Neural networks for predicting mass transfer parameters in supercritical extraction

NEURAL NETWORKS FOR PREDICTING MASS TRANSFER PARAMETERS IN SUPERCRITICAL EXTRACTION

A.P.Fonseca, J.V.Oliveira and E.L.Lima^*

Programa de Engenharia Química / COPPE / Universidade Federal do Rio de Janeiro,

Cidade Universitária, C.P. 68502, 21945-970, Fax: [phone],

Phone: [phone], Rio de Janeiro - RJ, Brazil

E-mail: [e-mail]

(Received: November 3, 1999 ; Acepted: April 6, 2000 )

Abstract - Neural networks have been investigated for predicting mass transfer coefficients from supercritical Carbon Dioxide/Ethanol/Water system. To avoid the difficulties associated with reduce experimental data set available for supercritical extraction in question, it was chosen to use a technique to generate new semi-empirical data. It combines experimental mass transfer coefficient with those obtained from correlation available in literature, producing an extended data set enough for efficient neural network identification. With respect to available experimental data, the results obtained to benefit neural networks in comparing with empirical correlations for predicting mass transfer parameters.

Keywords: Neural network, Mass transfer coefficients, Supercritical carbon dioxide, Ethanol.

INTRODUCTION

Reliable and simple mass transfer models are necessary to design supercritical extraction (SCE) plants and determine the optimum operating conditions. The mass transfer performance of a small-scale countercurrent contactor is a first step in a long-term project aimed at developing reliable models for describing the performance of supercritical extraction columns. These models can be useful in the design and simulation of high-pressure processes dealing with liquid feeds such as hazardous wastes, contaminated waters, heavy oil residues, and organic/water mixtures (Lahiere and Fair, 1987).

However, compared with the number of thermodynamic and phase equilibrium studies, publications involving mass transport in supercritical media and correlations are very scarce (Lahiere and Fair, 1987; Tan and Liou, 1989; Bernard et al., 1993; Lim et al., 1995; Puiggené et al., 1997; Yu et al., 1999).

Due to the lack of mass transfer correlations, in the past, authors (Brunner, 1984; Madras et al., 1994) have been forced to use existing low-pressure correlations, such as that of Wakao-Kaguei (1982), which may not apply for packed beds with supercritical fluids. Other authors (Bernard et al., 1993) have proposed the use of logarithmic equations based on the two-film mass transfer model (Treybal, 1988) to simulate an extraction column and, through the numbers of liquid and gas transfer units, calculate the volumetric mass transfer coefficients. However, in this type of approach the model serves to estimate parameters and is not capable of extrapolating experimental information.

Neural networks (NNs) offer nice characteristics to deal with the estimation of many kinds of physical parameters. Although there is not any physical significance associated to NNs coefficients, they offer great flexibility to easily accommodate quite different situations. This represents their main advantage.

The purpose of the present work is to use NNs for predicting mass transfer coefficients for the supercritical carbon dioxide-ethanol-water system and it is organized as follows: first, in section 2, a description of the process and a classical modeling are shown, introducing a brief discussion of mass transfer and hydraulic characteristics in packed extraction columns. Then, in section 3, strategies to design a neural network model for predicting mass transfer coefficients are presented, considering sufficient and insufficient available data points. Section 4 is used to present the results and a discussion about the mass transfer coefficient predictions capability of the NNs compared to the one of an empirical correlation. Finally, some conclusions of this research are presented in section 5, underlying the most significant aspects of the contribution.

A CLASSICAL MODELING

Modeling of the SCE process is a complex task, mainly due to the difficulties associated to the obtention of experimental data near the critical point and to the scarcity of available empirical correlations.

Lim et al. (1995) investigated the applicability to SCE of a mathematical model developed for conventional spray and packed liquid-liquid extraction columns. The model accounts for flooding, dispersed-phase hold-up and mass transfer efficiency, and is based on the work of Seibert and Fair (1988) that correlates mass transfer coefficient and hydraulic characteristics (such as drop diameter, slip velocity, and operating hold-up).

The mass transfer model was derived using the two-film resistance theory for the calculus of the dispersed-phase overall mass transfer coefficient (K_od, m s^-1) in the form of the following equation,

(1)

where K_c and K_d are the mass transfer coefficient of the continuous and dispersed phase (m s^-1), respectively, and m_dc is a distribution coefficient. These parameters were calculated as described as follows.

For the dispersed-phase coefficient, many workers have used the predictive method of Handlos and Baron (1957),

(2)

where U_s is the slip velocity (m s^-1), and m_c and m_d the continuous and dispersed viscosities (Kg m^-1 s^-1), respectively. An alternative method, proposed by Laddha and Degaleesan (1978), has also been used,

(3)

where D is the molecular diffusion coefficient (m² s^-1), Sc_d is the Schmidt number based on the dispersed-phase (Sc_d = m_d / r_d.D), and r_d is the dispersed-phase density (Kg m^-3).

Depending on hydrodynamic conditions, either of these approaches can be valid. The function has been established from the intersection of curves based on Eqs. (2) and (3) and is used as a criterion of choice: when q < 6, Eq. (2) is used; when q > 6, Eq. (3) is used.

The continuous-phase coefficient can be extracted from the following correlation (Seibert and Fair, 1988),

(4)

where the ‘c’ subindex represents the continuous-phase, Re_c the Reynolds number (Re_c = U_s.r_c.d_vs / m_c), Sc_c is the Schmidt number, Sh_c is the Sherwood number (Sh_c = K_c.d_vs / D), d_vs is the Sauter mean drop diameter (m) and f_d is the fraction of free dispersed-phase hold-up in the contacting section.

The slip velocity is calculated by,

(5)

where U_d and U_c are the superficial dispersed and continuous phase velocities (m s^-1), respectively, and e is the packing void fraction.

The following equations are solved to calculate the operating hold-up,

(6)

(7)

where a_p is the total packing surface per volume of column (m^-1), g is the gravitational constant (m s^-2), Dr is the density difference between two phases (Kg m^-3), U₀ is the characteristic velocity (m s^-1) calculated through the Eq. (7) with C₀ = 1.63 for packed column (Knit mesh packing).

Lim et al. (1995) evaluated the model performance against experimental data for the supercritical carbon dioxide-ethanol-water system (Table 1). The physical properties of the experimental system, the geometries of the extraction column, the type of packing that were used and the efficiency of the model are summarized in the original work.

The experimental dispersed-phase overall mass transfer coefficients were computed from a relationship between the number of transfer units (NTU) and the height equivalent theoretical plates (HETP), that is also described in the original work.

MASS TRANSFER COEFFICIENTS PREDICTION

Because of their flexibility to accommodate different situations, NNs are natural candidate for complex SCE processes. Although NNs parameters have not any physical meaning, the flexibility characteristic of these paradigms easily justified their use in place of traditional correlations.

One problem that frequently arises during the identification process is related to the number of available experimental points. Many times, because of time and/or economical reasons, this number is not sufficient for an efficient identification.

Two situations were analyzed in this work: sufficient and insufficient amount of data points.

Sufficient Amount of Data Points

There are various considerations associated with the design of the neural network, the most essential are being to recognize the input and output variables.

As the objective of this work is to determine the dispersed-phase overall mass transfer coefficient (K_od) or the volumetric mass transfer coefficient (K_od.a, s^-1), where "a" is the specific packing surface (m^-1), it seems natural to choose this parameter as the output of the NN.

Analyzing the Eqs. (1-7), it can be observed that all the parameters are functions of U_d, U_c and f_d, or functions of T and P (physical and thermodinamics properties), where T is the temperature (K) and P is the pressure (bar). Therefore, the input variables can be chosen as T, P, U_d, U_c and f_d. However, analyzing the 23 experimental data set available in Table 1, it can be identified only 2 U_c values. It was made on purpose since one of the Lim’s conclusions is that K_od.a depends on U_d/ m_dc only for one U_c value. Then, adopting U_c = 0.022.10^-2 m.s^-1 (maximum available-data number; 18), it will be chosen T, P, U_d and f_d as input variables. After some attempts, it can be observed that to insert the m_dc (thermodynamic parameter) as input improved the NN’s performance. Thus, the resulting model structure is shown in Fig. 1. This model will be identified as "Re".

Insufficient Amount of Data Points

It can be seen on the literature, studies determining the mass transfer coefficients in optimum operating conditions, for example T=313.2 K and P=10.1 MPa (Bernard et al., 1993). In this case, the amount of experimental data is reduced (minimum available-data number; 7) and may not be high enough for an efficient NN identification. Thus, another neural network, which will be identified as R.Tn, has been proposed, using a technique to generate new semi-empirical data, named "Tn" (Fonseca et al., 1999).

This technique may generate a new semi-empirical point (y^se) from any kind of available phenomenological information (model, f^m), correcting this value in terms of the experimental data information (f^e), in the following way,

(8)

(9)

(10)

where are independent variables (NN inputs in this case) and w_i represents an appropriate weighting factor.

U_d, U_c and f_d represent the possible input variables, since T and P are remained fixed. It was verified that to insert the parameter S/F – solvent-to-feed ratio – as NN input, the NN’s performance is improved.

RESULTS AND DISCUSSION

The NNs built in this paper were of the feedforward type, with three layers of neurons, using a sigmoidal function as activation. As a result of the analysis in section 3, the number of input and output layer neurons for Re was 5 and 1, and for R.Tn was 4 and 1, respectively. The number of hidden neurons was determined by a trial and error procedure. The NNs were trained using backpropagation with a cross validation technique (Morris et al., 1994; Ted Su and Mc Avoy, 1997).

Figure 2 shows the relationship between K_od.a and U_d/ m_dc as predicted by Re and Lim’s model, compared with the experimental data used for the NN identification (runs: 1, 3, 4, 6, 7, 9, 10, 16, 17, 19, 20, 21 and 23).

It can be observed that Re reproduces the experimental data set more accurately than Lim’s model. The percentual the average absolute deviations (AAD), were 10.68% and 1.52% for Lim’s model and Re, respectively.

Figure 3 presents predicted values for Re and Lim’s model, compared with experimental data not used during the NN identification procedure (runs: 2, 5, 8, 18 and 22). These results confirm the superiority of the NN over Lim’s model; the AAD performance indexes are 7.32% and 24.71%, respectively.

For the R.Tn model identification (based on 7 experimental data, runs: 7, 9, 10 11, 12, 14 and 15) 53 new semi-empirical points were generated, covering the entire operational domain. The validation of the predicting capacity of each model was performed as runs 8 and 13.

Figure 4 shows the relation between K_od.a and S/F, as predicted by Re, R.Tn and Lim’s model, compared with the experimental data used for the NNs identification (runs: 7, 9, 10 11, 12, 14 and 15). The corresponding AAD values were 0.26%, 0.88% and 5.85%. These results again indicate the superiority of the NN approach. Although the number of experimental points was significantly reduced, the Re model shows the best performance.

On the other hand, as shown in Figure 5, when the NNs were used for predicting values not used in the identification stage, the R.Tn model was clearly superior (the AADs values were 3.76%, 11.49% and 0.76% for Lim’s model, Re and R.Tn, respectively).

CONCLUSIONS

The main objective of this work was to analyze the use of neural networks for the prediction of parameters in the complex supercritical fluid extraction process.

The motivation for this investigation was the scarcity model specific of the available correlations for this kind of parameters. The study was performed using the supercritical carbon dioxide-ethanol-water system as the supercritical extraction process.

Although based on a reduced number of experimental data (a quite frequent situation in the real world of the process engineering), results gave a clear indication that the NNs approach represents a nice alternative.

Two important observations must be made in relation to their investigation. Even using a number of experimental points below the recommended minims for NNs identification, these models behavior better than a model based on phenomenological concepts. When the number of experimental data is extremely reduced, but there is available some phenomenological information, it seems reasonable to use the semi-empirical data to identify NNs models.

A thorough investigation of the results here observed must be performed, but certainly they represent a clear indication of the potentiality of the NNs to substitute traditional correlations approaches for parameters prediction.

ACKNOWLEGMENTS

Financial support from Conselho Nacional de Pesquisas and Fundação Universitária José Bonifácio is gratefully acknowledged.

NOMENCLATURE

a specific packing surface [m^-1] a_p total packing surface per volume of column [m^-1] D molecular diffusion coefficient [m^-2 s^-1] d_vs Sauter mean drop diameter [m] f output variable; f^e experimental; f^m model g gravitational constant [m s^-2] K mass transfer coefficient [m s^-1] K_o overall mass transfer coefficient [m s^-1] m_dc distribution coefficient P pressure [bar] T temperature [K] U superficial velocity [m s^-1] U_s slip velocity [m s^-1] U₀ characteristic velocity [m s^-1] x_ji independent variables

Greek Letters

e packing void fraction m viscosity [Kg m^-1 s^-1] r density [Kg m^-3] f_d hold-up

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*

To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  16 Mar 2001
• Date of issue
  Dec 2000

History

• Received
  03 Nov 1999
• Accepted
  06 Apr 2000