Hybrid nonlinear autoregressive neural network—Weibull statistical model applied to the supercritical extraction of lanolin from raw wool

“This is a post-peer-review, pre-copyedit version of an article published in SN Applied Sciences. The final authenticated version is available online at: http://dx.doi.org/10.1007/s42452-020-03475-7” 1 Hybrid Nonlinear Autoregressive Neural Network – Weibull Statistical Model Applied to the Supercritical Extraction of Lanolin from Raw Wool Abel Valverde*, Jesús Alvarez-Florez, Francesc Recasens Chemical Engineering Department, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain *Corresponding author: abel.valver[email protected] Keywords. High pressure extraction, lanolin, neural networks, genetic algorithm, Weibull Abstract Supercritical extraction of lanolin from raw wool with modified CO2 (5% ethanol) at temperatures above the melting point of lanolin (T = 36-42 ºC) is difficult to model because of the multicomponent diffusion in the liquid layer. In this work, a neural network model is proposed based on the experiments previously published by our research group. Experimentally, the extraction of a 100-cm3 packed bed of raw wool depends on five variables, i.e., temperature (60-80 ºC), pressure (120-200 bar), solvent mass flow rate (3-5 kg/h), wool packing density (127-318 kg/m3), and time (~1h). A nonlinear autoregressive exogenous (5,3,1) neural network was designed and trained with the experimental data augmented using an empirical Weibull statistical function. This correctly predicts the lanolin breakthrough at the extractor exit with only ±0.42% error. The simple arithmetics of neural network allows a fast optimization with Genetic Algorithm to find optimum operation conditions for the extraction process. 2 Article Highlights —Statistical function successfully models lanolin production under four factors (pressure, temperature, flowrate and packing density). —Extraction yield vs time can be calculated using neural computing. The prediction of yield is excellent with an error is only ±0.42%. —The arithmetics of neural computing allows fast optimization of the production process. 1. Introduction The early attempts to extract lanolin from raw wool using supercritical carbon dioxide date back to 1990s, and are due to King and coworkers [1], Koo et al [2], and to New Zealand researchers [3]. In all cases, pure compressed CO2 was employed. The first experimental systematic study, published a few years later, is that due to Eychenne et al. [4]. In their work, CO2 modified with 5% ethanol was used under near-critical conditions; this allowed operation at lower pressure. Higher extraction yields were possible. Recently, we published an article dealing with the modeling of the extraction of solid lanolin from raw wool using modified CO2 [5]. At 30 ºC, the lanolin is a solid (mp. 38-44 ºC) and the fluid is a liquid. For these conditions, extraction kinetics could be interpreted with the shrinking-core concept [5]. In this case, the pure solid lanolin covering the wool fibers is in direct contact with the fluid and is progressively dissolved by direct contact with the flowing solvent. In this model, two solid lanolin fractions are assumed to be located one over the other on the wool fiber, as suggested by the experimental results obtained in the fractionation process developed by Bayona et al. in a patent [6,7], and chemical characterization of the fractions [8,9]. 3 The present work, based also on the same experiments [4], comprises wider ranges of temperature, pressure and other variables, as summarized on Table 1. As can be seen, in all the extraction runs lanolin is extracted from a liquid into a near-critical fluid at temperatures above the melting point of lanolin, i.e., 60 to 80ºC. For these temperatures, the conservation equations for lanolin become complicated by the fact that a multicomponent liquid phase is present, involving additional mass transfer resistances compared with the extraction at temperatures below the lanolin melting point where lanolin is a pure solid. For temperatures above 30 ºC, the lanolin diffusion equations from the liquid up to the liquidfluid interface makes solution very difficult. Table 1. Settings for lanolin extraction runs from raw wool [4] Temperature, T 60, 80 ºC Pressure, P 120, 150, 200 bar Solvent mass flow rate, Q 3, 4, 5 kg/h Solvent passed Up to 5 kg Wool packing densities, ρ B 127, 159, 227, 318 kg/m3 * Extraction solvent composition: 95% wt. CO2 – 5% ethanol; total extraction time 1 h. In [10], an exact analytical solution of the conservation equations with some simplifying assumptions was obtained. The corresponding theory is summarized below in the Conservation equations, paragraph 3.1. It is evident, however, that the conservation equations become very complex, because not only lanolin is extracted to the fluid, but also CO2 and ethanol are dissolved in the liquid phase, thus giving a very complicated situation to model. On the other hand, artificial neural networks (ANNs) can be used to solve the conservation equations in cases where the physical model is either very complex, the solution algorithm is difficult, or it does not exist [11,12]. 4 In the present case, we are interested in the effects of the factors that influence the yield and rate of extraction of lanolin from wool. The main feature of ANNs is their ability to learn by reproducing the output from a given input by observation and minimization of the error providing an optimal set of weighting parameters. In the prediction process, no explicit law is assumed that relates output from input. It will be apparent that our extraction runs can somehow be considered as a desorption process. As regards to a work on adsorption column operation, an early paper due to Bulsari and Palosaari [13,14] dealing with system identification was very interesting to us. In our case, lanolin is not held on wool fibers by adsorption. Instead, dissolution of lanolin in the solvent depends on how far the fluid is from saturation, so the rate of dissolution depends on a mass transfer coefficient and a concentration gradient. Our objective in this paper was to use neural network computing combined with the properties of the conservation equations for lanolin extraction from raw wool, as described below. To this end, experimental data about the variables that affect the extraction rate and yields are essential. In this sense, the systematic extraction study data [4] is most appropriate to try to model. In the following text, we first present the Experimental data available. Then, in the Theory section, after doing a deep statistical analysis of available data, we discuss the selection of the recurrent networks and the available tools in the literature, and present the details of the training data sets and test data sets. Finally, in the presentation of Results, we solve a few optimization problems relative to lanolin production. 5 2. Experimental background The experimental study behind this work was published elsewhere by Eychenne et al., [4]; the reader should refer to it for details. Here, we report only a few significant features. A Separex 200 unit was used for the high-pressure extraction. This is shown on Fig 1. A summary of the scope of the measurements, features of the extractor vessel and the wool stock used are given in Table 2. Fig 1. Separex 200 unit process flow diagram. P1, Milton-Royal CO2 pump; P2, Pulsa-Feeder ethanol pump; HE1, HE2, HE3, heat exchangers; BPR back-pressure control valve; E, Separex 200 extractor vessel; S, Separex cyclone separator; PI, pressure indicators; TIC temperature indicating controller; FC, flow controller 6 Table 2. Properties of extractor and wool used [4] Extractor vessel Shape Cylindrical Material Stainless steel AISI 316L Inside dimensions 145 mm x 30 mm (H x D) Volume 100 cm3 Cross section 706.8 mm2 Wool fibers Wool load in extractor 13 g Wool composition 60-65% wool proper, 10-15% wax and proteins, 10% soluble stains (salts), 1-20% soil and vegetable matter Fiber geometry Cylindrical Average fiber length (approx.) 15 cm Initial fiber radius, R 8.3 μm Initial fiber radius, r 0 10 μm Lanolin content 20 % wt Types of lanolin Two lanolin fractions (external and internal) Lanolin melting point 38 – 44 ºC 3. Theory 3.1 Conservation equations. Consider a vertical bed of packed wool from which the lanolin deposited on the fibers is extracted by the flowing solvent. See item E, in Fig 1. At t = 0, a step input of lanolin-free solvent is introduced in the bed at z = 0, while fluid saturated with lanolin is removed at the other end of the bed at z = zT. If pressure drop is neglected, the solvent passes through the bed in plug flow at constant pressure and temperature. The rate of extraction depends on two factors: the solubility of lanolin in the solvent and a mass transfer coefficient. Two assumptions are made: 1) since the amount of lanolin on the wool is small, the bed void fraction, ϵ, is about constant and equal to that of packed wool; and 2) since the layer of liquid lanolin covering de fibers is very thin, the total mass transfer area, a, can be considered constant; therefore, the holdup of liquid in the bed, ϵL, can be taken equal to its 7 average value during the extraction process. With these assumptions, the balance equations for lanolin in the flowing fluid and in the liquid phase, respectively, are: 𝜖𝜖𝜕𝜕𝐶𝐶𝑔𝑔 𝜕𝜕𝜕𝜕 +𝑢𝑢𝜕𝜕𝐶𝐶𝑔𝑔 𝜕𝜕𝜕𝜕 =𝑟𝑟𝑣𝑣 (1) −𝜕𝜕(𝑉𝑉𝐿𝐿𝐶𝐶𝐿𝐿) 𝜕𝜕𝜕𝜕 =𝑉𝑉𝑉𝑉𝐺𝐺𝑎𝑎 (𝐶𝐶𝑔𝑔∗−𝐶𝐶𝑔𝑔) (2) Where rv is the volumetric mass transfer rate at position z in the bed and time t, and kG is the solvent-side film mass transfer coefficient. In eqn. (2), Cg* is related to the concentration of lanolin in the liquid, CL, through a Henry-type equilibrium constant, as is usually done in high pressure studies. Then: 𝐶𝐶𝑔𝑔∗=𝐾𝐾𝐶𝐶𝐿𝐿 (3) Note that CL decreases with increasing z, because of lanolin dissolution in the solvent occurs for increasing values of z. Using the following relationship: 𝑉𝑉𝐿𝐿=𝑉𝑉 𝜖𝜖𝐿𝐿 (4) the gas and liquid holdups, ϵ and ϵL, and the Henry constant, K, eqns. (1) and (2), become: 𝜖𝜖𝜕𝜕𝐶𝐶𝑔𝑔 𝜕𝜕𝜕𝜕 +𝑢𝑢𝜕𝜕𝐶𝐶𝑔𝑔 𝜕𝜕𝜕𝜕 =𝐾𝐾𝑉𝑉𝐺𝐺 𝑎𝑎 (𝐶𝐶𝐿𝐿−𝐶𝐶𝑔𝑔 𝐾𝐾) (5) −𝜕𝜕𝐶𝐶𝐿𝐿 𝜕𝜕𝜕𝜕 = 𝐾𝐾𝑘𝑘𝐺𝐺𝑎𝑎 𝜀𝜀𝐿𝐿(𝐶𝐶𝐿𝐿−𝐶𝐶𝑔𝑔 𝐾𝐾) (6) Where ϵL is the liquid holdup taken as an average during a run. If the bed is initially loaded with new wool, and the inlet solvent is lanolin-free, the initial and boundary conditions for eqns. (5) and (6), are: 𝐶𝐶𝑔𝑔 (𝑡𝑡,𝑧𝑧= 0)= 𝐶𝐶𝑔𝑔 (𝑡𝑡= 0, 𝑧𝑧 )= 0 (7) 𝐶𝐶𝐿𝐿 (𝑡𝑡= 0, 𝑧𝑧)=𝐶𝐶𝐿𝐿0 (8) Where CL0 is the initial concentration of lanolin covering the fibers. As will be noted, CL0 equals the pure lanolin density. 8 3.2 Initial state of the bed. Lanoline breakthrough at bed exit. An expression for CL(t) is available if we consider the molecular diffusion of lanolin over a quiet liquid layer in contact with the solvent and a fluid-side mass transfer coefficient, D and kG, respectively [14]. It is not difficult to show [15] that the expression of CL (t) at bed inlet, z = 0, is: 𝐶𝐶𝐿𝐿= 𝐶𝐶𝐿𝐿0 𝑒𝑒𝑅𝑅𝑘𝑘𝐺𝐺𝐾𝐾 𝐷𝐷 �1−�1+𝑟𝑟0 2𝑘𝑘𝐺𝐺𝑎𝑎𝐾𝐾𝑎𝑎 𝑅𝑅2(1−𝜀𝜀) � (9) Eqn (9) indicates that CL0 drops rapidly to zero. In eqn (9), D is the molecular diffusivity of lanolin in the liquid, R is the final radius of the wool fiber, and r0 is the initial radius. kG is the solvent-side mass transfer coefficient. The mass transfer group, Bi, is a modified Biot number for mass transfer, 𝑅𝑅𝑘𝑘𝐺𝐺𝐾𝐾 𝐷𝐷= 𝐵𝐵𝐵𝐵 (10) Now, let us see what happens at the end of the bed. In our previous work, Fullana et al. [16] showed in Fig. 13 of their article the value of the extraction yield in the fluid Cg vs. time during solute breakthrough for three different velocities of extraction. The response of an extraction bed consists in a time delay, called here n later, of constant concentration, in which Cg decreases more or less rapidly depending on the solvent flow rate. After a step input pulse of solvent at bed inlet, z = 0, the shape of the breakthrough curve corresponds to an inverse step at z = zT. The shape of the breakthrough curve can be found from eqn (9) and can be fitted to an empirical function, as we discuss next. By performing the operations under the square root of eqn (9), it is found that, 15 closed-loop to predict the rest of the extracted fractions for the extraction curve at successive times, always holding constant the same external state of operation. Fig 2 illustrates schematically how the network works. Fig 2. Input, output and delays in a recurrent neural network. For predictions, not for training, the net works in closed-loop entering the previous output as the feedback input of the network. i, i+1, …, correspond to a different data set (or extracted fraction curve), and t0, t1, …, are the different time steps. 4.2. Training parameters and prediction error The settings for the training parameters used by Matlab are shown on Table 3. Table 3. Matlab training parameter values. Training Parameter Value Learning rate 0,001 Maximum iterations Infinite Goal (maximum mean squared error) 10-20 Maximum time 4 min Minimum gradient 10-20 Maximum number of validations Infinite 16 The index of performance of a NN used here is the absolute average of relative deviations, defined statistically [18] as, 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴% = 1 𝑁𝑁∑�𝑑𝑑𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒−𝑑𝑑𝑖𝑖𝑁𝑁𝑁𝑁� 𝑑𝑑𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒 100 𝑁𝑁 𝐵𝐵=1 (22) where N is the number of data in a curve, that is, the number of time steps evaluated, Xiexp is the extracted fraction regressed with the Weibull function, and XiNN is the fraction predicted by the network. As will be discussed in the Results and Discussion, for all the curves included in the training set and in the test set, their AARD% was calculated. In all cases, if this value is less than the tolerance, then the network is accepted, otherwise the training is repeated. The first step is the selection of the architecture of the network. As will be discussed later in the results, with 1 or 2 neurons in the hidden layer the result with the test is very good, but the training error increases. Conversely, a large number of neurons has the opposite effect. Clearly, there are an optimum number of neurons in the hidden layer. 5. Results and discussion Before we present the results on the neural net computing, we examine first the results of the statistical analysis given under the paragraph 3.1 5.1 Results of Statistical Study Because of the difficulties of working at high pressure, the measurements in an extraction run were not replicated. Before proceeding to the neuronal computing, two statistical tests were done, so we fitted the time response of the extraction runs (see Fig. 3) using expression (14) as regression equation, as suggested by the shape of the curves X (t), and 17 the theory. Both coefficients obtained from regression and their t-student statistical test results are summarized in the tables that follow. Table 4. Parameters of fitting Weibull's cumulative distribution function integral to experimental data points. P bar T ºC Q kg/ h ρ B kg/ m3 c (pValue) α (pValue) (i) b (pValue) n (ii) h n’ (iii) h c'' (iv) kg/m3 sol. kg/ m3 120 80 3 127 0,183 (0,103) 0,560 (0,13) 1,023 (0,516) 0,0111 0,057 0,33 120 80 3 227 0,174 (0,090) 0,510 (0,15) 1,216 (0,524) 0,0057 0,054 0,33 120 80 4 159 1,430 (2E-4) 1,524 1,738 (0,003) 0,1415 0,0065 0,33 0,33 120 80 4 227 1,227 (5E-5) 1,271 2,735 (0,018) 0,1305 0,0043 0,29 0,33 120 80 4 318 1,599 (6E-4) 1,837 1,713 (0,010) 0,1346 0,0028 0,37 0,33 120 80 5 127 2,194 (0,129) 2,279 1,097 (0,162) 0,1350 0,0092 0,41 0,33 120 80 5 227 1,983 (0,002) 2,175 1,276 (0,004) 0,0051 0,37 0,33 150 60 4 127 1,669 (2E-5) 1,772 1,340 (2E-4) 0,0155 0,73 0,74 150 80 4 127 2,114 (0,001) 2,338 1,255 (0,003) 0,0115 0,68 0,69 200 60 4 127 2,556 (8E-4) 3,073 1,309 (0,004) 0,0176 1,26 1,24 (i) For the first two operating conditions (Q = 3 kg/h) α has been fitted along with c and b because of the lack of data at higher extraction times. In all the other cases α has been calculated fixing the value of the function limit to infinity at 1 through the equation α=(Γ(1/b)c/b)b after finding c and b. (ii) n = zTϵ/u = ZTπRB 2ϵρs/Q, see terms in the nomenclature section. (iii) Note that n-n’ always results in the approximate constant 0,13 value, namely, the experimental error done when time was recorded. (iv) c’’ = cmLρs/Q as Cg(ZT,t) = (dX/dt)mLρs/Q, where mL is the mass of lanoline, Cg(zT,t) the concentration of lanolin in the solvent at the extractor outlet. In Table 4 we give the parameters obtained by fitting the data points with the integral of the Weibull function calculated analytically, eqn (18). In the table, the p-value from t-test of fitted parameters is provided. n is the time delay, eqn (17). At 120 bar, 80 ºC and 4 kg/h an evident error of experimental data consisting of an unusual initial delay has been overcome by setting a n value different from the theoretical one (n'). c’’ is the solubility value obtained from Weibull distribution, while sol. is the solubility extracted from Eychenne et al. [5]. 18 The results suggest that the exponent of the time variable (b) from eqn. (14) evolves from a value of 0,5 (equal to the theoretical one) to a value between 1 and 2 in most of cases at the extractor’s outlet. It is a reasonable behavior since both values are close and b increases as expected because of the profile of the curve. Solubility values obtained from the regressed equation (c’’) agree very well too those found experimentally by Eychenne et al. [5] for most of the operating conditions; besides, the first two regressions at 3 kg/h, are probably due to the lack of extracted fraction values at higher extraction times. The t-test applied to regression coefficients shown in Table 4 demonstrates the significance of these coefficients in the regression, as for most cases the p-value is lower than 0,05 and, therefore, they are significant factors of the regression with a 95% confidence level. Only the first two runs at 3 kg/h are out of this interval, likely because the available experimental values are those of the first linear slope of extraction; and at 120 bar, 80 ºC, 5 kg/h and 127 kg/m3, probably because the error on the experimental data (actually first value was excluded). The goodness of fit was also assessed through Fisher’s F statistical test, and for a further characterization of the regression, statistical parameters of the associated Weibull distribution were also found. These results are summarized on Table 5. Table 5. Results of Statistical Analysis P bar T ºC Q kg/h ρB kg/m3 Goodness of fit statistics Weibull parameters df R2 Adj. R 2 RMSE pVal μ (h) σ2 (h2) Me Mo 120 80 3 127 1 0,999 0,997 2,2E-03 0,014 1,747 2,917 1,244 0,054 120 80 3 227 1 0,999 0,997 2,3E-03 0,014 1,632 1,819 1,294 0,426 120 80 4 127 3 0,999 0,999 8.8E-03 1,3E-6 0,703 (*) 1,141 1,145 120 80 4 159 3 0,999 0,998 1,3E-02 6,2E-6 0,699 0,192 0,777 0,621 19 120 80 4 227 4 0,995 0,994 2,5E-02 1,5E-6 0,815 0,121 0,932 0,921 120 80 4 318 3 0,997 0,996 2,0E-02 1,9E-5 0,625 0,160 0,701 0,555 120 80 5 127 1 0,988 0,980 2,4E-02 0,019 0,456 0,191 0,473 0,187 120 80 5 227 2 0,999 0,998 8,7E-03 6,8E-5 0,504 0,159 0,413 0,169 150 60 4 127 4 0,998 0,998 1,1E-02 3,3E-8 0,599 0,204 0,512 0,250 150 80 4 127 3 0,997 0,996 1,7E-02 5,1E-6 0,473 0,144 0,391 0,154 200 60 4 127 4 0,991 0,988 2,7E-02 5,8E-7 0,391 0,091 0,338 0,158 (*) Incorrect value found. The fitting at these operating conditions was not considered for the neural network training because of the error in the experimental data. On one hand, goodness of fit parameters are shown, i. e. the degrees of freedom (df), the R- squared (R2), the adjusted R-squared (Adj. R2), the root mean squared error (RMSE), and the p-value (pVal) obtained from the F-test of the regression. On the other hand, the associated Weibull distribution parameters are given, i.e. the mean (μ), the variance (σ2), the median (Me) and the mode (Mo) of the distribution. As for the F-test results of the overall regression given on Table 5, besides the low RMSE values found, two parameters highlight. On one hand, most of adjusted R2 values are higher than 0,99, even for operating conditions with 3 and 4 degrees of freedom, suggesting a very good fitting of the model to the experimental data. On the other hand, in all cases the pvalues are lower than 0,05, proving the goodness of the fit at the 95% confidence level. The statistical parameters of the associated Weibull distribution of each fitting are also shown on Table 5. Those parameters describe how the extracted fractions and lanolin concentrations in the solvent phase are distributed over time. The mean values clearly decrease with increasing pressure, temperature, flow rate and packing density in most of the cases. As will be seen later, this trend agrees with the results obtained from neural network predictions, since the increase of those operating conditions leads to predicted extraction 20 curves with higher initial slope and an earlier end of extraction. As far as the variance is concerned, this shows the same relation with increasing operating conditions as the mean does, since when initial extraction slope is higher, extraction data is more concentrated around the mean extraction time. The median indicates the time halfway to the end of the extraction. Despite the median gives significant information about the total duration of a run, note that it is not the time where the extracted fraction is 0,5. It is always higher than the mean value, indicating that more than half of the lanolin present in the bed is extracted during the first half. Finally, mode is an interesting parameter as it indicates the time with the maximum reaction rate at the extractor outlet zT. This corresponds to the inflection point of the lanolin breakthrough curve. The low values of mode suggest that the extraction front quickly reaches the extractor outlet, deflecting in a slow extraction onwards until it is completed. In conclusion, the Weibull statistical distribution explains accurately the behavior of experimental data, so it is suitable for data augmentation. The values obtained will conform a semi-empirical training set for the design and implementation of the neural network. 5.2 Data augmentation The present work is based on the experimental data taken from the crude data reported in Fig 3. Only temperatures above 50ºC and pressures above 100 bar have been considered. On Table 1 the different states of operation in terms of P, T, Q and ρB, are given. For every state, measurements of the extracted fraction were done at only 6 times (sometimes only 4), within about 1 h for each run. As will be noted, for a given run the number of data points is very scarce. Therefore, a method for data augmentation is necessary. As discussed in 21 paragraph 3.1, the integral of the Weibull function can be used to regress the observed extraction values. To this purpose, an analytical integral for eqn. (19) was used. In practice, the parameters α and b, for each run were obtained. Fig 3. Experimental data from figures 5, 6b, 8b, 9b and 10 of Eychenne et al. [4]. Data available at 120 bar, 80ºC, 3 kg/h and 127 kg/m3 has not been represented as it matches with the exact same values of those at same conditions and 4 kg/h. In order to assess that data augmentation and that the fitted extraction curves are correct, the densities of solvent and other lanolin properties are necessary; these were taken from our previous work. 5.3 Fitting the experimental extraction data The fitting procedure used in data augmentation depends on the parameters c, α, b and n. The time n represents the time shift from which the X (t) starts to have a positive value, so that it corresponds to the point (n, 0). In order to calculate n, we apply its definition, that is, 22 n is the time taken by the solvent to reach the end of the bed. So n can be calculated simply as, 𝑛𝑛 = 𝜕𝜕𝑇𝑇𝜀𝜀 𝑢𝑢= 𝜕𝜕𝑇𝑇𝜋𝜋𝑅𝑅𝐵𝐵 2𝜀𝜀𝜌𝜌𝑆𝑆 𝑄𝑄 (23) Finally, when the parameters c, α, b and n for all the runs of Fig 3 are known, it is possible to calculate the value of Cg (t, zT) for the experimental runs. A sample of this calculation is shown on Fig 4. It is clear that the Weibull function fits the experimental data points very well. 23 Fig 4. Top curves are the fractions of lanolin extracted, X (t), fitted with the Weibull function after data augmentation (continuous lines), and compared with the measured extraction data. Bottom curves are the lanolin breakthrough at end of bed calculated by differentiation of the top curves. 5.4 Designing the network The method implemented in this work consists of an iterative training of the nonlinear autoregressive neural network until maximum AARD% goal both in training and test 24 predictions is satisfied (see Fig. 5). Most common training algorithms act as local optimization routines in order to adjust the weights with the goal to minimize the error between predicted values and experimental targets. Initial weight population is stochastic, namely takes a different value at each new run, which may lead to a different local optimum depending on the initial weights basin. In this case, iterating training is used as a means to achieve a multistart global optimization. Fig 5. Final hybrid nonlinear autoregressive neural network – Weibull statistical model scheme. As noted in paragraph 4. (Data treatment for neural networks), there is a substantial difference in the prediction error in terms of the AARD% for the training data and for the test data when the number of neurons in the hidden layer is changed. Recall that the ranges of the variables P,T, Q, ρB are those of Table 1. Our first result is that the default value applied in NARXNET (10 neurons) is unnecessarily large. Instead, we found that a reduced 31 Fig 9. First curve at left: extracted fraction curve for the optimum operating conditions (200 bar, 80 ºC, 4,56 kg/h and 318 kg/m3) to obtain the maximum extraction yield in 1 hour. 2Other two curves at right: extracted fraction curve for the optimum operating conditions to obtain at least 90% of lanolin extracted in 1 hour. Two possible solutions are shown; left: 120 bar, 60 ºC, 4,87 kg/h and 127 kg/m3; right: 150,4 bar, 60,46 ºC, 3,88 kg/h and 141,2 kg/m3. 6. Conclusions In this work, we have used neural network computing for modeling the behavior of the near-critical extraction of lanolin from raw wool, as a function of the variables that affect the extraction yield. These variables are: pressure, temperature, mass flow rate, packing density, and time. A recurrent neural network designed is capable to predict the extracted fraction for any set of variables, within the scope of the experiments available. Our reference experiments were those previously published by our research group. Once more, it has been shown that it is possible to design a neural computing model that has many applications in SCF extraction, without the need to develop and solve a complex 32 physical model of the extraction process, and without a knowledge of the laws that govern the mass transfer rate within the extraction vessel. The potential of this network has also been discussed, showing interesting applications, as it allows, among others, to develop studies to optimize the operating variables of the extraction process. A powerful global optimization algorithm, such as genetic algorithm, may be used with a low computing time cost, because once the network is trained, it works like a simple function with basic arithmetic operations. Moreover, it offers the possibility of applying weights to the variables so the desired variable is enhanced. Another development of this work has been to employ a statistical function that allows to model the lanolin breakthrough from a packed bed of wool from which lanolin is extracted. The function used is the Weibull cumulative probability distribution function. The integral of the Weibull function allows to regress the rather scarce extraction data and to produce a sufficient data augmentation to use in the neural algorithms. The use of the Weibull function is dictated by a physical model of extraction available to us from previous work. Conflict of Interest: The authors declare that they have no conflict of interest. 33 Nomenclature a Mass transfer area, 1/m A Const. defined by eq (12), 1/s AARD% Absolute average of relative deviations, eqn. (22) b Weibull parameter, Bi Biot number for mass, B Const. eqn. (16), kg bi NN bias neuron c Const eqn. (17) c' Const eqn. (14) Cg Fluid-phase concentration, kg/m3 C L Liquid-phase concentration, kg/m3 C L0 Initial lanolin liquid, kg/m3 D Lanolin diffusivity, m2/s K Henry equilibrium const. k G Fluid-side mass transfer coef, m/s Me, Mo Incomplete gamma function argument, eqn. (19) Median and mode, h n Time, s P Pressure, bar Q Mass flowrate, kg/s R Final radius, m 34 R B Bed radius, m r 0 Initial radius, m r v Mass transfer rate, kg/m3/s t Time, s T Temperature, ºC u Superficial velocity, m/s U External input in NN V Total bed volume, m3 V L Liquid volume, m3 w ij NN weights x Variable eqn. (18) X Extracted lanolin fraction, Y NN output z Length coordinate, m z T Total bed length, m Greek symbols and acronyms α Weibull parameter ε Porosity, ρ B Packing density, kg/m3 ρ L Lanoline density, kg/m3 35 ρs Solvent density, kg/m3 ρ w Wool density, kg/m3 σ2 Parameter eqn. (18) Variance, h2 µ Mean, h ANN Artificial neural network LM Levenberg-Marquardt SCFE Supercritical fluid extraction NARXNET Non-linear autoregressive with external NN 36 References [1] Cygnarowicz-Provost, M.; King, J.W.; Marmer, W.M.; Magidman, P. Extraction of Woolgrease with Supercritical Carbon Dioxide. J. Am. Oil Chem. Soc., 1994, 71, 222–225. [2] Kuo, B.S.; Kim, J.C.; Jeon, J.H.;, Bae, H.K. Desorption of Wool Grease from Greasy Wool with Supercritical Carbon Dioxide, Hwahak Konghak, 1992 , 30, 491-498. [3] Jones, F.W.; Bateup, D.R.; Dixon, S.R.; Gray, S.R. Solubility of Wool Wax in Supercritical Carbon Dioxide, J. Supercrit. Fluids, 1995, 10, 100-111. [4] Eychenne, V.; Sáiz, S.; Trabelsi, F.; Recasens, F. Near-critical Solvent Extraction of Wool with Modified Carbon Dioxide—Experimental results, J. Supercrit. Fluids, 2001, 21, 23-31. [5] Valverde, A.; Recasens, F., Extraction of Solid Lanolin from Raw Wool with Nearcritical Ethanol-modified CO2—A Mass Transfer Model, J. Supercrit. Fluids, 2019, 14. 151-161 [6] Bayona, J.M.; Erra P.; Moldovan Z.; Domínguez C.; Jover E.; Recasens F.; Larrayoz M.A., Method for Obtaining Lipid Fractions from Wool or Lanolin using Pressurized Carbon Dioxide, Patent WO 2002100990A1, Spanish Patent and Trademark Office, Madrid, filed 16.07.2005 [7] Domínguez, C.; Jover, E.; Garde, F.; Bayona, J.M.; Erra, P. Characterization of Supercritical Fluid Extracts from Raw Wool by TLC-FID and GC-MS, J. Am. Chem. Oil Soc., 2003, 80, 717-724. [8] Domínguez, C.; Erra, P.; Bayona, J.M. Physico-chemical and Dyieng Properties of Raw Wool Extracted by Pressurized CO2/Modifiers, Textile research, 2010, 80, 651-659. 37 [9] López-Mesas, M.; Carrillo, F.; Gutiérrez, M.C.; Crespi, M. Alternative methods for wool wax extraction from wool scouring wastes, Aceites y grasas, 2007, 58, 402-407. [10] Puiggené, J; Larrayoz, M.A.; Recasens F. Free Liquid-to-Supercritical Fluid Mass Transfer in Packed-beds, Chem. Eng. Sci., 1997, 52, 195-312. [11] Bhat, N.V.; McAvoy, T.J. Use of neural nets for dynamic modelling and control of chemical process systems, Comput. Chem. Eng., 1989, 14, 573-583. [12] Blayo, F.; Verleysen, M. Les réseaux de neurones artificiels, Presses Universitaires de France, Paris, 1996. [13] Bulsari, A.B. (Ed.), Neural networks for chemical engineers, Amsterdam: Elsevier , 1995. [14] Bulsari, A.B.; Palosaari, S. Application of Neural Networks for System Identification of an Adsorption Column, Neural Comput. & Applic.,1993 1 160-165. [15] A. Valverde, PhD thesis, to be published. [16] Fullana, M.; Trabelsi, F.;, Recasens, F. Use of Neural Net Computing for Statistical and Kinetic Modelling and Simulation of Supercritical Fluid Extractors, Chem. Eng. Sci., 2000, 55, 79-85. [17] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran 77, Cambridge University Press, 1993. [18] Gupta, A. Introduction to Deep Learning: Part 1, Chem. Eng. Prog. , 2018, 114, 22-29.