Mass distribution in a dynamic sample zone inside a flow injection manifold: modelling integrated conductimetric profiles

Analytica Chimica Acta 477 (2003) 59–71

Mass distribution in a dynamic sample zone inside a flow injection manifold: modelling integrated conductimetric profiles Fernando A. Iñón, Francisco J. Andrade, Mabel B. Tudino∗ Laboratorio de Análisis de Trazas, Departamento de Qu´ımica Inorgánica, Anal´ıtica y Qu´ımica F´ısica, Universidad de Buenos Aires, Pabellón 2, Ciudad Universitaria, 1428 Buenos Aires, Argentina Received 16 July 2002; received in revised form 21 October 2002; accepted 23 October 2002

Abstract A mathematical model for fitting the experimental ICM (integrated conductimetric method) curves developed by the authors in a previous work, is presented for the first time in this study. The proposed model fits the experimental curves with great precision and allows to predict physical dispersion for single-line flow injection system. The correlation of the model’s parameters with typical reactionless FIA peak parameters is also assessed. The IDQ coefficient—a novel dispersion estimator previously reported by the authors—can also be predicted when operational FIA variables are changed. Experimental and modelled profiles are compared as a function of the system’s variables, showing an excellent agreement. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Physical dispersion; Flow injection; Integrated conductimetric detection; Mathematical model

1. Introduction There is no doubt about the important role that flow injection analysis (FIA) plays in the modern chemical analysis. Since R˚užiˇcka and Hansen [1] first introduced the technique, more than 10,000 contributions have been published [2]. Although some other techniques, such as sequential injection analysis (SIA) [3] and multicommutation (MC) [4], have evolved from FIA and have brought in many additional advantages, the foundations of these techniques are the same, namely, controlled sample dispersion. Consequently, several efforts have been done to study the influence of the system variables on the dis∗ Corresponding author. Tel.: +54-11-4576-3360; fax: +54-11-4576-3341. E-mail address: [email protected] (M.B. Tudino).

persion process as a way of improving the analytical performance of flow systems. Dispersion arises from the mass transport phenomena (i.e. convection, diffusion and chemical reactions) occurring into the FIA tube and the way in which the detector integrates the signal in time and space. Based on the classical studies of Taylor [5,6] devoted to theoretical aspects of the dispersion phenomena of an injected tracer in a flow system, several works have been reported and reviewed in the field of FIA [7–11]. In a system with no chemical reactions, the main factor concerning dispersion is the mass redistribution produced by the convective-diffusive transport. If a chemical reaction is involved, the kinetics of mixing of the reactants and the kinetics of the reaction itself will also contribute to dispersion [10,12,13]. From a physico-chemical point of view, the process can be described through the general diffusive-convective transport equation [5]. Unfortunately, it does not have an

0003-2670/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 3 - 2 6 7 0 ( 0 2 ) 0 1 3 9 9 - 5

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Nomenclature ±95% C
tb τ a D Dm Dr G G∗ h h0 IDQ κ L l PeL Per q r Re SV t ta tR u u¯

confidence interval for a 95% significance level concentration (M) elapsed time between peak apearence and the returning to baseline time (Peak width in time unit) (s) Fourier number Dm tm a−2 tube radius (cm) R˚užiˇcka and Hansen dispersion coefficient hh−1 0 molecular diffusion coefficient (m2 s−1 ) radial dispersion coefficient conductivity (S) normalised conductivity GG−1 0 peak height (cm) height of the steady state signal (cm) integrated dispersion quotient specific conductance (S) reactor length (cm) loop length (cm) length based Peclet number uLD ¯ −1 m molecular diffusion and tube radius based Peclet number uaD ¯ −1 m volumetric flow rate (cm3 s−1 ) radial coordinate (cm) −1 Reynolds number 2uaδη ¯ 2 injection volumen πa l (cm3 ) time (s) time of appearance (s) mean residence time (s) linear velocity of flow (cm s−1 ) mean velocity of flow in the axial direction qa−1 (cm s−1 )

exact solution in the working range of regular FI systems [10]. Several models have been proposed in order to describe and/or predict dispersion in FI systems. These models range from the more theoretical partial differential equations [7,13–21] to the more pragmatic neural networks (i.e. see [22]). Most of them are able to describe dispersion in a wide variety of FI systems. In some cases it is also possible to optimise the manifold with the use of expert systems (i.e. see [23]). DeLon

Hull [10] and Kolev [11] have carried out a classification of the existing mathematical models and have summarised their range of applicability. Unlike this work, the models (and/or their outputs) have been derived considering a single-point (punctual) detector. For those models based on the description of the physical processes, punctual detection conditions their validation. This is because the association of the shape and size of experimental curves profiles with the mass transport process occurring into the tube is not straightforward, as it was stated by van der Linden [24]: “it is difficult, if not impossible, to relate the experimentally obtained response curves in an unequivocal way to the actual spatial distribution inside the system”. In order to advance on the study of dispersion, the authors have reported a new methodology different from the single-line point detection: the integrated conductimetric method (ICM) [25]. ICM allows to follow the mass redistribution process of the injected plug as a function of time. It is based upon monitoring the electrical conductance (G) of the whole single-line flow system as time elapses by placing platinum electrodes in both ends of the manifold: before the injection valve and after the reactor. The injection of a dielectric fluid (water) into an electrolytic carrier (e.g. HNO3 ) in a single-line manifold produces a characteristic profile G versus t which depends on the hydrodynamic characteristic of the FI system and can be assigned to the mass redistribution of the injected pulse along the tube. From these curves it is possible to define a new dispersion descriptor (IDQ, integrated dispersion quotient) [25] that is closely related to R˚užiˇcka and Hansen dispersion coefficient (D). Although it is being studied in systems involving chemical reactions, the present work will be only dedicated to those systems in which no chemical reaction occurs. ICM presents a series of advantages in comparison to the classical stimulus–response method. Firstly, ICM is sensitive to the mass redistribution process (and thus dispersion) as a function of time, making unnecessary to deconvolute the transport mechanism from the typical FI signal. Additionally, the ICM detection does not contribute to dispersion and it is possible to measure the redistribution process in each separate component of the system and to estimate the global response through a suitable combination of each one of them.

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A limitation of the ICM methodology is that, up to date, it has been only tested for single-line manifolds. Nevertheless, the actual influence of manifold geometry (single, double, and reverse line configuration) on the sensitivity of a given technique has been reviewed by Chalk and Tyson [26]. These authors have shown that, under appropriate conditions, all the systems give the same sensitivity. For example, comparing the double and single line manifolds, the better mixing achieved by the formers is counterbalanced by an unavoidable higher degree of dilution of the sample pulse, which yields no significant improvements in sensitivity. Consequently, a single-line manifold should be more convenient in terms of simplicity. Considering that ICM methodology provides useful information on dispersion process and that the existing mathematical models consider only the single-point detection, the authors present a phenomenological mathematical model developed in order to fit and predict experimental ICM curves profiles. Once an analytical expression for G(t) is available, IDQ can be estimated a priori, providing a novel way for the evaluation of dispersion with no contribution of the detection system. Moreover this new model allows to derive the estimators of dispersion in conventional FIA systems.

2. Experimental Experimental set-up and regents were the same reported in a previous work [25]. For better understanding of the study, they are summarised further. 2.1. Reagents Analytical reagent grade chemicals and double deionized water (DDW, 18 M cm−1 MilliQ water system (Millipore, Milwaukee, USA) were used. 0.75 M solutions of HNO3 were prepared from analytical grade concentrated nitric acid and employed as carrier solutions. Cobalt sulphate solutions (1% w/v) were prepared for spectrophotometric measurements. 2.2. Apparatus ICM measurements were performed with the single line flow arrangement shown in Fig. 1. An automatic

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Fig. 1. Experimental set-up. (a) FIA manifold: P pump, l loop, L reactor, W waste; A, A , B detection points. Single-point detection: suitable detector in point B; integrated detection: Pt wires located at points A–B, A–A or A –B. (b) Detailed view of the conductimetric integrated detection cell; E platinum wire.

injection valve (Valco Instruments Co., Austin, TX, USA), a variable speed peristaltic pump Ismatec MS Reglo (Cole-Parmer, Chicago, IL, USA) and PTFE tubing of 0.8 and 0.5 mm i.d. were employed for the experiments. Conductimetric measurements were performed by using a digital conductimeter (Wissenschaftlich Technische Werkstatten LF 521) with a couple of Pt wires (0.03 cm diameter) located at points A–B, A–A , A –B, or B (Fig. 1) in home-made flow cells. Depending on the location of the detection points, it is possible to perform one-point (B) or integrated (A–B, A –B, A–A ) conductimetric measurements. Single-point conductimetric measurements were performed with a conductimetric cell similar to the one described by Taylor and Nieman [27]. The employed T-shape connectors are shown in Fig. 1(b). Spectrophotometric measurements were carried out with a spectrophotometer (Hitachi U-1100) equipped with a flow cell (Carl Zeiss, 80 ␮l volume and 1.00 cm optical path length) located at point B (Fig. 1). The signals were acquired via a data acquisition system (Keithley 1 DAS 801) and processed with a personal computer. A trigger was connected between the valve and the computer in order to start the acquisition phase when the valve is commuted, with a typical repro-

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ducibility of 0.2 s. This value is considered acceptable since it is significantly lower than other characteristic times (t1 , ta , etc.). 2.3. Procedure 2.3.1. Integrated conductimetric detection (ICD) The FI system was assembled as described in Fig. 1(a). The conductance for the injection of water in the nitric acid carrier was measured as function of time. An air bubble injection was also performed for comparative purposes. Several experiments were carried out in order to obtain G versus t profiles for different values of carrier flow rate (q), loop (l) and reactor (L) length and tube radius (a) for two different reactor geometry. Further discussion on these topics was provided elsewhere [25]. These values were varied as follows: q = 0.33–2 cm3 min−1 ; L = 30–145 cm; a = 0.025 and 0.040 cm; l = 5–61 cm. The equivalent injected volumes ranged between 9 and 110 ␮l for a = 0.025 cm, and 25 and 306 ␮l for a = 0.040 cm. The loop length was chosen instead of the injected volume as the former is better related with the observed ICM curves. 2.3.2. Normalization of the experimental curves For comparative purposes, the conductivity axis (y axis) of ICM curves was normalized as described before [25] and G∗ values obtained. G∗ ranges from 0 to 1 and represents the evolution (recovered fraction) of the carrier conductivity after the disruption.

2.3.5. Curve fitting SigmaPlot® and MathLab® were used for curve fitting purposes. In all the cases the tolerance was fixed at 0.000001. Obtained R2 values were typically >0.9985, indicating the goodness of the regressions. Nevertheless, the fitting was evaluated in terms of residual values and their distribution. The estimation error of each parameter was also considered. 3. A model for ICM curves 3.1. The ICM curve profile Fig. 2 shows a typical G∗ versus t profile obtained for the injection of water in the single line system, with the Pt wires located at points A and B. Results for the injection of an air bubble (which represents the “plug” flow behaviour of the injected pulse) are also given. As it was previously discussed, the profile obtained for the injection of water is ascribed to the mass redistribution process within the FI tube. Since the sample and the carrier solutions have different conductivities, the sample plug changes its physical characteristics while it travels along the tube and thus, the monitored property approaches the carrier conductivity as time elapses. It should be noted that, even when the sample pulse is still inside the manifold (t lower than ta , the time of arrival), there is a variation in conductance. This variation is assigned to the mass redistribution of the

2.3.3. Punctual (single-point) conductimetric detection The experiments described in the previous section were repeated with the FI system shown in Fig. 1 by locating both Pt electrodes at point B, and then the arrival times of the injected pulse for the assayed systems were obtained. 2.3.4. Spectrophotometric detection Classical response curves (single-point) were obtained by using the FI configuration described in the previous section but changing the detector. In this case, the carrier solution was DDW and the sample loop was filled with the CoSO4 solutions. Absorbance at 520 nm was recorded as a function of time with a spectrometer furnished with a 80 ␮l flow cell.

Fig. 2. Typical ICM profile. (a) Water Injection; (b) air-bubble Injection. Experimental conditions: L = 80 cm, l = 25 cm, a = 0.04 cm, q = 1.00 cm3 min−1 , straight reactor.

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injected sample within the tube. Three different zones can be distinguished: zone I, elapsed between t = 0 (the injection point) and t1 (time when the injected pulse leaves the plug-flow type behaviour); zone II, between t1 and ta , corresponding to the evolution of the measured conductance towards the value obtained for the carrier; and zone III, from ta onwards, showing the conductivity changes when the injected pulse leaves the system and the solution recovers the conductivity of the carrier solution. 3.2. The proposed model The injected plug is considered as an insulating cylinder whose length (Leq (t)) and area (Seq (t) = π [aeq (t)]2 ) change as time elapses (Fig. 3 above), but its integrated concentration remains equal to the original concentration of the injected pulse (Ceq (t) = C0 ). It should be noticed that this does not necessary imply that the mean concentration in any cross section of the system is equal to C0 . On the contrary, part of the ICM profile can be ascribed to the reduction of the mean concentration in a given cross section, as it will be explained further. The cylinder blocks the electrical field and the measured signal is assumed to be related mainly to the re-

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duction of the disc area, or, which is the same, the radial dispersion is considered the most important contribution to the ICM profile between t1 and ta . Once the sample begins to leave the system, the measured values depend on the amount of dielectric remaining in the system. With the purpose of developing an analytical expression of G∗ (t), the equivalent electrical circuit representing the single-line flow system (Fig. 3) was deduced. The total resistance of the system can be calculated from the resistance (Ri ) of each section by applying the general electricity laws. The conductance of each section Gi can be calculated as the inverse of the resistance of the section (Ri−1 ) and its value will depend on the cylinder dimensions as function of time. G(t)−1 = R(t) = R1 (t) + + R3 (t)

1 (1/Rcarrier2 (t)) + (1/RH2 O (t)) (1)

When the whole system is filled with an homogeneous electrolytic solution linear relationships between G and the carrier concentration (L and l held constant) and between G and (L + l)−1 (carrier concentration held constant) are found. Moreover, Eq. (2)

Fig. 3. Simplified model. A and B location of the platinum wires; Ri resistance of the different sections of the system (for more information, see text).

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is also verified [25,28]. G = R −1 = κ

S0 L+l

(2)

where S0 is equal to π a2 , and κ is a constant that can be regarded as the specific conductance measured under these conditions. Considering that each resistance in Eq. (1) can be expressed in terms of Eq. (2), with the corresponding area and length of each section, and Rcarrier2  RH2 O , Eq. (3) is derived. G∗ (t) =

S0 − Seq (t) (S0 − Seq (t))(1 + (Leq (t)/L + l)) + S0 (Leq (t)/L + l)

(3)

In order to fit experimental data using Eq. (3), Seq (t) and Leq (t) were modelled as given further. 3.2.1. Model for Seq (t) As the detection system has proved to be independent of the linear velocity (u), it can be assumed as a mean value detector [29–31]:
a C(r)2πr Cmean = dr (4) πa 2 0 where Cmean is the mean concentration measured by the detector and C(r) is the concentration at the radial coordinate r. By applying the concept of the insulated plug as an equivalent cylinder, this equation is equal to aeq (t)a−1 aeq (t) has been modelled considering that: • for t = 0 (injection time), aeq equals a (Seq equals S0 ); • the flow pattern is laminar. Re numbers are lower than 100, and thus, far below the critical value of 2100 at which the transition from laminar to turbulent flow pattern starts [32]; • the axial velocity profile (u(r)) produces a radial concentration gradient which forces the walls of the equivalent cylinder to be released from the tubing walls, causing the relaxation process. So, the disc area is reduced to a given Seq value. On the knowledge that the velocity profile does not vary either with t or z (longitudinal coordinate), and assuming that the radial concentration gradient (dC/dr) is proportional to the axial velocity profile (du/dr), the relaxation force is directly related to the radial coordinate. Thus, the change of aeq with time can be

written as: ∂aeq ∂u = k1 (5) ∂t ∂r Eq. (5) can be integrated considering the parabolic profile developed under FIA conditions [33] and the relaxation process starting at t = t1 , yielding Eq. (6): Seq (t) = π [aeq (t)]2 = π(a e−2k1 umax (t−t1 )/a )2 2

(6)

where umax is the maximum axial velocity (twice the average linear velocity u). ¯ The product k1 umax represents the net mobility of the cylinder wall in the radial coordinate. The greater the mobility of the equivalent cylinder walls towards the center of the tube (which is reflected as an increse in the product k1 umax a−2 ), the sharper the change in G∗ (t) at t1 . Wentzell et al. [34] have proposed an expression of a generalised laminar velocity profile in which u(r) is proportional to rp (with p = 2 in the case of parabolic profile). Our results show that if p is included as another model parameter, the values of p obtained after the fitting procedure are statistically [35] equal to 2 in most cases. Moreover, the values obtained for the other parameters (t1 , k1 , etc.) do not change significantly if p is included as a parameter to fixed to 2, but their (S.E.) increase. Since the dependence between parameters and their S.E. sharply increases when the model is overdimensioned (by the inclusion of many variables) [36] the parameter p was fixed and equalled to 2. 3.2.2. Model for Leq (t) In the case of the length of the equivalent cylinder (Leq (t)), two models were investigated. The first one is just phenomenological since the shape of zone III resembles a F-curve (a near sigmoidal profile [12,32]). The second one takes into account the stretching of the sample plug due to the velocity profile and thus, it gives a better insight of the process involved. However, the non-linear fitting of the function to experimental data hardly converges and is highly dependent of the initial parameter’s values. Thus, the first model was assumed, and the following sigmoidal function was tested: k2 l Leq (t) = (7) 1 + k4 ek3 (t−t2 ) where k2 , k3 and k4 are model’s parameters whose physical meaning and dependence with FI variables will be discussed further.

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After an analysis of the effect of each parameter on Eq. (7), it was concluded that k4 can be avoided [28] as its influence is close to that of t2 (see further). The inclusion of k4 overdimensions the model, and it was equalled to 1. Preliminary results show that the combination of Leq (t) and Seq (t) in Eq. (3) reveals a high dependence between k1 and k2 . This dependence is explainable: k2 is related to the difference between 1 and G∗ value at the beginning of zone III and k1 determines the magnitude of the relaxation and hence, determines G∗ value at the end of zone II. Further details on k1 and k2 will be given further. The different strategies analysed for overcoming this interdependence, derived in the following expression for Leq (t): √ l − k5 t Leq (t) = (8) 1 + ek3 (t−t2 )

be seen, the goodness of fit is excellent and the residuals are not greater than the experimental measurement error. Table 1 summarises a typical regression result. Model CI% means the confidence interval (95%, CI95% ) [35] estimated through the non-linear regression divided by the estimated mean value of the parameter. Experimental CI% is the CI95% found by averaging the parameters values obtained through the curve fitting to six replicates of identical injections (i.e. performed under identical boundary conditions) divided by the parameter mean value. As can be seen in all the cases, the error of the estimated parameters is lower than the experimental error. An acceptable dependence value [36] is observed for t1 , t2 and k3 , and a slight dependence between k1 and k5 (see further) needs to be accepted. Nevertheless, this dependence does not obscure the correlation with FI variables.

where k5 is a new model’s parameter that replaces k2 . The physical meaning of all model’s parameter (t1 , t2 , k1 , k3 and k5 ) will be described in the next sections.

4.2. Model’s parameters: physical explanation and its dependence with FIA variables

4. Results and discussion 4.1. Goodness of fit Fig. 4 shows a typical experimental curve fitted by the model and the residual of the regression. As can

4.2.1. t1 The parameter t1 estimates the time at which relaxation begins (see Fig. 2). A direct correlation (slope 1.0002, intercept = 0.0001, R 2 = 0.9998) between model t1 and experimental t1 was found. This parameter only depends on hydrodynamics conditions inside

Fig. 4. Curve fitting. Experimental conditions: L = 80 cm, l = 25 cm, q = 1.00 cm3 min−1 , a = 0.040 cm.

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Table 1 Typical regression results Parameter

Mean value

Dependence

Experimental CI%

Model CI%

t1 (s) t2 (s) k1 (cm) k5 (cm s−0.5 ) k3 (s) R2

2.50 11.6 2.8 × 10−6 2.6 0.77 0.9993

0.4757 0.5486 0.8855 0.9368 0.5432

1.1 1.0 3.9 4.1 0.5

0.2 0.4 1.9 3.2 1.9

L = 65 cm, l = 24 cm, q = 0.73 cm3 min−1 , a = 0.025 cm.

the sample loop as it was discussed in our previous work [25]. It is independent of L, proportional to l and to u¯ −1 , and almost independent of a for the two tubing radii tested (0.08 and 0.05 cm i.d.). Under these conditions, t1 can be written as (0.049 ± 0.04)lu¯ −1 . Simulations based on the random walk method [34,37] were carried out [28], showing that t1 is the time at which the more concentrated segment of the sample plug has no longer the initial concentration. Consequently, it can be said that for any single-line FI system, D and IDQ will be >1 for t values higher than t1 . A deeper analysis of these simulations is beyond the scope of this contribution and it deserves a separate discussion that will be submitted for publication soon. 4.2.2. k1 This parameter is related to the radial mass redistribution in the system: the greater k1 , the greater is the dispersion. As it was mentioned before, taking into account Eq. (6), the relaxation force is proportional to k1 ua ¯ −2 . Moreover, as the product k1 u¯ has the same units of a diffusion coefficient, it can be analysed in terms of the adimensional numbers of Fourier (τ = Dm tR a−2 ) and Peclet (Per = uaD ¯ −1 m ):  • k1 ua ¯ −2 tt−1 R will be referred as τ , and it resembles a Fourier number τ evaluated at a given time t. So, τ  evaluated at the residence time, equals k1 ua ¯ −2 . −1  −1 • ua(k ¯ 1 u) ¯ = ak1 will be referred as Per , as it resembles a radial Peclet number.

For those systems where u, ¯ tR , and a are similar, it is equivalent to analyse k1 or τ  . Fig. 5 shows the variation of k1 with the linear flow velocity for coiled and straight reactors. As can be seen, the difference between both geometries increases as u¯ increases, and

k1 values are always lower for coiled reactors than for straight ones. In the case of straight reactors, k1 initially increases as u¯ increases, but at higher u¯ no dependence is noticed. The behaviour of coiled reactors is similar, but at high u¯ values a slight reduction of k1 is observed. The differences between the performances of coiled and straight reactors are stressed when L and u¯ are increased. This observation is in agreement with the general theory about the effect of coiling reactors on dispersion: the higher the number of times the parabolic velocity profile is modified, the more uniform the distribution of velocities of the fluid streamlines and, the more similar to the plug-flow type is the transport process [33,38]. The number of times that the flow changes is given by the reactor shape (e.g. coiling factor) and the frequency of changing is given by the mean flow velocity. Fig. 6 shows the effect of changing k1 on Per and  τ values. Although theoretical τ decreases with the increment of u, ¯ it is observed that τ  increases (Fig. 6 inset) as the product k1 ·u¯ becomes larger. It can be be seen that, even when all FI variables remain the same, the hydrodynamic conditions change when the spatial configuration of the reactor is changed. As it is well known, coiling the reactor increases the convective contribution to radial mass transport (Per increases, τ  decreases) due to the development of a secondary flow pattern. Results show that as L increases, k1 is not affected, τ  increases (larger axial stretching), but Per remains constant. It can be said that increasing L does not change the hydrodynamics boundary conditions, and it only allows the dispersion process to keep going on. Consequently, an increment in Per can be associated to a larger radial mass reposition, whilst an increase in τ  can be associated to a larger axial stretching.

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Fig. 5. Variation of k1 with u¯ at two different reactor spatial configurations. Experimental conditions: a = 0.04 cm, L = 128 cm, l = 25 cm, coiled and straight reactors.

Regarding the loop length, k1 decreases exponentially as l increases, reflecting the decrease in dispersion with the increment of l. In the case of the tube radii, k1 decreases sharply as a decreases, showing a better radial mass redistribution as a is lowered. Moreover, for the same values of u¯ and lL−1 , those systems with lower a yield larger Per and lower τ  . These

facts can be understood as a better preservation of the sample-plug. 4.2.3. k5 This parameter relates the reduction of the equivalent cylinder length with the elapsed time. The lower is the value of k5 , the smoother is the reduction of

Fig. 6. Per y τ  for coiled and straight reactors. Experimental conditions: a = 0.04 cm, L = 128 cm, l = 25 cm.

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Leq (t), which can be translated as a lesser degree of stretching. A linear relationship between k5 and l and a practical independence of k5 with L and a were observed regardless the reactor geometry (note that k5 in coils is lower than in straight reactors). With respect to the effect of u¯ on k5 , it was noticed a linear relationship up to 6 cm s−1 and a complete independence onwards. At lower linear flow velocities, the diffusive component for mass redistribution prevents the axial stretching (lower k5 values) but, as u¯ increases, the axial stretching becomes more important until convection dominates the process completely. Here the dispersion becomes independent of u, ¯ and Taylor theory can be applied [5,39]. Under the boundary conditions tested, k5 becomes equal to 0.077l–2.02 exp(−0.48u), ¯ where the S.E. of each parameter is around 3%. 4.2.4. k3 An increment in k3 is directly related to the sharpness of the change in conductance with time in zone III, which is inversely related to
tb . Nevertheless,
tb also depends on other parameters of the model, as it will be shown further. It was found that k3 is proportional to L−x , where x increases as a decreases (x = 0.50 ± 0.06 and 0.36 ± 0.03 cm for a = 0.04 and

0.025 cm, respectively), that is, as the tube radius decreases, the dependence of
tb with L also decreases. Additionally, in coiled reactors, not only k3 is higher (and
tb lower) in comparison with straight reactors, but at lower u, ¯ x values are nearly the same and decrease with the increment of u. ¯ It is clear that a low x value is always related to a lower contribution of the transport to the whole dispersion. In the case of l, k3 slightly decreases as l increases, probably because the dispersion in the reactor turns negligible the influence of l on the peak width. Fig. 7 shows a linear relationship between k3 and u¯ for straight reactors, and an increment of k3 when a is decreased. Before carrying out the log–log regression for confirming the linear relationship k3 versus u, it is necessary to know if the intersection to the y axis is significantly different from zero. This is an important fact (which it is not frequently taken into account in the literature [40–44]) since a constant term does affect the linearity of log–log regressions. In our case, it is expected that k3 approaches to zero as u¯ decreases, that is, the sample keeps its original distribution under stopped-flow conditions (assuming that the size of the sample-plug is large enough as to neglect the axial diffusion at its boundaries). Regarding the slope of the log–log regression, it is statistically equal to 1 (0.99 ± 0.07 and 1.06±0.07, for a = 0.025 and 0.040 cm, respectively).

Fig. 7. Effect of u¯ on k3 values. Experimental conditions: L = 128 cm, l = 25 cm, a = 0.04 cm, q = 1.00 cm3 min−1 , straight reactor.

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4.2.5. t2 This parameter estimates the time at which the inflection point of zone III is observed. As it will be shown, it is directly correlated with the time at which the conventional FIA peak reaches its maximum value. Under the investigated conditions t2 can be written as Lu¯ + k7 , where k7 is directly proportional to t1 (k7 = (1.18 ± 0.2)t1 , R 2 = 0.9892). 4.3. Estimation of the attributes of classical FI transient 4.3.1. Time attributes: ta , ∆tb and tR A good linear relationship between t2 and tR (t2 = 1.004tR –1.02, R 2 = 0.9989, n = 18) was found, so t2 estimates the time at which the maximum of the conventional FIA peak is observed. In order to obtain a complete estimation of the FIA peak, it is necessary to evaluate the baseline-to-baseline time (
tb ) and the time of arrival ta . Note that in G∗ (t), Leq (t) is the function related to the sample length, therefore, ta and
tb must be related to Leq (t). As shown in Fig. 2, G∗ (t) presents an inflection point at t = ta . On the other hand, Leq (t) is composed by two functions: F 1 = l − k5 t 0.5 and F 2 = (1 + exp(k3 (t − t2 )))−1 . A careful analysis reveals that the inflexion point of G∗ (t) at ta corresponds to a local minimum of the function (derivate(F 1) + derivate(F 2)). Algebraically, ta is the time at which Eq. (9) is satisfied. (k5 e2k3 t + (4k32 t 1.5 + 3k5 )e(k3 t2 +k3 t) + (3k5 − 4k32 t 1.5 )e(2k3 t2 ) )ek3 t = −k5 e3k3 t2

(9)

By means of successive iterations of Eq. (9) and starting from t = 0.8·t2 , convergence is quickly achieved (usually <2 s) and its solution is easily achieved. It was found an excellent linear relationship between experimental and calculated ta , as the slope is statistically equal to 1 and the interception equal to 0 (slope = 0.985 ± 0.028, intercept −0.6 ± 0.8 s, R 2 = 0.9963). This fact is true for each tested system.
tb can be calculated as the difference between the time at which Leq (t) equals 0.0001l(= t2 +ln(9999)k3−1 ), and ta . A linear correlation (slope = 0.67 ± 0.2, intercept −0.8 ± 0.9 s, R 2 = 0.9963) was found between the
tb value estimated by the model and the experimental
tb measured by using an spectrophotometric flow-cell of 80 ␮l. Experimental peak

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widths are higher than the estimated ones because there is a contribution of the flow-cell to the peak broadening in the first case [45]. 4.3.2. Height attributes: integrated detection quotient (IDQ) As it was mentioned above, IDQ can be directly related to D [25]. So, we will only deal here with the evaluation and prediction of IDQ. In the previous section, it was shown that it is possible to estimate all the time-parameters of the classical FIA peak via the application of the proposed model to ICM measurements. A complete description of the single-point peak is attainable if the model is employed for IDQ evaluation. In fact, IDQ is defined as: IDQ(t) =

tG∗dm t tG∗dm − 0 G∗ (t) dt

(10)

where G∗dm is defined as G∗ at the maximum degree of radial dispersion and it is equal to L(L+l)−1 . Since the model provides an expression for G∗ (t), Eq. (10) can be written as: IDQ(t) =

t

tG∗dm

tG∗dm − (L + l) 0 ((e2k1 t + e2k1 t1 )/ √ (e2k1 t (k5 t − L − 2l) + e2k1 t1 (L + l))) dt (11)

Although there is no analytical solution for the equation, it can be easily and quickly integrated by computer software. The comparison between IDQ values calculated through the experimental curves and by the model shows an excellent agreement under all the tested boundary conditions (slope = 1.003 ± 0.001, intercept = 0.00 ± 0.01, n = 40, R 2 = 0.9998). Thus, once the dependence of all the model parameters on flow variables is established, it is possible to generate algorithms in order to reduce dispersion (smaller
tb and IDQ) fixing ta . 4.4. Range of applicability It should be noticed that the model has been validated in the range of the tested experimental conditions mentioned above. Further work should be conducted in order to extrapolate these results beyond

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Table 2 Summary of correlations between model’s parameters and FIA variables for straight reactors Parameter

Correlation

Observations

t1 (s) t2 (s) k1 (cm) k5 (cm s−0.5 ) k3 (s)

0.049lu¯ −1

Independent of L and a, under tested conditions

Lu¯ + 1.18t1 Interpolated from figures 0.077l–2.02·exp(−0.48u), ¯ Proportional to L−x (x = 0.36 for a = 0.025, x = 0.50 for a = 0.04) Proportional to u¯ −1 (Fig. 7)

these experimental conditions. Under the tested conditions, ICM curves were adjusted and predicted with great precision. In order to evaluate the prediction capabilities of the model, new six systems working at the regular operational variables were assembled. Each experimental curve was fitted using the proposed model (“adjusted values”) and the values obtained for each parameter (t1 , k1 , t2 , k5 , k3 ) were compared with those predicted (“predicted values”) by the correlations found in Section 4.2 and summarised in Table 2. The mean prediction error for each parameter and for IDQ on the basis of ‘n’ evaluated systems, can be calculated as a mean prediction error [36]:   n  (oˆ i − oi )2 MPRES(o) =  i=1 (12) 2 x n n i=1 oi where oi is the “adjusted value” for a given parameter “o” in the system i and oˆ i is the “predicted value” for that parameter. The MPRES values obtained for each model parameter are: t1 = 0.6%, k1 = 1%, k5 = 1.2%, k3 = 1%, t2 = 0.8%. The MPRES for IDQ is 1%. When analysing the prediction capability of the model for the estimation of the different time attributes of a conventional single-line FIA profile, the contribution of the detection module to the overall dispersion (negligible in case of ICM) has to be incorporated (Section 4.3). For doing so, equations arrived in Section 4.3 for our 80 ␮l flow-cell were used for predicting the typical FIA peak parameters obtaining a MPRES of 1.1% and 1.2% for tR and
tb , respectively. Regarding height attributes, the contribution to the overall dispersion due to the detection module becomes relevant in comparison to the contribution of the transport and injection, D becomes higher than IDQ, as expected. Unfortunately, not an exact calculation of these differences can be performed as the

Interpolated from figures

dispersion due to the detection module depends on all the FIA operational variables and on the hydrodynamic characteristics of the flow-cell [45–49]. Further work is being conducted in order to incorporate a flow cell in the ICM manifold by, for example, placing the second Pt electrode after the spectrophotometer flow cell. Since there is a plethora of flow-cells designs, the influence of the cell volume and features on dispersion is not included here. 5. Conclusions This contribution presents the development of a mathematical model that successfully fits experimental ICM curves profiles and enables to extract useful analytical information by predicting the attributes of a typical reactionless FI response under tested operational conditions. Although some correlation between model’s parameters and FIA variables has to be interpolated from experimental data, the model allows the study and evaluation of dispersion via ICM, which has some advantages with respect to the stimulus-response alternative. Time-attributes (residence time, peak width and time of arrival) of the classical single-line FIA response were also predicted. Furthermore, the model should bring in the possibility of optimising systems, as a mathematical expression for IDQ is provided, and hence, a way for improving a sensitivity. Work on extending the model to other reactor geometries and to the incorporation of the flow cell is being carried out in our laboratory but more experimental input is required before reporting these new results. Acknowledgements The authors thank the program for Science and Technology of the University of Buenos Aires

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