Mathematical modeling of dispersion in single interface flow analysis

Analytica Chimica Acta 663 (2010) 178–183

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Mathematical modeling of dispersion in single interface flow analysis S. Sofia M. Rodrigues, Karine L. Marques, João A. Lopes ∗ , João L.M. Santos, José L.F.C. Lima REQUIMTE, Servic¸o de Química-Física, Faculdade de Farmácia, Universidade do Porto, Rua Anibal Cunha, 164, 4099-030 Porto, Portugal

a r t i c l e

i n f o

Article history: Received 8 September 2009 Received in revised form 22 January 2010 Accepted 28 January 2010 Available online 4 February 2010 Keywords: Single interface flow analysis Optimization Experimental design Multivariate linear regression Feed-forward neural networks

a b s t r a c t This work describes the optimization of the recently proposed fluid management methodology single interface flow analysis (SIFA) using chemometrics modelling. The influence of the most important physical and hydrodynamic flow parameters of SIFA systems on the axial dispersion coefficients estimated with the axially dispersed plug-flow model, was evaluated with chemometrics linear (multivariate linear regression) and non-linear (simple multiplicative and feed-forward neural networks) models. A Doptimal experimental design built with three reaction coil properties (length, configuration and internal diameter), flow-cell volume and flow rate, was adopted to generate the experimental data. Bromocresol green was used as the dye solution and the analytical signals were monitored by spectrophotometric detection at 614 nm. Results demonstrate that, independent of the model type, the statistically relevant parameters were the reactor coil length and internal diameter and the flow rate. The linear and non-linear multiplicative models were able to estimate the axial dispersion coefficient with validation r2 = 0.86. Artificial neural networks estimated the same parameter with an increased accuracy (r2 = 0.93), demonstrating that relations between the physical parameters and the dispersion phenomena are highly non-linear. The analysis of the response surface control charts simulated with the developed models allowed the interpretation of the relationships between the physical parameters and the dispersion processes. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Since the introduction of non-segmented continuous flow analysis by Skeggs [1], and particularly with the emergence of flow injection analysis (FIA) [2], flow-based methodologies have proved to be valuable tools in analytical chemistry. Recently, a new strategy for fluids management called single interface flow analysis (SIFA) was proposed [3]. This strategy differs from the traditional concept of continuous flow analysis because it does not depend on the insertion of defined volumes of sample and reagent in the analytical path. SIFA is instead based on the mutual penetration of sample and reagent zones at the single interface reaction created when sample and reagent solutions meet before the detection system. The optimization of the experimental conditions is a critical step for developing a flow-based methodology. Although the classical trial and error method may be attempted to carry out this task it requires a large number of independent experiments and sometimes the optimal experimental conditions are never identified. In previous works involving FIA [4,5] and sequential injection analysis (SIA) [6] it was already shown that the use of mathematical models

∗ Corresponding author. Tel.: +351 222078994; fax: +351 222078961. E-mail address: [email protected] (J.A. Lopes). 0003-2670/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.aca.2010.01.051

could be helpful for the development and optimization of the analytical system. Mathematical models are helpful, especially when coupled with optimization algorithms to devise configurations for specific purposes [7]. In this context, and mostly for FIA, different models have been proposed comprising both “black-box” and first principles models (probabilistic and deterministic) as described in a review by Kolev [8]. More recently by Pai et al. adopted a dispersion–convolution model for simulating peak shapes in a single-line FIA system [9]. These models are essentially based on the axially dispersed plugflow model that has demonstrated to offer the best compromise between the mathematical simplicity and precision [10]. However, due to the different nature of the SIFA method, mainly the particular reaction zone formation process, available flow injection mathematical models based on the axially dispersed plug-flow model are of limited utility. Therefore, to carry out faster and finest SIFA optimization procedures new mathematical models are required. Taking into account the specificities of SIFA, the importance of the physical and hydrodynamic flow properties of each SIFA system and the mutual sample/reagent inter-dispersion process that occur at the dynamic single interface the estimation of the axial dispersion coefficient is of utmost importance to model the behavior of the SIFA system and to predict analytical signal magnitude. In this way, it would be possible to anticipate the perspective of gathering enough information as to enable the implementation of simplified

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calibration routines based, for instance, on the utilization of a single standard solution. The aim of this work was to estimate and compare empirical models of distinct structure (e.g., linear and non-linear) to describe the effect of different system operational parameters in the dispersion process that takes place at the single interface. Five physical and hydrodynamic flow parameters were evaluated: reactor coil length, reactor configuration, reactor internal diameter, flow-cell volume and flow rate. The reaction interface was defined by the axial dispersion coefficient estimated from experimental profiles using the axially dispersed plug-flow model described by Kolev [8]. Linear (multivariate linear regression) and non-linear (simple multiplicative and feed-forward neural network) models were estimated and the corresponding performance in terms of accuracy and precision was compared. 2. Experimental 2.1. Reagents Bromocresol green was used as a model reagent. A 2.0 g L−1 bromocresol green solution was prepared by dissolving 0.1 g in 50 mL of water. Working standard solutions were prepared by suitable dilution with water. Doubly deionized water (conductivity ≤ 0.1 ␮S cm−1 ) was used as carrier and in solutions preparation. 2.2. System setup The single interface flow system comprised two syringes pumps (Burette 1 S, Crison, Alella, Spain), one three-way solenoid valve 161 T031 (NResearch Inc., West Caldwell, USA), a USB-4000 fiber optic wavelength scanning spectrophotometer (Ocean Optics, Dunedin, USA) equipped with 10-mm optical path flow-cells with different inner volumes (8, 30 or 60 ␮L), reaction coils with different configuration and internal diameter (0.5 and 0.8 mm) and flow lines with 0.5 or 0.8 mm internal diameter made of PTFE tubing. Home made connectors and confluence points were also used. A PC was used to control the flow system by means of dedicate programs developed with Microsoft Quick-Basic 4.5, which enabled to operate the syringe pumps by RS-232C serial binary data signals and the solenoid valve by TTL signals. The computer was equipped with a PC-LABCard model PCL-711 B interface card from Advantech (Taipei, Taiwan). A CoolDrive power drive (NResearch Inc., West Caldwell, NJ) was used to actuate the solenoid valve. The Spectrasuite software version 2007 (OceanOptics, Dunedin, USA) was used for the spectrophotometric signal acquisition. 2.3. SIFA manifold The developed flow manifold, pictured in Fig. 1(A), comprised two syringes pumps (B1 and B2 ), which were used to insert and propel the dye and carrier (water) solutions, respectively. The three-way solenoid valve (Vs) allowed the alternate introduction of dye or carrier into the analytical path being synchronised with the corresponding syringe pump actuation. The detector was placed in the terminal position of the flow manifold. The reaction coil (R) was positioned on the analytical path between the solenoid valve and the detector. The analytical cycle started by establishing baseline, which coincide with the system cleaning step, that was accomplished with the carrier solution: Vs was closed and B2 was actuated, propelling the carrier solution through the reactor and the detector; subsequently, Vs and B1 were opened and the dye solution was inserted into the analytical path. The analytical signal resulting from the mutual inter-dispersion of the dye and carrier was measured at the detector. The interface was subsequently sent to waste and a new analytical cycle began.

Fig. 1. (A) Schematic representation of the implemented SIFA manifold (S: bromocresol green solution; C: carrier solution; B1 and B2 : syringes pumps; Vs: solenoid valve; R: reaction coil; D: detector; E: waste) and (B) interface signal (black line) and the corresponding first derivative (grey line) for the system and hydrodynamic conditions L = 250 mm, V = 8 ␮L, D = 0.5 mm, C = “linear” and Q = 100 mm3 min−1 .

2.4. Experimental design A D-optimal experimental design was selected to produce the experiments suitable for chemometrics modeling [11]. This design is suitable for developing interaction models. Different levels of the operation parameters (factors) were varied: reactor coil length (L), reactor configuration (C), reactor internal diameter (D), flow-cell volume (V) and flow rate (Q). Table 1 summarizes the factors with the corresponding levels. A total of 60 experiments were produced varying the five parameters. The experimental design was produced using Modde version 6.0 (Umetrics, Umea, Sweden). Experiments were run according to the designed experiments, maintaining the same dye concentration (15.0 mg L−1 ) in all experiments. For each experiment, six replicate interface signals were obtained by inserting six dye segments sequentially. The resulting interface signals were initially smoothed using a bidirectional exponential filter to remove noise while preventing the introduction of artificial shifts in the original signals [12]. Signal processing was performed with Matlab version 6.5 (MathWorks, Natick, MA).

Table 1 Summary of the physical and hydrodynamics flow parameters and corresponding variation levels used for the experimental design. Abb.

Factor

Type

Levels

L D C V Q

Reactor length (mm) Reactor inner diameter (mm) Reactor configuration Flow-cell inner volume (mm3 ) Flow rate (mm3 min−1 )

Quantitative Quantitative Qualitative Quantitative Quantitative

250, 500, 1000, 2000 0.5, 0.8 Linear, spiral, “8” shape 8, 30, 60 100, 500, 3000

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2.5. Mathematical modeling 2.5.1. SIFA signal modeling The principles of which FIA is based are relatively simple and have been described thoroughly by many authors [2,8]. Mass transfer in FIA systems can have convective or diffusion origin. The mass transfer parameters can be linked through different expressions depending on the assumptions made. One of the most common expressions is the convective diffusion equation for laminar flow [13]. This expression accounts for axial and radial dispersion in open tubes. A simplification of this model assumed that the radial dispersion is negligible and the process is mainly dominated by axial dispersion (this is common in open tubes of limited length). The relation between the axial dispersion coefficient DL and the molecular diffusion coefficient (D) is given also by Kolev [10]. The SIFA interface between the carrier and the analyte solution can be viewed as an equivalent to the leading edge of the sample plug in FIA. Considering that the development of the SIFA interface is a step-function input, the solution of the axial dispersion equation is given by Eq. (1). c(t) =

 2





˛−t



erfc



2

t/Pe

=

 2





1 + erf

t−˛



2

t/Pe



(1)

¯ In Eq. (1), Pe is the Péclet number (Pe = uL/D L ) where L is the tube length, ˛ is the detection point (in time units) and  is a scaling factor that depends on the absorbance. The axially dispersed plug-flow model assumes that the SIFA system can be reduced to a uniform tube. This is clearly an oversimplification, since SIFA systems are composed by multiple sections of with different geometric and dispersion properties, as outlined by Kolev [13]. Eq. (1) was adjusted to each SIFA signal by optimizing the axial dispersion coefficient DL and ˛, since all other parameters are known ( is constant since solutions of equal concentration were used). The parameter ˛ was necessary to model the signal (location parameter). A non-linear least squares optimization algorithm was selected to adjust the parameters ˛ and DL to each experimental SIFA signal. Resulting coefficients were calculated as the average value considering five different replicates for each experiment. The first replicate was never considered in the analysis due to some variability. 2.5.2. Axial dispersion coefficient modeling The axial dispersion coefficient (DL ) was modeled from the system configuration and hydrodynamic flow parameters L, C, D, V and Q using linear and non-linear models. Three different model structures were tested: a linear model obtained by multivariate linear regression (MLR), a non-linear model proposed by Montesinos et al. [14] here called non-linear multiplicative model and an artificial neural network, designated from now on by models A, B and C, respectively. The MLR method (model A) was applied to establish a linear model between the regressors (L, C, D, V and Q) and the dependent variable DL , considering a structure with simple, interaction and quadratic terms. The model was estimated according to the structure in Eq. (2), where x are the regressors (L, C, D, V and Q), I is the number of regressors and ˇ are the regression coefficients [12]. DL = ˇ0 +

i

i=1

ˇi xi +

I I i=1

j=1

ˇi,j xi xj + ε

(2)

The regression coefficients were estimated according to the ordinary least squares solution [11]. After estimating the coefficients, the model was refined by sequentially excluding non-significant terms considering the corresponding p-values, establishing the threshold at the 95% confidence level (each term is preserved when p-value < 0.025).

A non-linear model (model B) proposed by Montesinos et al. [14] to model the axial dispersion coefficients of a FIA system was also considered (Eq. (3)) (the originally proposed model did not included the internal tube diameter, holding coil configuration and the flow-cell volume). DL = ˇ0 + ˇ1 Lˇ2 Q ˇ3

(3)

In this work, an updated form of Eq. (3), including all evaluated configuration and hydrodynamic flow parameters was tested (including additionally D, V and C). Artificial neural networks (ANN) have the ability to capture unknown non-linear relationships between regressors and dependent variables (model C). Despite of this advantage, ANN are also very easily affected by the overfitting phenomena and their coefficients (weights and bias) are difficult to interpret [15]. A feedforward ANN can be seen as a non-linear transformation (ϕ) of a series of I regressors (xi ) as indicated in Eq. (4). DL = ϕ(L, Q, C, V, D) + ε

(4)

The network topology considered only three layers of neurons. The first layer was simply the input layer (number of neurons equal to the number of regressors), the second layer was a nonlinear layer (hyperbolic tangent as the activation function) and the third layer was the output layer (one neuron and a linear activation function) [15]. The network parameters (weights and bias) were estimated iteratively using back-propagation [16]. An earlystopping method was applied to prevent overfitting. Calibration data were randomly divided in a calibration (70%) and validation dataset (30%). The network training was stopped whenever the error obtained for the validation dataset increased. An unseen testing set was then evaluated in the network and the corresponding testing error evaluated as a model generalization indicator. 2.5.3. Models validation Models estimation was performed using least squares (model A), non-linear least squares (model B) and Levenberg–Marquardt with back-propagation algorithms (model C). All models were developed with Matlab version 6.5 (MathWorks, Natick, MA) and the Neural Network Toolbox for Matlab. For all models, the error between the experimental axial dispersion coefficient and the estimated one was reported for a testing dataset. For the case of model C, the data selected for calibration (70%) was further divided in two sets: 70% for calibrating the network and 30% for checking overfitting (early-stopping) as described in Section 2.5.2. Modeling errors were reported considering unseen datasets according to the following procedure: (1) the 60 experiments were randomly divided in calibration (70%) and testing (30%) sets and (2) each model was estimated using the calibration set and tested with the corresponding testing set. Since results might depend on a particular division between calibration and testing, model predictions were evaluated considering 500 models estimated with different calibration/testing splits. Models accuracy was reported considering the coefficient of determination average estimated for the 500 different testing datasets. 3. Results and discussion 3.1. Screening design The experimental design produced 60 experiments, from which 3 were run in duplicate and 1 in triplicate. These replicated experiments were useful to assess the SIFA method reproducibility (lack-of-fit error). The 60 experiments produced according to the experimental design were tested experimentally using the SIFA manifold with spectrophotometric detection in random order. For

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2

181



100 i (yi − yˆ i ) / i yi2 (yi is the experimental absorbance and y ˆi the estimated absorbance for time point t = i) was used to verify the adjustment of the simulations using the estimated DL values and the experimental data. Results showed that e was lower than 3% for all sixty SIFA profiles. This result demonstrates the good adjustment of Eq. (1) to the SIFA signals. For illustrative purposes, a representation of three experimental SIFA signals obtained with different system and hydrodynamic flow conditions and the corresponding simulations using the estimated DL values was shown in Fig. 2. The simulated profiles are in very good agreement with the experimental profiles (e < 3%). Since the quality of a statistical inference depends on the way the data approaches a normal distribution, it was found to be adequate to apply a transformation to the axial dispersion coefficient. The experimental DL values were transformed using the natural logarithm which resulted in a more symmetrical distribution better adjusted for modeling purposes (Fig. 3). 3.3. Axial dispersion modeling Fig. 2. Simulation using the axial dispersed plug-flow model with corresponding estimated values of DL for three SIFA experiments executed with different physical and hydrodynamic conditions. Markers are experimental data and solid lines are simulated profiles. Experimental conditions were () L = 250 mm, V = 8 ␮L, D = 0.5 mm, C = linear, Q = 100 mm3 min−1 ; () L = 1000 mm, V = 60 ␮L, D = 0.8 mm, C = linear, Q = 500 mm3 min−1 ; () L = 2000 mm, V = 8 ␮L, D = 0.8 mm, C = “8” shape, Q = 3000 mm3 min−1 .

each experiment the consecutive replicate signals were processed according to the procedure explained before and the axial dispersion coefficient averaged. A representation of one experimental signal and corresponding first derivative of a particular experiment was illustrated in Fig. 1(B). 3.2. Axial dispersion estimation The replicates of each experimental SIFA run were averaged and modeled using Eq. (1). For each experiment a DL value was estimated. The percentage error variance calculated as e =

The DL estimates were modeled with the three modeling approaches (models A, B and C). For each approach models were optimized in order to maximize the validation correlation coefficient. The parameter reactor configuration is a dummy variable and therefore was coded as a three element vector ([1 0 0] for linear, [0 1 0] for spiral and [0 0 1] for “8” shape) yielding three derived variables. For model A (MLR) the interaction structure as shown in Eq. (2) was initially considered. The actual values of variables and the corresponding natural logarithm transformation were also evaluated. In the evaluation of the contribution of the five independent terms and interactions, it was found that many terms were not statistically significant according to the t-test p-values (superior to 0.05 for a confidence interval of 95%). It was particularly verified that the reaction coil configuration and the flow-cell inner volume were not significant. Non-significant terms were removed and the model recalibrated. This procedure was repeated until only significant terms were preserved. The presence of the terms log(Q) and log(L) was very important for ensuring the model linearity since it

Fig. 3. Histogram of estimated DL (left) and histogram of the natural logarithm of the estimated DL (right) for the 60 experiments.

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Table 2 Model types A and B coefficients estimated with all experiments. Regression coefficients

Model A (MLR)

Model B (non-linear multiplicative)

Value

Standard deviation

p-Value

Value

Standard deviation

p-Value

ˇ0 ˇ1 ˇ2 ˇ3 ˇ4

−8.897 0.9838 −4.306 0.2659 –

0.635 0.0822 0.4079 0.0412 –

<1 × 10−5 <1 × 10−5 0.006 <1 × 10−5 –

2.859 −23.22 −0.1250 −0.1610 0.3978

0.275 4.42 0.0147 0.00697 0.0844

<1 × 10−5 <1 × 10−4 <1 × 10−3 <1 × 10−4 0.012

was not possible to model log(DL ) using Q and L directly. Eq. (5) represents the optimized structure of model A. Log(DL ) = ˇ0 + ˇ1 log(L) + ˇ2 D + ˇ3 log(Q ) + ε

(5)

The coefficients for this model were estimated using the 60 experiments and the values and respective standard deviations are shown in Table 2. Model validity was guaranteed by the analysis of variance (ANOVA). The p-values for the Snedecor’s F statistic were lower than the 0.05 limit for the regression significance and higher than 0.4 for the lack-of-fit test (for both parameters). The comparison between the experimental values of DL and the predicted values using the optimized model type A reveals a coefficient of determination of 0.861. The validation methodology explained in Section 2.5.3 was adopted to this model structure and results demonstrate that the average coefficient of determination for the calibration sets (0.861) is very similar to the corresponding average coefficient for the testing sets (0.851), which ensures a good model validity. This model was further used to develop a surface control region. The model was simulated varying the values of L and Q within their experimental bounds and keeping D = 0.5 mm. The resulting response surface control chart (Fig. 4) shows the evolution of the simulated values for the axial dispersion coefficient in function of the tube length and flow rate. The optimization of the number of terms included in model type B followed a strategy similar to the one adopted for model type A. Each statistically non-significant term was removed until only significant terms were preserved. The optimized structure for this model is represented in Eq. (6) and the estimated parameters values are listed in Table 2. Log(DL ) = ˇ0 + ˇ1 Lˇ2 Q ˇ3 Dˇ4 + ε

in coefficients of determination for calibration and testing sets very similar (0.860 and 0843, respectively) and close to the corresponding coefficients obtained for model type A. The response surface control chart is also very similar to the already obtained for the model type A (not shown). Therefore, the non-linear model (type B) does not appear to represent any advantage over the linear model type A. In order to develop the ANN model (model type C) the inputs and number of hidden nodes had to be optimized. The adopted procedure was to evaluate the performance of ANN models considering different inputs combinations. For each combination the ANN was trained and tested considering different number of nodes (from 1 to 10 nodes). The identification of each ANN was described in Section 2.5.2. The coefficients of determination were averaged yielding an estimator of model accuracy. The best network was selected based on the maximum testing correlation coefficient (Fig. 5). For the optimized input variables (L, Q and D), it is clear that the optimal number of hidden nodes as shown in Fig. 5 was 3. This result was observed both in terms of r2 but also in terms of mean squared error (not shown). An ANN was estimated using the optimized structure and results were shown in Fig. 6. The calibration set was used to estimate the network parameters (white circles), the validation set was used to assess the number of training iterations (black circles) and the unseen test dataset (grey circles) was used to estimate the network error. The coefficient of determination for the test set was 0.929 (0.961 for the calibration data) which is substantially higher than the corresponding coefficients obtained for model types A and B. This result suggests that the ANN was able to capture the non-linearity in the data, which was not possible using the simpler models A and B.

(6)

The coefficient of determination between experimental and predicted values of log(DL ) is 0.860 which is very similar to the already obtained for the model type A. The validation of this model resulted

Fig. 4. Response surface for DL built from the estimated MLR model and considering D = 0.5 mm.

Fig. 5. Average testing coefficient of determination (r2 ) considering neural networks with different number of hidden layer nodes for predicting log(DL ) (figure insertion illustrates the optimized model C (artificial neural network) for modeling DL ).

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posed methodology it is possible to mathematically simulate SIFA profiles based on a set of desired operatory conditions (e.g., system configuration and hydrodynamic parameters) without proceeding with any experimental work. A natural evolution of this work would be the introduction of chemical reaction for further SIFA dispersion phenomena analyses. Acknowledgment The authors are grateful to Fundac¸ão para a Ciência e Tecnologia (FCT) for the financial support under the project PTDC/QUIQUI/105514/2008. References

Fig. 6. Experimental log(DL ) versus model C (ANN) predictions for the calibration set (), validation set (䊉) and testing set ( ).

4. Conclusions The influence of physical and hydrodynamic flow properties on the analytical signals of the recently proposed strategy designated by single interface flow analysis (SIFA) was analyzed with empirical models. By using the axially dispersion plug-flow model it was concluded that, the axial dispersion coefficient is mostly dependent of the length, the internal diameter of the reaction coil and flow rate. Among the empirical models analyzed, neural networks provided the best results. This is in agreement with the hypothesis that relations between the analyzed physical parameters and the axial dispersion coefficient are highly non-linear. With the pro-

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