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Dispersion in cylindrical channels on the laminar flow at low Fourier numbers Witold Kucza * , Juliusz Da˛browa, Katarzyna Nawara AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Al. Mickiewicza 30, 30-059 Cracow, Poland
H I G H L I G H T S
G R A P H I C A L A B S T R A C T
Simulations and measurements of dispersion at low Fourier numbers are presented. Double-humped peaks were measured by impedance detection. Numerical responses were fitted to experimental ones by an optimization method. The determined diffusion coefficients of salts agree well with the literature data.
A R T I C L E I N F O
A B S T R A C T
Article history: Received 20 February 2015 Received in revised form 19 April 2015 Accepted 23 April 2015 Available online xxx
A numerical solution of the uniform dispersion model in cylindrical channels at low Fourier numbers is presented. The presented setup allowed to eliminate experimental non-idealities interfering the laminar flow. Double-humped responses measured in a flow injection system with impedance detection agreed with those predicted by theory. Simulated concentration profiles as well as flow injection analysis (FIA) responses show the predictive and descriptive power of the numerical approach. A strong dependence of peak shapes on Fourier numbers, at its low values, makes the approach suitable for determination of diffusion coefficients. In the work, the uniform dispersion model coupled with the Levenberg–Marquardt method of optimization allowed to determine the salt diffusion coefficient for KCl, NaCl, KMnO4 and CuSO4 in water. The determined values (1.83, 1.53, 1.57 and 0.90) 109 m2 s1, respectively, agree well with the literature data. ã 2015 Elsevier B.V. All rights reserved.
Keywords: Flow injection system Impedance measurements Numerical modeling Optimization Diffusion coefficient
1. Introduction
* Corresponding author. Tel.: +48 126 172 466; fax: +48 126 172 493. E-mail address: [email protected] (W. Kucza).
Flow injection analysis (FIA) is based on an injection of a sample solution into a carrier stream, where controllable dispersion and generation of a reproducible signal at a detector occur. Since its introduction in early 1970s by Hansen and Ruzicka, FIA has become one of the primary techniques in
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Please cite this article in press as: W. Kucza, et al., Dispersion in cylindrical channels on the laminar flow at low Fourier numbers, Anal. Chim. Acta (2015), http://dx.doi.org/10.1016/j.aca.2015.04.049
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chemical analysis [1–4]. The main characteristics of FIA are: signal reproducibility, accuracy and extremely short times of analysis. The most popular methods of detection used in FIA are: photometry, luminescence and electrochemical detection [3]. Conductivity measurements, although not selective ones due to the contribution of all ions to the electrical conductivity, are an attractive choice for electrochemical detection [3,5–9]. Their advantages are: speed of measurements, simplicity of the detector construction [4,7,8] and capability of lowering the detection limit when combined with gas diffusion separation [8]. FIA is usually used in chemical analysis, but it can be also used for kinetic parameters determination, e.g., diffusion coefficients. The use of FIA in this field is limited to systems of high Fourier numbers (t ¼ Dtm =r20 where: D denotes the diffusion coefficient; tm: the mean residence time; r0: the channel radius) where Taylor’s analytical solution [10,11] is valid [12–15] or requires a calibration technique, which is based on the Vanderslice numerical calculations [16,17]. The first approach requires low flow rates (long residence times) whilst the second one requires using standards of known diffusion coefficients (for which calibration factors are determined) [17]. These problems have been recently evaded by
using the inverse methods based on deterministic or stochastic numerical computations, allowing to determine diffusion coefficients based upon the shape of FIA response curves [18]. In the present work, the numerical solution of the uniform dispersion model is used for simulations of dispersion in cylindrical channels at low Fourier numbers. It is well known that such condition requires application of numerical methods [16,19,20]. Analytical solutions of diffusion–convection equation are valid when convection (for t < 0.1) or diffusion (for t > 0.6) dominates the overall dispersion [20]. At Fourier numbers between 0.1 and 0.3 asymmetrical, double-humped peaks are theoretically predicted [16,21]. However, such behavior is rarely observed due to experimental non-idealities: turbulence generated by pump pulsation, mismatches between sample loop and tubing and between tubing and detector cell as well as coiling and other deformation of channels, which are all influencing the ideal laminar flow [16,20,22]. From the above, flow injection analysis at low Fourier numbers is a challenge in regards to both experimental implementation as well as mathematical modeling. However, a success reached in both fields may result in accelerating the analysis since experiments would be carried for higher flow rates.
Fig. 1. A schematic illustration of the homemade FIA setup. Insets on the right side illustrate: injection systems (at the top) and the impedance detector (at the bottom).
Please cite this article in press as: W. Kucza, et al., Dispersion in cylindrical channels on the laminar flow at low Fourier numbers, Anal. Chim. Acta (2015), http://dx.doi.org/10.1016/j.aca.2015.04.049
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Fig. 2. Numerically simulated FIA responses for different numbers of (axial nodes)/(radial nodes)/(temporal steps) for diffusion coefficients: (a) 1.8 109 and (b) 9 1010 m2 s1.
The paper presents the homemade setup in which effects disturbing FIA peaks shape have been minimized. The change of conductance of solutions of KCl, NaCl, KMnO4 or CuSO4 during dispersion in water allowed determination of diffusion coefficients of the salts using the optimization method coupled with numerical computations. The best-fit dependencies agree with the experimental responses and the determined diffusion coefficients agree with the literature values determined by other methods. 2. Materials and methods Sample solutions of concentrations of 102 M used in experiments were prepared by dissolving four salts: KCl, NaCl, KMnO4 and CuSO4 of an analytical-reagent grade of purity in distilled water, which was also used as a carrier. Dispersion was studied using a homemade setup that is illustrated in Fig. 1. The carrier flow was forced by means of gravitation, which allowed to eliminate pulsations disturbing shape of FIA peaks [16,22]. Sample solutions (of volume of 223 mL) were introduced to a vertically placed, straight PVC channel of an internal diameter of 2.87 mm by using a two-position syringe with two channels formed on a piston surface, as illustrated in Fig. 1. Configuration of the detection cell was similar to the one presented by Hohercakova and co-workers [8,9]. Pt wires of 0.05 mm diameter were placed transversally to the channel axis on the distance of 1 mm, see Fig. 1. It is worth noting that such configuration of the detector maintained the channel geometry, thus eliminating a mismatch potentially
disturbing the peaks shape [16,22]. The Pt wires were stabilized in positions by the hot-melt glue. For detection, impedance measurements of the solution between Pt wires were carried out. The Autolab PGSTAT 302N equipped with a frequency response analyzer for the amplitude 0.35 V over the frequency range 1000– 1010 s1 was used. The distance between the injection and the detection positions was 2 m. The flow rate was preliminary fixed with a flowmeter (FM) placed between the detector and the outlet, and verified gravimetrically. All experiments were carried out at an ambient temperature, varying from 21.5 to 22.5 C. 3. Theory and calculations For low concentrations ( 1000, can be neglected [16,19]). Under the above assumptions
the diffusion–convection equation can be written in cylindrical coordinates as: @c D @ @c @c ¼ r (2) v @t r @r @r @x The diffusion–convection problem is solved numerically using the recently presented approach [18]. Numerical computations
Fig. 4. On the left: concentration profiles for (a) t = 0.1, (b) 0.2, (c) 0.3 and (d) 0.8 calculated after the half-residence time (the channel dimensions range from an initial plug front to the detector position in the axial direction and from r0 to r0 in the radial direction). On the right: predicted peak shapes, presented here as relative concentrations.
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were divided into steps during which the following actions occur: (1) computations of concentration accumulation in the radial direction, (2) translation of nodes in the axial direction according to the parabolic velocity profile (expressed by Eq. (1)), and (3) interpolation of concentration values for a new nodes distribution (allowing a subsequent integration in the radial direction). Such partitioning accelerates calculations, lowers memory usage and allows adapting a grid to the sample plug. The radial diffusion equation is solved using the method of lines and the RADAU5 solver (an implicit Runge–Kutta method of order 5) built-in Mathcad 14. Accumulations in the first and last radial nodes are conformed to the Neumann boundary condition (zero flux at r = 0 and r = r0). For the total number of time steps of 70 and 1400 spatial nodes, grid-independent solutions were obtained. The impedance measured at high frequencies are related to the bulk properties of electrolytes. In terms of equivalent circuits, they can be interpreted as the charging of the bulk capacitance C in parallel to the resistance R (inversely proportional to salt concentration). In this case real Z0 and complex Z00 components of impedance are given by: Z0 ¼
RðvC Þ2
(3)
2
R þ ðvC Þ 2
1
R ðvC Þ Z 00 ¼ 2 R þ ðvC Þ2 2
(4)
The solution of the above set of equations is the bulk resistance R and capacitance C of the carrier with a dispersed sample: R¼
Z 02 þ Z 002 Z0
C¼
Z 00
v Z02 þ Z 002
(5)
(6)
Fig. 5. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for KCl at flow rate 2.05 mL min1.
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Resistance of solution flowing through the detector changes in the radial direction. For very narrow electrodes in close proximity multiple single resistors in parallel can approximate the equivalent circuit. This condition was fulfilled since separation between Pt wires (of 0.05 mm diameter) was 1 mm, much lower than the initial width of sample plug (34 mm), which increase by about two orders of magnitude when passing the detector due to dispersion. For parallel connection of resistors their conductances add, therefore the measured change of solution conductance (inverse of resistance) will be proportional to the average concentration in the detector: Z 1 r0 c¼ cðrÞdr (7) r0 0 Fluid carrier contribution to the overall conductance was eliminated by subtraction of the conductance of carrier (initial values) from the conductance of dispersed solution. Because the radial nodes were distributed uniformly Eq. (7) had the following numerical form: c¼
1 Nr S cn 1 if 1 < n < Nr 0:5 otherwise Nr 1n¼1
(8)
where Nr denotes the total number of radial nodes. Simulated results were normalized similarly as in the works published by van Akker and co-workers [23,24]: (numerical response) (average experimental response)/(average numerical response). Numerical responses were fitted to the experimental ones using the Levenberg–Marquardt method of minimization of residuals, which combines the steepest descent and the Gauss–Newton methods. The value of the estimate of diffusion coefficients (an initial value where optimization starts) was 109 m2 s1 for all considered cases. Numerical calculations as well as optimizations were carried out using programing tools and built-in functions in Mathcad 14 by PTC. The mean time required for the determination of a single diffusion coefficient was 20 min on a computer with a 2.53 GHz Core 2 Duo processor and 4 GB RAM. The Mathcad files
Fig. 6. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for NaCl at flow rate 2.25 mL min1.
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In order to verify accuracy of numerical solutions, computations for different grid densities were carried out. A usual criterion for grid independent solutions is doubling of nodes number resulting in an error below a certain threshold (e.g., below an experimental error). In the work it was assumed that the normalized root mean square error (NRMSE) below 1% satisfies the condition. Numerical results for different numbers of (axial nodes)/(radial nodes)/ (temporal steps) calculated for diffusion coefficients 1.8 109 and 9 1010 m2 s1 are presented in Fig. 2a and b, respectively. Calculations referred to experimental conditions are presented in Section 2. The flow rate was 2.3 mL min1. The NRMSEs for responses calculated for defined diffusion coefficients for configuration 50/15/50 were 1.14 and 0.51%, respectively, for 70/20/70: 0.58 and 0.32% and for 100/30/100: 0.23 and 0.12%, in relation to configuration 140/40/140, taken as a reference. Therefore, further simulations and determination of diffusion coefficients were conducted for configuration 70/20/70 that met the established criterion. These results were also confronted with those from an independent method, i.e., the random walking model (RW), as compared in Fig. 3a and b. The RW calculations are based on the method presented by Betteridge et al. [25], exploited later by Wentzell et al. [22] and improved recently by Kucza [18]. In the approach a sample solution is considered as a number of individual molecules (here 106 molecules) that take random steps (dependent on the diffusion coefficient) and are translated downstream due to convection. The results of both methods coincide, the only difference is presence of a noise on FIA responses, inherent for the random walking model. Concentration profiles for t = 0.1, 0.2 and 0.3 calculated after the half-residence time are illustrated in Fig. 4a–c on the left. Additionally, the concentration profile for Taylor dispersion (for t = 0.8) is presented in Fig. 4d. The channel dimensions range from an initial plug front to the detector position in the axial direction,
and from r0 to r0 in the radial direction. The predicted peak shapes, presented here as relative concentrations, are shown in Fig. 4 on the right. FIA responses strongly depend on the Fourier number; for lower values the first peak is more pronounced whilst for the higher ones, the second peak increasingly dominates. The first peak is very sharp, it starts at the half-residence time when first sample molecules (placed initially on the channel axis and thus having maximal velocity) reach the detector. Its shape is characteristic for high flow rates and/or low diffusion coefficients on convection-driven dispersion (when the radial diffusion plays a minimal role). For higher values of Fourier numbers the radial diffusion equalizes concentration in the sample plug and the second peak becomes more pronounced with a maximum at the residence time. For high Fourier numbers diffusion plays a key role, however even for t = 0.8 concentration in sample plug is still unaligned, as reflected by asymmetrical FIA response in Fig. 4d. It is worth noting that the present experimental setup is characterized by relatively large sensitivity, as the dispersion (ratio of the initial concentration to a maximal one in the detector) was from 25 to 55. Such dilution of the sample solutions (of initial concentration 102 M) should results in the linear dependence between measured electrical conductance changes and salt concentration. Animations of concentration profiles (AVI files) for the four values of Fourier numbers are available as Supplementary material. A strong dependence of peak shapes on Fourier numbers, make the approach suitable for determination of diffusion coefficients. Measurements were carried out eight times for each salt for flow rates 2.0–2.6 mL min1. The conductance changes of KCl, NaCl, KMnO4 and CuSO4 solutions measured by the impedance technique are presented in Figs. 5–8, respectively. For the sake of clarity, only five from eight responses are presented in figures. Experimental responses (points) and the best-fit dependencies (lines) calculated numerically coincide showing the doublehumped peak behavior, as expected for low Fourier numbers. The experimental FI conditions are collected in Table 1 in the first four columns. The literature diffusion coefficients are shown in the fifth column. These values were calculated on the grounds of the literature data [26–29] and Stokes–Einstein equation (since all
Fig. 7. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for KMnO4 at flow rate 2.42 mL min1.
Fig. 8. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for CuSO4 at flow rate 2.60 mL min1.
allowing numerical simulation and optimization can be obtained free from the author via e-mail. 4. Results and discussion
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Table 1 Experimental FI conditions on impedance measurements and the determined diffusion coefficients of salts in water. Literature values in column 5 are presented for comparison. Salt
Mean flow rate (mL min1)
Mean temperature ( C)
Number of determinations
Dlit 109 (m2 s1)
Dexp 109 (m2 s1)
Fourier number
Peclet number
KCl NaCl KMnO4 CuSO4
2.05 2.25 2.42 2.60
21.6 22.0 22.5 22.5
8 8 8 8
1.79 [28] 1.49 [27] 1.30–1.47 [10] 0.80 [25,26]
1.83 0.01 1.53 0.01 1.57 0.01 0.90 0.01
0.34 0.26 0.25 0.13
4141 5437 5688 10,654
solutions were diluted, the dynamic viscosity of water [30] in respective temperatures was taken for calculations). The values of diffusion coefficients determined using the present method are collected in column 6. The experimental values are in good agreement with the literature ones, determined by other methods, proving the validity of the presented approach. The relative error of the present diffusion coefficients determinations is 1.5% at most, which is lower than while using e.g., calibration method: 4% [17]. Moreover, the present experimental procedure took less than 15 min (35 with calculations), which compares favorably with about 1 h for the Vanderslice method [17], more than 2 h for Taylor dispersion analyses [10,12,31] and several hours for the conductometric [32] and diaphragm methods [26]. The data in Table 1 are completed with the Fourier t and Peclet Pe numbers with values confirming the present model assumptions. The agreement between experimental and best-fit responses slightly differs for each of the salts. The best-fit quality is obtained in Fig. 6 for NaCl, which exhibits extremely small variation of diffusion coefficient on concentration (about 0.6% in concentration range from 0.75 103 to 102 M) [28], The difference between literature diffusion coefficient value and the present one is about 2.5%. Slightly worse fit qualities were obtained for the other two 1:1 salts, KCl in Fig. 5 and KMnO4 in Fig. 7. Also in these cases, differences could be partially attributed to composition-dependent diffusion coefficients. In the case of KCl, literature value of diffusion coefficient vary from 1.74 109 m2 s1 for concentration of 102 M to 1.82 109 m2 s1 at infinite dilution, with character of this dependence being rather complex. The difference between literature diffusion coefficient value and that obtained by our method is about 4.0%. In the case of KMnO4 (Fig. 7) the dependence between diffusion coefficient value and concentration is much stronger; basing on the results obtained by Taylor [10], the value of diffusion coefficient of KMnO4 changes from 4.90 1010 m2 s1 for concentration of 6.3 102 M to 1.698 109 m2 s1 at infinite dilution. Still, the quality fit can be considered as satisfactory, with the obtained value being relatively high, due to high dilution of the initial sample. The biggest differences between experimental results and fitted responses can be seen in Fig. 8 for CuSO4, which is the only 2:2 salt considered in this work. Literature values of diffusion coefficient vary from 6.27 1010 m2 s1 for concentration of 6.3 102 M to 7.99 1010 m2 s1 at infinite dilution [26,27]. The value of diffusion coefficient obtained by our method was found to be 9.02 1010 m2 s1, 13% more than expected value. This is probably due to the fact that the CuSO4 does not meet all criteria of the model, especially the one about ideality of the solution. From the four salts the best agreement between experimental and theoretical results (regarding both FIA responses and diffusion coefficients) was obtained for NaCl, which diffusion coefficient is almost concentration-independent. Therefore dependence or independence of diffusion coefficient on concentration seems to be critical regarding the validity of this uniform dispersion approach. For other cases agreement is good, evidencing that main experimental non-idealities were eliminated in the presented flow-injection system by avoiding pulsations and
maintaining the same geometry for the channel and the detector. Still present differences between the experimental and the calculated responses can be credited to: different from unity activity coefficients (especially for CuSO4 being the 2:2 salt), deviation from sharp carrier/sample/carrier initial interfaces assumed in computations, minimal changes in a cross section area at the detector and the lack of thermostating. 5. Conclusions The presented approach allows modeling of dispersion in cylindrical channels in many aspects. Here concentration profiles have been compared and related to resulting FIA responses at low Fourier numbers. Good agreement between experimental results and theoretical predictions shows usefulness of the experimental setup and the numerical method. The determined diffusion coefficients of KCl, NaCl, KMnO4 and CuSO4 are in accordance with the values determined by other methods. It is worth noting that time needed for determination of diffusion coefficients can be reduced by using a tubing of lower internal diameters. The measurements at low Fourier numbers (for 0.1 < t < 0.3) will always be at least 3–8 times faster than those on Taylor dispersion (t > 0.8). The impedance detection used in the work, with inherent speed of measurements and simplicity of the detector construction, make the approach suitable for determination of diffusion coefficients of wide range of substances (not only electrolytes) of even very high specific resistances.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.aca.2015.04.049. References [1] J. Ruzicka, E.H. Hansen, Flow injection analyses: part I. A new concept of fast continuous flow analysis, Anal. Chim. Acta 78 (1975) 145–157. [2] E.H. Hansen, Bibliography of flow based micro-analytical techniques. http:// www.flowinjection.com/tutorials/database, 2014 (accessed 13.11.14). [3] S.D. Kolev, I.D. McKelvie, Advances in Flow Injection Analysis and Related Techniques, Elsevier, Hungary, 2008. [4] E.H. Hansen, M. Miro, How flow-injection analysis (FIA) over the past 25 years has changed our way of performing chemical analyses, Trends Anal. Chem. 26 (2007) 18–26. [5] M.T. Oms, A. Cerda, V. Cerda, Preconcentration by flow reversal in conductometric sequential injection analysis of ammonium, Electroanalysis 8 (1996) 387–390. [6] J.R. Farrell, P.J. Iles, Y.J. Yuan, Determination of arsenic by hydride generation gas diffusion flow injection analysis with electrochemical detection, Anal. Chim. Acta 334 (1996) 193–197. [7] P.C. Hauser, Z. Zhang, Flow-injection determination of lead by hydride generation and conductometric detection, Fresen. J. Anal. Chem. 355 (1996) 141–143. [8] Z. Hohercakova, F. Opekar, A contactless conductivity detection cell for flow injection analysis: determination of total inorganic carbon, Anal. Chim. Acta 551 (2005) 132–136. [9] Z. Hohercakova, F. Opekar, K. Stulik, Thinly insulated wire cells – a new device for sensitive contactless conductivity detection in flow analyses, Electroanalysis 17 (2005) 1924–1930.
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[10] G. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. Lond. A 219 (1953) 186–203. [11] G. Taylor, Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion, Proc. R. Soc. Lond. A 225 (1954) 473–477. [12] D.G. Leaist, Determination of ternary diffusion coefficients by the Taylor dispersion method, J. Phys. Chem. 94 (1990) 5180–5183. [13] H. Cottet, J. Biron, L. Cipelletti, R. Matmour, M. Martin, Determination of individual diffusion coefficients in evolving binary mixtures by Taylor dispersion analysis: application to the monitoring of polymer reaction, Anal. Chem. 82 (2010) 1793–1802. [14] E.D. Snijder, M. te Riele, G.F. Versteeg, W. van Swaaij, Diffusion coefficients of CO, CO2 N2O, and N2 in ethanol and toluene, J. Chem. Eng. Data 40 (1996) 37–39. [15] H. Cottet, M. Martin, A. Papillaud, E. Souaid, H. Collet, A. Commeyras, Determination of dendrigraft poly-L-lysine diffusion coefficients by Taylor dispersion analysis, Biomacromolecules 8 (2007) 3235–3243. [16] J.T. Vanderslice, K.K. Stewart, A.G. Rosenfeld, D.J. Higgs, Laminar dispersion in flow-injection analysis, Talanta 28 (1981) 11–18. [17] G. Gerhardt, R.N. Adams, Determination of diffusion coefficients by flow injection analysis, Anal. Chem. 54 (1982) 2618–2620. [18] W. Kucza, Flow injection analysis simulations and diffusion coefficient determination by stochastic and deterministic optimization methods, Anal. Chim. Acta 788 (2013) 74–80. [19] H. Bate, S. Rowlands, J.A. Sirs, H.W. Thomas, The dispersion of diffusible ions in fluid flow through a cylindrical tube, J. Phys. D: Appl. Phys. 2 (1969) 1447–1456. [20] T. Korenaga, Aspects of sample dispersion for optimizing flow-injection analysis systems, Anal. Chim. Acta 261 (1992) 539–548. [21] S. Kolev, Mathematical modelling of flow-injection systems, Anal. Chim. Acta 308 (1995) 36–66.
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Analytica Chimica Acta xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Analytica Chimica Acta journal homepage: www.elsevier.com/locate/aca
Dispersion in cylindrical channels on the laminar flow at low Fourier numbers Witold Kucza * , Juliusz Da˛browa, Katarzyna Nawara AGH University of Science and Technology, Faculty of Materials Science and Ceramics, Al. Mickiewicza 30, 30-059 Cracow, Poland
H I G H L I G H T S
G R A P H I C A L A B S T R A C T
Simulations and measurements of dispersion at low Fourier numbers are presented. Double-humped peaks were measured by impedance detection. Numerical responses were fitted to experimental ones by an optimization method. The determined diffusion coefficients of salts agree well with the literature data.
A R T I C L E I N F O
A B S T R A C T
Article history: Received 20 February 2015 Received in revised form 19 April 2015 Accepted 23 April 2015 Available online xxx
A numerical solution of the uniform dispersion model in cylindrical channels at low Fourier numbers is presented. The presented setup allowed to eliminate experimental non-idealities interfering the laminar flow. Double-humped responses measured in a flow injection system with impedance detection agreed with those predicted by theory. Simulated concentration profiles as well as flow injection analysis (FIA) responses show the predictive and descriptive power of the numerical approach. A strong dependence of peak shapes on Fourier numbers, at its low values, makes the approach suitable for determination of diffusion coefficients. In the work, the uniform dispersion model coupled with the Levenberg–Marquardt method of optimization allowed to determine the salt diffusion coefficient for KCl, NaCl, KMnO4 and CuSO4 in water. The determined values (1.83, 1.53, 1.57 and 0.90) 109 m2 s1, respectively, agree well with the literature data. ã 2015 Elsevier B.V. All rights reserved.
Keywords: Flow injection system Impedance measurements Numerical modeling Optimization Diffusion coefficient
1. Introduction
* Corresponding author. Tel.: +48 126 172 466; fax: +48 126 172 493. E-mail address: [email protected] (W. Kucza).
Flow injection analysis (FIA) is based on an injection of a sample solution into a carrier stream, where controllable dispersion and generation of a reproducible signal at a detector occur. Since its introduction in early 1970s by Hansen and Ruzicka, FIA has become one of the primary techniques in
http://dx.doi.org/10.1016/j.aca.2015.04.049 0003-2670/ ã 2015 Elsevier B.V. All rights reserved.
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chemical analysis [1–4]. The main characteristics of FIA are: signal reproducibility, accuracy and extremely short times of analysis. The most popular methods of detection used in FIA are: photometry, luminescence and electrochemical detection [3]. Conductivity measurements, although not selective ones due to the contribution of all ions to the electrical conductivity, are an attractive choice for electrochemical detection [3,5–9]. Their advantages are: speed of measurements, simplicity of the detector construction [4,7,8] and capability of lowering the detection limit when combined with gas diffusion separation [8]. FIA is usually used in chemical analysis, but it can be also used for kinetic parameters determination, e.g., diffusion coefficients. The use of FIA in this field is limited to systems of high Fourier numbers (t ¼ Dtm =r20 where: D denotes the diffusion coefficient; tm: the mean residence time; r0: the channel radius) where Taylor’s analytical solution [10,11] is valid [12–15] or requires a calibration technique, which is based on the Vanderslice numerical calculations [16,17]. The first approach requires low flow rates (long residence times) whilst the second one requires using standards of known diffusion coefficients (for which calibration factors are determined) [17]. These problems have been recently evaded by
using the inverse methods based on deterministic or stochastic numerical computations, allowing to determine diffusion coefficients based upon the shape of FIA response curves [18]. In the present work, the numerical solution of the uniform dispersion model is used for simulations of dispersion in cylindrical channels at low Fourier numbers. It is well known that such condition requires application of numerical methods [16,19,20]. Analytical solutions of diffusion–convection equation are valid when convection (for t < 0.1) or diffusion (for t > 0.6) dominates the overall dispersion [20]. At Fourier numbers between 0.1 and 0.3 asymmetrical, double-humped peaks are theoretically predicted [16,21]. However, such behavior is rarely observed due to experimental non-idealities: turbulence generated by pump pulsation, mismatches between sample loop and tubing and between tubing and detector cell as well as coiling and other deformation of channels, which are all influencing the ideal laminar flow [16,20,22]. From the above, flow injection analysis at low Fourier numbers is a challenge in regards to both experimental implementation as well as mathematical modeling. However, a success reached in both fields may result in accelerating the analysis since experiments would be carried for higher flow rates.
Fig. 1. A schematic illustration of the homemade FIA setup. Insets on the right side illustrate: injection systems (at the top) and the impedance detector (at the bottom).
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Fig. 2. Numerically simulated FIA responses for different numbers of (axial nodes)/(radial nodes)/(temporal steps) for diffusion coefficients: (a) 1.8 109 and (b) 9 1010 m2 s1.
The paper presents the homemade setup in which effects disturbing FIA peaks shape have been minimized. The change of conductance of solutions of KCl, NaCl, KMnO4 or CuSO4 during dispersion in water allowed determination of diffusion coefficients of the salts using the optimization method coupled with numerical computations. The best-fit dependencies agree with the experimental responses and the determined diffusion coefficients agree with the literature values determined by other methods. 2. Materials and methods Sample solutions of concentrations of 102 M used in experiments were prepared by dissolving four salts: KCl, NaCl, KMnO4 and CuSO4 of an analytical-reagent grade of purity in distilled water, which was also used as a carrier. Dispersion was studied using a homemade setup that is illustrated in Fig. 1. The carrier flow was forced by means of gravitation, which allowed to eliminate pulsations disturbing shape of FIA peaks [16,22]. Sample solutions (of volume of 223 mL) were introduced to a vertically placed, straight PVC channel of an internal diameter of 2.87 mm by using a two-position syringe with two channels formed on a piston surface, as illustrated in Fig. 1. Configuration of the detection cell was similar to the one presented by Hohercakova and co-workers [8,9]. Pt wires of 0.05 mm diameter were placed transversally to the channel axis on the distance of 1 mm, see Fig. 1. It is worth noting that such configuration of the detector maintained the channel geometry, thus eliminating a mismatch potentially
disturbing the peaks shape [16,22]. The Pt wires were stabilized in positions by the hot-melt glue. For detection, impedance measurements of the solution between Pt wires were carried out. The Autolab PGSTAT 302N equipped with a frequency response analyzer for the amplitude 0.35 V over the frequency range 1000– 1010 s1 was used. The distance between the injection and the detection positions was 2 m. The flow rate was preliminary fixed with a flowmeter (FM) placed between the detector and the outlet, and verified gravimetrically. All experiments were carried out at an ambient temperature, varying from 21.5 to 22.5 C. 3. Theory and calculations For low concentrations ( 1000, can be neglected [16,19]). Under the above assumptions
the diffusion–convection equation can be written in cylindrical coordinates as: @c D @ @c @c ¼ r (2) v @t r @r @r @x The diffusion–convection problem is solved numerically using the recently presented approach [18]. Numerical computations
Fig. 4. On the left: concentration profiles for (a) t = 0.1, (b) 0.2, (c) 0.3 and (d) 0.8 calculated after the half-residence time (the channel dimensions range from an initial plug front to the detector position in the axial direction and from r0 to r0 in the radial direction). On the right: predicted peak shapes, presented here as relative concentrations.
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were divided into steps during which the following actions occur: (1) computations of concentration accumulation in the radial direction, (2) translation of nodes in the axial direction according to the parabolic velocity profile (expressed by Eq. (1)), and (3) interpolation of concentration values for a new nodes distribution (allowing a subsequent integration in the radial direction). Such partitioning accelerates calculations, lowers memory usage and allows adapting a grid to the sample plug. The radial diffusion equation is solved using the method of lines and the RADAU5 solver (an implicit Runge–Kutta method of order 5) built-in Mathcad 14. Accumulations in the first and last radial nodes are conformed to the Neumann boundary condition (zero flux at r = 0 and r = r0). For the total number of time steps of 70 and 1400 spatial nodes, grid-independent solutions were obtained. The impedance measured at high frequencies are related to the bulk properties of electrolytes. In terms of equivalent circuits, they can be interpreted as the charging of the bulk capacitance C in parallel to the resistance R (inversely proportional to salt concentration). In this case real Z0 and complex Z00 components of impedance are given by: Z0 ¼
RðvC Þ2
(3)
2
R þ ðvC Þ 2
1
R ðvC Þ Z 00 ¼ 2 R þ ðvC Þ2 2
(4)
The solution of the above set of equations is the bulk resistance R and capacitance C of the carrier with a dispersed sample: R¼
Z 02 þ Z 002 Z0
C¼
Z 00
v Z02 þ Z 002
(5)
(6)
Fig. 5. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for KCl at flow rate 2.05 mL min1.
5
Resistance of solution flowing through the detector changes in the radial direction. For very narrow electrodes in close proximity multiple single resistors in parallel can approximate the equivalent circuit. This condition was fulfilled since separation between Pt wires (of 0.05 mm diameter) was 1 mm, much lower than the initial width of sample plug (34 mm), which increase by about two orders of magnitude when passing the detector due to dispersion. For parallel connection of resistors their conductances add, therefore the measured change of solution conductance (inverse of resistance) will be proportional to the average concentration in the detector: Z 1 r0 c¼ cðrÞdr (7) r0 0 Fluid carrier contribution to the overall conductance was eliminated by subtraction of the conductance of carrier (initial values) from the conductance of dispersed solution. Because the radial nodes were distributed uniformly Eq. (7) had the following numerical form: c¼
1 Nr S cn 1 if 1 < n < Nr 0:5 otherwise Nr 1n¼1
(8)
where Nr denotes the total number of radial nodes. Simulated results were normalized similarly as in the works published by van Akker and co-workers [23,24]: (numerical response) (average experimental response)/(average numerical response). Numerical responses were fitted to the experimental ones using the Levenberg–Marquardt method of minimization of residuals, which combines the steepest descent and the Gauss–Newton methods. The value of the estimate of diffusion coefficients (an initial value where optimization starts) was 109 m2 s1 for all considered cases. Numerical calculations as well as optimizations were carried out using programing tools and built-in functions in Mathcad 14 by PTC. The mean time required for the determination of a single diffusion coefficient was 20 min on a computer with a 2.53 GHz Core 2 Duo processor and 4 GB RAM. The Mathcad files
Fig. 6. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for NaCl at flow rate 2.25 mL min1.
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In order to verify accuracy of numerical solutions, computations for different grid densities were carried out. A usual criterion for grid independent solutions is doubling of nodes number resulting in an error below a certain threshold (e.g., below an experimental error). In the work it was assumed that the normalized root mean square error (NRMSE) below 1% satisfies the condition. Numerical results for different numbers of (axial nodes)/(radial nodes)/ (temporal steps) calculated for diffusion coefficients 1.8 109 and 9 1010 m2 s1 are presented in Fig. 2a and b, respectively. Calculations referred to experimental conditions are presented in Section 2. The flow rate was 2.3 mL min1. The NRMSEs for responses calculated for defined diffusion coefficients for configuration 50/15/50 were 1.14 and 0.51%, respectively, for 70/20/70: 0.58 and 0.32% and for 100/30/100: 0.23 and 0.12%, in relation to configuration 140/40/140, taken as a reference. Therefore, further simulations and determination of diffusion coefficients were conducted for configuration 70/20/70 that met the established criterion. These results were also confronted with those from an independent method, i.e., the random walking model (RW), as compared in Fig. 3a and b. The RW calculations are based on the method presented by Betteridge et al. [25], exploited later by Wentzell et al. [22] and improved recently by Kucza [18]. In the approach a sample solution is considered as a number of individual molecules (here 106 molecules) that take random steps (dependent on the diffusion coefficient) and are translated downstream due to convection. The results of both methods coincide, the only difference is presence of a noise on FIA responses, inherent for the random walking model. Concentration profiles for t = 0.1, 0.2 and 0.3 calculated after the half-residence time are illustrated in Fig. 4a–c on the left. Additionally, the concentration profile for Taylor dispersion (for t = 0.8) is presented in Fig. 4d. The channel dimensions range from an initial plug front to the detector position in the axial direction,
and from r0 to r0 in the radial direction. The predicted peak shapes, presented here as relative concentrations, are shown in Fig. 4 on the right. FIA responses strongly depend on the Fourier number; for lower values the first peak is more pronounced whilst for the higher ones, the second peak increasingly dominates. The first peak is very sharp, it starts at the half-residence time when first sample molecules (placed initially on the channel axis and thus having maximal velocity) reach the detector. Its shape is characteristic for high flow rates and/or low diffusion coefficients on convection-driven dispersion (when the radial diffusion plays a minimal role). For higher values of Fourier numbers the radial diffusion equalizes concentration in the sample plug and the second peak becomes more pronounced with a maximum at the residence time. For high Fourier numbers diffusion plays a key role, however even for t = 0.8 concentration in sample plug is still unaligned, as reflected by asymmetrical FIA response in Fig. 4d. It is worth noting that the present experimental setup is characterized by relatively large sensitivity, as the dispersion (ratio of the initial concentration to a maximal one in the detector) was from 25 to 55. Such dilution of the sample solutions (of initial concentration 102 M) should results in the linear dependence between measured electrical conductance changes and salt concentration. Animations of concentration profiles (AVI files) for the four values of Fourier numbers are available as Supplementary material. A strong dependence of peak shapes on Fourier numbers, make the approach suitable for determination of diffusion coefficients. Measurements were carried out eight times for each salt for flow rates 2.0–2.6 mL min1. The conductance changes of KCl, NaCl, KMnO4 and CuSO4 solutions measured by the impedance technique are presented in Figs. 5–8, respectively. For the sake of clarity, only five from eight responses are presented in figures. Experimental responses (points) and the best-fit dependencies (lines) calculated numerically coincide showing the doublehumped peak behavior, as expected for low Fourier numbers. The experimental FI conditions are collected in Table 1 in the first four columns. The literature diffusion coefficients are shown in the fifth column. These values were calculated on the grounds of the literature data [26–29] and Stokes–Einstein equation (since all
Fig. 7. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for KMnO4 at flow rate 2.42 mL min1.
Fig. 8. The measured FIA responses (conductance changes) as points and the bestfit dependencies (lines) obtained by the present approach for CuSO4 at flow rate 2.60 mL min1.
allowing numerical simulation and optimization can be obtained free from the author via e-mail. 4. Results and discussion
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Table 1 Experimental FI conditions on impedance measurements and the determined diffusion coefficients of salts in water. Literature values in column 5 are presented for comparison. Salt
Mean flow rate (mL min1)
Mean temperature ( C)
Number of determinations
Dlit 109 (m2 s1)
Dexp 109 (m2 s1)
Fourier number
Peclet number
KCl NaCl KMnO4 CuSO4
2.05 2.25 2.42 2.60
21.6 22.0 22.5 22.5
8 8 8 8
1.79 [28] 1.49 [27] 1.30–1.47 [10] 0.80 [25,26]
1.83 0.01 1.53 0.01 1.57 0.01 0.90 0.01
0.34 0.26 0.25 0.13
4141 5437 5688 10,654
solutions were diluted, the dynamic viscosity of water [30] in respective temperatures was taken for calculations). The values of diffusion coefficients determined using the present method are collected in column 6. The experimental values are in good agreement with the literature ones, determined by other methods, proving the validity of the presented approach. The relative error of the present diffusion coefficients determinations is 1.5% at most, which is lower than while using e.g., calibration method: 4% [17]. Moreover, the present experimental procedure took less than 15 min (35 with calculations), which compares favorably with about 1 h for the Vanderslice method [17], more than 2 h for Taylor dispersion analyses [10,12,31] and several hours for the conductometric [32] and diaphragm methods [26]. The data in Table 1 are completed with the Fourier t and Peclet Pe numbers with values confirming the present model assumptions. The agreement between experimental and best-fit responses slightly differs for each of the salts. The best-fit quality is obtained in Fig. 6 for NaCl, which exhibits extremely small variation of diffusion coefficient on concentration (about 0.6% in concentration range from 0.75 103 to 102 M) [28], The difference between literature diffusion coefficient value and the present one is about 2.5%. Slightly worse fit qualities were obtained for the other two 1:1 salts, KCl in Fig. 5 and KMnO4 in Fig. 7. Also in these cases, differences could be partially attributed to composition-dependent diffusion coefficients. In the case of KCl, literature value of diffusion coefficient vary from 1.74 109 m2 s1 for concentration of 102 M to 1.82 109 m2 s1 at infinite dilution, with character of this dependence being rather complex. The difference between literature diffusion coefficient value and that obtained by our method is about 4.0%. In the case of KMnO4 (Fig. 7) the dependence between diffusion coefficient value and concentration is much stronger; basing on the results obtained by Taylor [10], the value of diffusion coefficient of KMnO4 changes from 4.90 1010 m2 s1 for concentration of 6.3 102 M to 1.698 109 m2 s1 at infinite dilution. Still, the quality fit can be considered as satisfactory, with the obtained value being relatively high, due to high dilution of the initial sample. The biggest differences between experimental results and fitted responses can be seen in Fig. 8 for CuSO4, which is the only 2:2 salt considered in this work. Literature values of diffusion coefficient vary from 6.27 1010 m2 s1 for concentration of 6.3 102 M to 7.99 1010 m2 s1 at infinite dilution [26,27]. The value of diffusion coefficient obtained by our method was found to be 9.02 1010 m2 s1, 13% more than expected value. This is probably due to the fact that the CuSO4 does not meet all criteria of the model, especially the one about ideality of the solution. From the four salts the best agreement between experimental and theoretical results (regarding both FIA responses and diffusion coefficients) was obtained for NaCl, which diffusion coefficient is almost concentration-independent. Therefore dependence or independence of diffusion coefficient on concentration seems to be critical regarding the validity of this uniform dispersion approach. For other cases agreement is good, evidencing that main experimental non-idealities were eliminated in the presented flow-injection system by avoiding pulsations and
maintaining the same geometry for the channel and the detector. Still present differences between the experimental and the calculated responses can be credited to: different from unity activity coefficients (especially for CuSO4 being the 2:2 salt), deviation from sharp carrier/sample/carrier initial interfaces assumed in computations, minimal changes in a cross section area at the detector and the lack of thermostating. 5. Conclusions The presented approach allows modeling of dispersion in cylindrical channels in many aspects. Here concentration profiles have been compared and related to resulting FIA responses at low Fourier numbers. Good agreement between experimental results and theoretical predictions shows usefulness of the experimental setup and the numerical method. The determined diffusion coefficients of KCl, NaCl, KMnO4 and CuSO4 are in accordance with the values determined by other methods. It is worth noting that time needed for determination of diffusion coefficients can be reduced by using a tubing of lower internal diameters. The measurements at low Fourier numbers (for 0.1 < t < 0.3) will always be at least 3–8 times faster than those on Taylor dispersion (t > 0.8). The impedance detection used in the work, with inherent speed of measurements and simplicity of the detector construction, make the approach suitable for determination of diffusion coefficients of wide range of substances (not only electrolytes) of even very high specific resistances.
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