortest computing times, they exhibited lower correlation coefficients, often neglecting the influence of flow rate and making use of one or more corrective factors to account for part vertical locations and attitude inside the fluidized bed during the coating process (Barletta et al., 2002a, 2006). At the same time, even finite difference models reported in the literature suffered of several drawbacks, which were related to the need for several limiting assumptions (physical and thermal properties of the coating constant during the process, coating surface temperature equal to the powder melting temperature, molten plastic powder not flowing down the wall, temperature within the bed uniform and constant, heat

transfer coefficient between the cylinder wall and the bed independent of location and direction and constant during coating), to the tricky mathematical formulations of the governing equations, to the employment of unreliable heat transfer coefficients to improve fitting between numerical and experimental data and to calibration procedures based upon a restricted number of experimental data (Leong et al., 1999b, 2001, 2002). Finally, finite element models, even suffering of less limiting assumptions in comparison with finite difference models, being very accurate and flexible in predicting coating thickness trends and thermal fields for a multitude of possible scenarios and using reliable calibration procedures based upon experimental data coming form wide and ad hoc developed experimental campaigns, they are strongly limited by the slow set-up time as well as by the onerous and recursive solution methods, which make them not suitable in industrial practice (Barletta et al., 2005, 2002b).

Values of R2R^{2} around 83%83 \% were found for EFB coating process, 8%8 \% lower than for CHDFB coating process. Furthermore, a larger standard error of 0.14 and a lower correlation factor of 0.65 were calculated for EFB coating process. This poorer results can be probably ascribed to the non linear behaviour of coating thickness versus operative parameters in EFB (Barletta and Tagliaferri, 2006a), differently from what happens in CHDFB coating process where relationship definitely linear can be found between the response ‘coating thickness’ and all the operative variables (Barletta et al., 2006). However, the proposed regression model can be defined as useful and only one of its kind instrument in predicting coating thickness trends in EFB coating process, if compared with other models reported in the literature (Barletta and Tagliaferri, 2006b; Ali and Inculet, 2000a, b). In fact, just one analytical model is proposed in the literature and it is referred to thermoset coatings (Barletta and Tagliaferri, 2006b). Besides, the available finite elements models are mostly focused on the determination of electrical fields and they do not provide any coating thickness trends (Ali and Inculet, 2000a). Anyway, they can be assessed as first approximation models, plenty of limiting assumptions and lacking of accurate calibration procedures based upon reliable experimental data (Ali and Inculet, 2000a, b).

The analysis of residuals for the two regression models showed residuals normally distributed with the RyanJoiner’s statistics close to 1 and PP-value larger than 0.1 for both CHDFB and EFB coating processes, hence confirming the good overall performance of the regression models developed.

4.3. Neural network approach

The simple experimental factors of CHDFB and EFB coating processes were chosen as input PEs even in the neural network model in order to comparatively evaluate the performance of neural network solution with both the regression model and available experimental data.

Figs. 8 and 9 display the performance of the neural network model developed for both CHDFB and EFB coating processes, respectively. In particular, the effect on RMSE of the number of PEs in the hidden layer is reported. As shown, RMSE is minimized when the number of PEs is set at 9 for both the coating processes.

Consequently, such PEs values were adopted in the hidden layer to find the best topology of the neural networks. For this purpose, the epochs number for each neural network model was fixed, at least, at 10000 and training was repeated, at least, 10 times to minimize the results variability. Figs. 10 and 11 show the RMSE trends according to the number of epochs for both CHDFB and EFB coating processes, respectively. The network weights, which minimized the RMSE, were chosen as the best weights. In CHDFB coating process, the training was interrupted after about 1500 epochs to avoid the overtraining of the neural network. A quite low RMSE (less than 0.06 ) was detected (Fig. 10). In EFB, the training

Fig. 8. Optimization of PEs in CHDFB coating process.

Fig. 9. Optimization of PEs in EFB coating process.

Fig. 10. Optimization of number of epochs in CHDFB coating process.

Fig. 11. Optimization of number of epochs in EFB coating process.
lasted all the 10000 epochs, as no overtraining problems occurred. Very low RMSEs, less than 0.04 for cross validation and close to 0.03 for training, were calculated. The better results achieved for EFB could be probably ascribed to the lack of overtraining, which allowed to complete the training through all the prescribed epochs, hence minimizing the error.

The performance of the developed neural network was subsequently checked in testing mode. Figs. 12 and 13 reports the neural network performance splitting the results in training, cross validation and testing set. All the data set examined revealed the good capability of the neural network to match the related experimental data (Tables 6 and 7), with no significant differences arising among them. In particular, Table 6 shows that correlation factors in the range of 0.92 (testing set) to 0.97 (training set) characterizes the neural network model in the prediction of coating thickness trends for CHDFB coating process. In each set, the maximum error in prediction does not overcome 10%10 \%, with MSE very close to 0.002 , thus demonstrating the good generalization capability of the developed model. Similar

Fig. 12. Generalization capabilities of neural network in CHDFB coating process.

Fig. 13. Generalization capabilities of neural network in EFB coating process.

Table 6
Generalization capabilities of neural network model in CHDFB coating process

considerations could be done for EFB coating process. Table 7 reports correlation factors varying in the range of 0.96 (testing and cross validation set) to 0.98 (training set), with maximum error very close to 8%8 \% and MSE around 0.001−0.00150.001-0.0015, hence confirming, once more, the good generalization capability of the proposed neural network model on both the coating processes.

Table 7
Generalization capabilities of neural network model in EFB coating process

Fig. 14. Comparison between neural network and regression model in CHDFB coating process.

Fig. 15. Comparison between neural network and regression model in EFB coating process.

An overall comparison among the MLP model, the experimental data coming from full factorial experimental plan and the built ad hoc regression model was operated. Figs. 14 and 15 summarize the results, showing the comparable performance of neural network and regression in modelling the experimental data coming from CHDFB
coating process (correlation factor of about 0.96 and standard error around 0.04 for both neural network and regression model) and the better performance of neural network in predicting experimental data from EFB coating process (correlation factor of 0.98 and standard error of about 0.027 for neural network and correlation factor of 0.89 and standard error of about 0.07 for regression model, respectively). This result could be ascribed to the better suitability of neural network solution to model non-linear trends, as those typical of coating thickness versus operative parameters in EFB coating process (see Fig. 5 and data reported in the literature Barletta and Tagliaferri, 2006b), where, oppositely, regression model typically experiences the worst behaviour. In particular, in Fig. 15, it is worth noting how regression model fails to predict experimental data in EFB for larger response values (close to 1 ), where experimental data report a significant bending of coating thickness trend according all the operative parameters (see Fig. 5).

Finally, Tables 8 and 9 report a comparison between the capability of the neural network, applied in production mode (Haykin, 1994), and regression models in predicting coating thickness in both CHDFB and EFB outside the operative ranges they were designed for. It can be noted that neural network model produced predictions very close to the experimental values for both coating processes, differently from the regression models, which, mostly, produced an underestimation of response for low values of coating thickness and an overestimation of response for large values of coating thickness. However, correlation factors and standard errors are reported in Table 10 for both the coating processes. In particular, standard errors from two to three times lower are calculated for the neural network model if compared with the regression model. Besides, neural network model exhibited correlation factors very close to 1 for both CHDFB and EFB, that is, quite larger than correlation factors reported for regression models. Therefore, the neural network model exhibits a quite better overall performance than regression models, with significant improvements in prediction and, above all, in generalization capabilities.

5. Conclusions

In this investigation, the development of a multi-layer perceptron (MLP) neural network based upon gradient descent rule with momentum and adaptive learning rate back-propagation (BP) algorithm used to predict coating thickness trends in CHDFB and EFB coating process is discussed. A comparison of neural network model with regression models is also reported.

First, a preliminary experimentation has shown coating thickness trends in CHDFB and EFB processes according to operational parameters consistent with the results reported in the literature and the theoretical expectations. Regression models were found able to predict coating thickness trends, with higher accuracy than analytical,

Table 8
Comparison between neural network and regression model in predicting coating thickness trends outside the range they were designed for: CHDFB scenario

Table 9
Comparison between neural network and regression model in predicting coating thickness trends outside the range they were designed for: EFB scenario

Table 10
Correlation and standard errors factors

finite difference and finite element models reported in the literature. Accordingly, even the neural network solution was found able to accurately predict the coating thickness trends in both the examined fluidized bed coating processes. As to the analysis of neural network settings, it was found: (i) the best number of PEs in the hidden and output layer was set at 9, (ii) in EFB, 20000 epochs had to be employed to get the network thoroughly trained, (iii) in CHDFB, not more than 1500 epochs had to be used to avoid overtraining the network, (iv) in both the coating processes, the network was found to have good generalization capability, stated by the good correlation factors found for all the training, cross validation and testing set.

Second, the developed neural network model was found to fit better the available experimental data rather than the connected regression models in both coating processes. This result was particularly evident in EFB, hence demonstrating the better flexibility of the developed neural network solution, when non-linear behaviour must be accounted for.

Finally, applying neural network and regression models outside the ranges they were designed for, it was deducted that neural network solution possesses surprising generalization capability. In particular, in both EFB and CHDFB coating processes, the developed neural network model exhibits better performance than the connected regression model.

All these results make the neural network model the best solution in predicting coating thickness trends in fluidized bed coating process, thereby stating the basis for a wide employment of such tools in the industrial practice.

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