Using hybrid neural models to describe supercritical fluid extraction processes

Using hybrid neural models to describe supercritical fluid extraction processes

A. P. FONSECA, G. STUART, J. V. OLIVEIRA and E. LIMA¹

Programa de Engenharia Química / COPPE / Universidade Federal do Rio de Janeiro, Cidade Universitária, C.P. Box 68502, 21945-970, Rio de Janeiro - RJ Brazil, Phone [phone], Fax: [phone], ¹E-mail: [e-mail];

(Received: February 8, 1999; Accepted: May 19, 1999)

Abstract - This work presents the results of a hybrid neural model (HNM) technique as applied to modeling supercritical fluid extraction (SCFE) curves obtained from two Brazilian vegetable matrices. The serial HNM employed uses a neural network to estimate parameters of a phenomenological model. A small set of SCFE data for each vegetable was used to generate a semi-empirical extended data set, large enough for efficient network training, using three different approaches. Afterwards, other sets of experimental data, not used during the training procedure, were used to validate each approach. The HNM correlates well withthe experimental data, and it is shown that the predictions accomplished with this technique may be promising for SCFE purposes.

Keywords: Supercritical fluid extraction, Modeling, Artificial neural network, Brazilian rosemary oil, pepper oil.

INTRODUCTION

Recently much attention has been given to modeling the supercritical fluid extraction (SCFE) of natural products from a variety of plants. Most models are based on a simplified phenomenological description of the process, making use of rigid assumptions (Reverchon, 1997). Few parameters are fitted using a set of experimental data and, generally, the model obtained is not capable of extrapolating experimental information with good accuracy, an undesirable situation from an engineering point of view. During the last ten years different research groups and companies have been working on the application of artificial neural networks (NNs) for modeling chemical engineering processes, especially those presenting highly nonlinear behavior (Morris et al., l994; Su and McAvoy, l997; Baughman and Liu, l995).

Though NNs present some advantages, like generalization characteristics and low computing time, they often have some drawbacks, such as too many nonphysical meaning parameters, a long training period and the need for a considerable amount of experimental data, which is not always available.

In order to overcome these problems, hybrid neural models have been proposed, since they ensure the physical meaning of some model parameters by incorporating process constraints given by the knowledge of some kind of physical model (Thompson and Kramer, l994; Psichogios and Ungar, l992). Although these approaches require smaller data sets than "black box" networks, they still require data sets big enough to cause problems in many practical situations. In this case a solution can be found by creating a certain amount of semi-empirical data, based on a weighted combination of experimental and modeling data (Tsen et al., l996).

SCFE is an interesting field for the application of HNMs due primarily to the high sensitivity of the phenomenological models with regard to small changes in the process variables, affording poor results. In this work, an HNM construction using a semi-empirical extended data set, generated by three different approaches, and the model proposed by Lack, as presented by Sovová (Sovová, l994), are shown to be good techniques for correlating and predicting the extraction curves for the SCFE of Brazilian rosemary oil (Rosmarinus officianalis L.) and pepper oil (Piper nigrum L.)(Coelho et al., l996; Coelho et al., l997; Ferreira, l996). These systems were chosen due to the availability of experimental data and empirical correlations for the mass transfer coefficient.

THEORY

The Hybrid Neural Model

Neural networks represent one of the most prominent alternatives for the identification of nonlinear systems in multidimensional spaces. To overcome some of the drawbacks of these paradigms - difficulties in physical interpretation of the parameter, high structure dimension, long training periods, requirement of a lot of experimental data - different approaches have been proposed to include a priori knowledge in the network structure (Psichogios and Ungar, l992; Thompson and Kramer, l994; Scott and Ray, l993), resulting in HNMs. One that seems to be quite efficient combines phenomenological models with neural networks. As pointed out by Thompson and Kramer (1994), two types of combination can be built for this approach: the phenomenological process model in parallel with the NN or the phenomenological model in series with the NN. The first strategy only corrects for model deviations, while the second one estimates some model parameters.

As a priori knowledge, we have adopted in this work the phenomenological model proposed by Lack to represent the experimental extraction curves. A detailed derivation of this model has been presented in the literature (Sovová, l994). After some simplifications, the material balances for an element of the extraction bed are given by:

(1)

(2)

where r is the solvent density (Kg m^-3), e the bed porosity, y the solvent-phase concentration related to solute-free solvent, x the solid-phase concentration related to solute-free solid, u the superficial velocity of solvent (m s^-1), h the axial coordinate (m), r_s the density of solid-phase (Kg m^-3), t the time (s) and J(x,y) the mass transfer rate (Kg m^-3 s^-1).

The boundary and initial conditions are

(3)

(4)

respectively, where x₀ is the concentration related to the solute-free solid-phase at t = 0.

Then, the extraction curve can be determined by the following equation:

(5)

where e(t) is the ratio of the mass of extract relative to the mass of the solute-free solid-phase and H is the height of bed.

This model was chosen because all the extraction curves are described as a function of the solvent-phase mass transfer coefficient and it seems to be suitable whenever information about the solid-phase mass transfer is not available.

According to Lack, the mass transfer term J(x,y) must represent two different mechanisms: diffusion in the solvent-phase (easily accessible solute) and diffusion in the solid-phase (after the solid-phase concentration has decreased to a certain x_k value, the easily accessible solute solid-phase concentration). The description of this situation has motivated intense research activity and several expressions have been proposed for the corresponding mass transfer rate.

Assuming some relatively simple expressions for the mass transfer term in each different region, Lack obtained analytical solutions for the above equations. Later, introducing a slight modification to Lack’s original work, Sovová obtained the following analytical expressions to describe the extraction curve:

(6)

where

where K_a is the volumetric overall mass transfer coefficient (Kg m^-3 s^-1), K_m is the mass transfer coefficient (m s^-1), y_r is the solubility, r₀ is defined as x₀ / x_k, t_m represents the time when the easily accessible solute becomes exhausted at the solvent entrance and t_n is the time when this situation reaches the end of the system. Also, k is an adjustable parameter used to modify mass transfer expressions. In accordance with Lack, parameter k introduces the resistance to the solid-phase mass transfer observed with the gradual exhaustion of the easily accessible oil from the particle surface.

Analyzing the parameters involved in Eq. (6), it can be observed that r₀ and t_m have fixed values; all the other parameters are functions of K_a (or K_m ), y_r and k. Therefore, it seems natural to choose these three parameters as the outputs of NNs in a HNM. The resulting model structure is shown in Figure 1.

Figure 1:
Schematic diagram of a hybrid neural network structure.

Considering a supervised training procedure for the NNs, a target data set must be available. In this case the set must include known values of the three outputs: K_m, y_r and k. The availability of these values is different for each case.

The mass transfer coefficient describes the transfer rate when all three mechanisms, diffusion and natural and forced convection, are taken into account. Parameter K_m is calculated from experimental data using Fick’s law that considers the concentration gradient in the solvent-phase (logarithmic mean) as the driving force for mass transfer.

The development of semi-empirical correlations to describe the mass transfer between solid and solvent phases represents an attempt to explain the complexity presented by these systems. These correlations have normally dimensionless numbers like the Sherwood number (Sh = K_md_p / D), which relates the mass transfer coefficient (K_m) to the diffusion coefficient (D, m^-2 s^-1) and a length parameter (d_p, particle diameter, m). In this work the diffusion coefficient was estimated using the correlation proposed by Wilke and Chang (l955).

One of the basic characteristics of mass transfer phenomena is the kind of convection involved in the transfer process. Verifying by experimental results from pepper oil the presence of natural and mixed convection, Ferreira (l996) proposed the following correlation for this system:

(7)

where Ra is the Rayleigh number (Ra = GrSc), Re the Reynolds number (Re = u_sd_p / n), Sc the Schmidt number (Sc = u / D), Gr the Grashof number (Gr = d_p³grDr / m²), u_s the interstitial velocity (m s^-1), n the dynamic viscosity (m² s^-1), g the gravitational constant (m s^-2), Dr the absolute difference (r_m - r), r_m the density of the solute-solvent mixture considering a pseudo-binary mixture (Kg m^-3) and m the viscosity (Kg m^-1 s^-1).

For rosemary oil, also using experimental results, Cremasco (1997) observed the presence of mixed convection and proposed the following correlation :

(8)

The existence of natural convection is a criterion to be used when choosing a mass transfer correlation to be applied in a model suchas Lack’s.

Semi-Empirical Extended Data Generation Methods

One important drawback of the neural network paradigm is the huge amount of information that is required for an efficient parameter fitting. Although the significance of this problem is reduced in the case of the HNMs, it can still cause some practical difficulties. Tsen et al. (1996) recently proposed a methodology to overcome this drawback by generating an extended set of semi-empirical data large enough for appropriate network training. Basically, it is based on a first-order Taylor series expansion, generating new data through interpolation and extrapolation using gradient information from a phenomenological model.

Results obtained by Tsen et al. (1996) and later by Milanic et al. (1997) indicate that this technique, when appropriately used, can be very efficient for neural network construction.

The methodology proposed by Tsen et al., here denominated MST1, can generate semi-empirical data sets combining experimental data with information from a phenomenological model, using the linear part of a Taylor series expansion.

Let x₁, x₂,..., x_M represent, for a certain nonlinear mapping, M independent input variables and f the corresponding output variable. Also, let f₁^e(x₁₁^e, x₂₁^e,..., x_M1^e), f₁^e(x₁₂^e, x₂₂^e,..., x_M2^e), and f_N^e(x_1N^e, x_2N^e,..., x_MN^e) represent N experimental data points. Given a new set of M independent input variables (x₁^se, x₂^se,..., x_M^se), for which the corresponding semi-empirical output value is sought, f^se(x₁^se, x₂^se,..., x_M^se), the expression for this value is given as a linear combination of first-order Taylor series expansions of the available experimental data, as follows:

(9)

where x_j^se is the semi-empirical input, x_jk^e the empirical input, f_k^e the empirical output in experimental coordinate, f_k^m the model output in experimental coordinate, f^se the semi-empirical output and w_k^se a weighting factor which is inversely proportional to the distance between semi-empirical and actual experimental points, given by

(10)

When the phenomenological model is not simple enough, this approach could result in some computational difficulties, associated with the numerical calculation of the derivatives involved. In order to overcome this possible disadvantage for certain cases, data generation can be carried out by the following simplification (MST0) of Eq. (9):

(11)

This approach represents a direct interpolation of the experimental data, without using any phenomenological information. In this case it is evident that the available experimental data must reasonably cove the operating region.

Another alternative to avoid numerical differentiation, but still using the phenomenological model information, is a new methodology, which we are calling MEM (Modeling Error Methodology) in this paper. In this case the phenomenological information is taken into account by a term that represents the difference, at each experimental point, between the experimental and the model output variable values: a modeling error.

Thus, the generation of a new semi-empirical point will be made by adding to the theoretical output the value d^se, as indicated in the following equations:

RESULTS AND DISCUSSION

As any other modeling approach, hybrid neural modeling is more an art than a science. Each new problem must be analyzed carefully, looking for the available a priori information in the form of first principles models, established empirical correlations or even ad-hoc empirical correlations. These considerations are especially important when we intend to generate some kind of extended data set based on an approximated model (MST1, MEM).

The HNMs for the SCFE of pepper and Brazilian rosemary oil are based on Eq. (6), where K_m, y_r and k are described by NNs. The available experimental data are presented in Table 1, for pepper oil (Ferreira, l996), and Table 2, for Brazilian rosemary oil (Coelho et al., l996; Coelho et al., l997). As can be seen, the amount of experimental data is reduced and certainly not high enough for the efficient training of the NNs. In this work we extended the available data, applying a different solution to each case.

Table 1: Empirical parameters for pepper oil

Run	Na	T (K)	P (bar)	r (Kg/m3)	rm (Kg/m3)	u (m/s).105	yr (g oil/g CO2)	Km (m/s).104	k
1	14	303.15	150	830	861	23.27	0.089	1.087	0.03
2	15	303.15	150	830	858	20.85	0.089	0.853	0.026
3	15	313.15	150	748	797	19.43	0.0932	0.735	0.05
4	15	313.15	150	748	796	16.54	0.0932	0.600	0.015
5	16	313.15	150	748	802	8.11	0.0932	0.594	0.002
6	15	323.15	150	656	723	21.37	0.0858	0.864	0.018
7	18	313.15	200	830	859	8.25	0.1044	0.398	0.044
8	13	313.15	200	830	852	68.45	0.1044	1.485	0.008
9	19	313.15	200	830	862	7.60	0.1044	0.816	0.008
10	19	313.15	300	928	935	5.01	0.1365	0.639	0.055
11	16	313.15	300	928	934	54.64	0.1365	1.446	0.12
12	19	323.15	300	880	895	6.06	0.1255	0.249	0.17
13	11	323.15	300	880	892	71.50	0.1255	1.250	0.035

^a

Number of data points

The new extended points are formed by the input variables (T, P, r, r_m, u), where T is the temperature (K) and P the pressure (bar), and the corresponding outputs for each case are K_m, y_r and k.

To avoid any inconsistency during the generation of a new input value (for example, an unrealistic density value), we generated temperature, pressure and velocity values covering all the operational domain and used empirical correlations, based on density values presented in Tables 1 and 2, to calculate the corresponding densities.

Table 2: Empirical parameters for rosemary oil

Run	N	T (K)	P (bar)	r (Kg/m3)	u (m/s).105	yr (g oil/g CO2).103	Km (m/s).104	k
1	17	310.15	100	686.5	2.967	1.72	0.5160	0.00198
2	19	310.15	120	750.4	2.283	2.45	0.2569	0.01175
3	20	310.15	140	788.2	2.371	2.15	0.2514	0.01372
4	18	310.15	160	816.5	2.250	2.00	0.2285	0.00615
5	16	320.15	100	449.4	4.067	0.55	0.9716	0.0001
6	19	320.15	120	633.7	2.667	1.28	0.5378	0.00039
7	14	320.15	140	704.2	2.000	2.50	0.4675	0.00214
8	20	320.15	160	746.8	2.450	2.80	0.2368	0.01687

As already mentioned, for each output we have used a different approach to generate the extended data, because the quality of the information available was completely different in each case.

Eqs. (7) and (8) were used to calculate extended data for K_m in the pepper oil and Brazilian rosemary oil systems, respectively.

Because there is not a phenomenological model or any kind of established correlation for the solubility variable, y_r, the adopted solution was to build a nonlinear model using the experimental data. This model was then used to generate extended data.

The calculation of extended values for k seems to be more complicated because it is not a variable, but an adjustable parameter. Despite this drawback, after fitting the parameter k for each experimental run, empirical correlations were developed for both essential oils.

All the six NNs built in this work were of the feedforward type, with three layers of neurons, using a sigmoidal function as the activation. The number of input and output layer neurons was determined by the problem: 5 input neurons for the pepper oil problem; 4 input neurons for the Brazilian rosemary oil problem; 1 output neuron in all cases. The number of hidden neurons was specific for each case and it was determined by trial and error. The NNs were trained using backpropagation with a cross validation mechanism (Morris et al., l994; Su and McAvoy, l997). For this validation procedure we used runs 2, 6 and 8 and runs 2 and 7 for pepper and Brazilian rosemary oil, respectively.

Considering that the available number of experimental data represent only 10% of the total number of points required for efficient NN training, the pepper oil data set was extended by 90 points and the Brazilian rosemary oil data set by 54 points.

The predicted extraction curves obtained from the HNMs for both systems, using different approaches for data extension, were compared with the predicted values obtained from Lack’s model and with the experimental values. The main difference between both modeling approaches is that for the HNM the only information we need for prediction are the input variables T, P, u, r and r_m(when applied), but for Lack’s model, we also need values for K_m, y_r and k. This characteristic of the HNM represents more flexibility to deal with different new situations.

The average absolute deviations from the experimental values of the predicted values for the extraction curves, using the model proposed by Lack and the HNMs for each methodology presented in this work, are shown in Table 3 for pepper oil and Table 4 for Brazilian rosemary oil. It can be observed from these tables that the HNMs correlate with the experimental data as accurately as Lack’s model, but using much less information, as already indicated. It can also be seen that the MEM procedure generally presents the best performance.

Table 3: Average absolute deviations from experimental e(t) (AAD%)

Run	Lack model	MST0	MST1	MEM
1	24.85	29.93	18.34	18.34
2	7.78	9.22	9.22	7.70
3	3.78	5.14	5.14	6.13
4	4.51	12.50	12.50	11.10
5	4.50	8.57	8.57	4.68
6	6.01	6.87	6.87	4.10
7	4.12	9.69	9.69	7.72
8	6.22	8.16	7.11	6.38
9	4.57	6.00	6.00	6.50
10	5.30	5.51	5.51	6.37
11	2.36	8.47	8.47	10.43
12	4.17	9.85	9.85	6.99
13	3.18	6.98	6.98	4.60

a

Table 4: Average absolute deviations from experimental e(t) (AAD%) for rosemary oil

In Figure 2 the extraction curves for the pepper oil system, predicted by the HNM and correlated with Lack’s model, are compared with experimental data for one representative run #12 and a satisfactory behavior can be observed. As expected from Table 3, HNN-MST1 and HNN-MST0 have been represented by a single curve.

Figure 2:
Experimental and correlated extraction curves for pepper oil, run #12.

The best data extension approach (MEM) has a predicted quality similar to that of the phenomenological Lack model, but it offers more flexibility because it does not explicitly depend on K_m, y_r and k values, as confirmed by Figure 3 for run #2 and Figure 4 for run #6.

Figure 3:
Comparison of Lack’s model and HNM-MEM predictions for pepper oil, run #2. AAD%: 7.78 and 7.70 for Lack and HNM-MEM, respectively.

Figure 4:
Comparison of Lack’s model and HNM-MEM predictions for pepper oil, run #6. AAD%: 6.01 and 4.10 for Lack and HNM-MEM, respectively.

Finally, Figure 5 shows the results obtained using the HNM-MST0 approach for run #2 in the Brazilian rosemary oil system. Although it seems to be a poor result, we must emphasize that in this case there was a very small amount of experimental data. Also, the unusual oscillation of these data is an indication of some experimental problems. Even in this case it is clear that the HNM, based on an extended data set obtained via MST0, leads to satisfactory predictions that compare favorably with those of the phenomenological model.

Figure 5:
Comparison of Lack’s model and HNM-MST0 predictions for rosemary oil, run #2. AAD%: 12.45 and 9.50 for Lack and HNM-MST0, respectively.

CONCLUSIONS

The main objective of the research work here reported was to explore some modeling alternatives to improve extraction curve prediction for SCFE processes. Hybrid neural models were chosen because of their favorable practical characteristics: they allow physical interpretation of the model parameters, result in low dimension structures, require smaller data sets for training stages etc. The normal technique used for extraction curve prediction is based on phenomenological models. Many of them can be found in the scientific literature, and in this work we have used the model proposed by Lack (Sovová, l994), which has provided satisfactory results for the systems studied: Brazilian rosemary and pepper oils. The main problem is that for each new situation it requires the determination of three parameters. A serial HNM based on this model, but with neural networks instead of the required parameters, was successfully applied.

The frequently ocurring practical problem of reduced number of available experimental data was solved by three different approaches, based on different principles. One uses a weighted sum of first-order extrapolations, with derivatives calculated from an approximated phenomenological model (MST1). Another is based on a weighted sum of modeling errors (MEM). The last and most elementary one is a simple zero-order interpolation between experimental data (MST0).

Analyzing the results obtained, it can be concluded that the HNM approach is a good alternative for the description of the SCFE process. The main advantage is that, after a satisfactory model adjustment, the information required to describe a new situation is much less than that required when using more rigorous phenomenological models.

During this research it became clear that modeling, and especially hybrid modeling, is an art, and good results depend on choosing the best tools for each case, a selection for which the quality of the information available is crucial.

A new approach to data extension, MEM, proved to be quite efficient for the generation of semi-empirical points. This method uses first-principle information in a very direct way, avoiding the need for the derivative calculations involved in Taylor series expansions.

These ideas must be examined more extensively before they can be considered a standard modeling approach, and our results must be considered a good example of their potentiality for the SCFE area.

NOMENCLATURE

d_p particle diameter [m]

D diffusion coefficient [m^-2 s^-1]

e ratio of the mass of extract relative to the mass of solute-free solid-phase

f output variable; f^e experimental; f^se semi-empirical

g gravitational constant [m s^-2]

h axial coordinate [m]

H height of the bed

J mass transfer rate [Kg m^-3 s^-1]

k adjustable parameter in Equation (6)

K_a volumetric overall mass transfer coefficient [Kg m^-3 s^-1]

K_m mass transfer coefficient [m s^-1]

M number of independent input variables

N number of experimental data points

P pressure [bar]

r₀ x₀/x_k

t time [s]

T temperature [K]

u superficial velocity of solvent [m s^-1]

x solid-phase concentration related to solute-free solid

x_i independent input variable; x_i^e experimental; x_i^se semi-empirical

x_k easily accessible solute solid-phase concentration

x₀ solid-phase concentration related to solute-free solid at t=0

sy solvent-phase concentration related to solute-free solvent

y_r solubility

z_w defined in Equation (6c)

Z defined in Equation(6b)

Greek Letters

d^se parameter in Equation (13)

e bed porosity

m viscocity [Kg m^-1 s^-1]

n dynamic viscocity [m² s^-1]

r solvent density [Kg m³]

r_m density of a pseudo-binary solute-solvent mixture [Kg m^-1 s^-1]

r_s solid-phase density [Kg m³]

t dimensionless time defined in Equation (6d)

t_m dimensionless time when the easily accessible solute becomes exhausted at the solvent entrance; at t_n this situation reachs the end of the system

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Publication Dates

• Publication in this collection
  16 Dec 1999
• Date of issue
  Sept 1999

History

• Accepted
  19 May 1999
• Received
  08 Feb 1999