Anal. Chem. 1999, 71, 2199-2204
Hydrodynamics and Mass Transfer of the Coaxial Jet Mixer in Flow Injection Analysis V. P. Andreev,† S. B. Koleshko,‡ D. A. Holman,§ L. D. Scampavia,§ and G. D. Christian*,§
Institute for Analytical Instrumentation, Russian Academy of Sciences, 26 Rigsky pr., St. Petersburg 198103, Russia, Department of Hydrodynamics, St. Petersburg State Technical University, 29 Polytechnicheskaya st., St. Petersburg 195251, Russia, and Department of Chemistry, University of Washington, Box 351700, Seattle, Washington 98195-1700
A coaxial jet mixer that was previously proposed for rapid and efficient mixing under laminar flow conditions has been studied both theoretically and experimentally. A mathematical model that consists of a set of Navie-Stokes equations that determine the flow velocities and three diffusion-convection reaction equations that determine the reactant and product concentrations has been developed. Equations are solved with the help of finite difference techniques for different flow conditions. The quality of sample and reagent mixing is characterized by the mean product concentration and the amount of product produced. Theoretical results are compared with experimental ones for the mixing of bromothymol blue (a pH indicator) in the outer capillary with NaOH in the inner capillary of the jet mixer. One of the important problems of flow injection analysis is the improvement of quality of sample and reagent mixing under the laminar flow conditions. A simple coaxial jet mixer for rapid and efficient mixing was proposed in ref 1 and was experimentally shown to be capable of efficient mixing in as little as 0.055 s. The cross-section parameters of a coaxial mixer are reported in Figure 1, with one of the reactants coming through the inner capillary while another one is coming through the outer capillary. In this previous study,1 mixing was performed by introducing two solutions of Bromothymol Blue indicator (BTB) in pH 6.01 and 7.56 buffers through the inner and outer capillaries of the mixer, respectively. The quality of mixing was estimated by the closeness of the absorbance value to that of the ideally mixed and equilibrated BTB solutions. The objective of the present study was to investigate the mixing process in more detail, both by mathematical modeling and experimentation. The practically instantaneous acid/base reaction was chosen in order to reveal the real dynamics of mixing. For the same reason, reactants and the detector wavelength were chosen in such a way that absorbance was due only to the product of the chemical reaction. Thus, the measured absorbance could be directly compared with the calculated values of the mean product concentration. * Corresponding author. Tel.: 206-543-5340 or -1635. Fax: 206-543-6506. E-mail:
[email protected]. † Russian Academy of Sciences. ‡ St. Petersburg State Technical University. § University of Washington. (1) Scampavia, L. D.; Blankenstein, G.; Ruzicka, J.; Christian, G. D. Anal. Chem. 1995, 67, 2743-2749. 10.1021/ac981037t CCC: $18.00 Published on Web 04/30/1999
© 1999 American Chemical Society
Figure 1. Cross section of coaxial jet mixer, where 2a ) 580 µm, 2b ) 247 µm, and 2c ) 350 µm.
THEORY Consider a straight cylindrical tube of inner diameter 2a with the coaxial inner tube with the outer diameter 2b and inner diameter 2c (Figure 1). The outer tube is considered to be of infinite length, while the inner tube is considered to be semiinfinite. This is quite natural as the practical values of these diameters are about 0.5 mm and the lengths of the tubes before and after the mixing point are on the order of 10 cm. The inner tube introduces the first reactant (NaOH), while the annular flow through the outer tube introduces the second reactant (BTB). The solutions are dilute enough to consider their viscosity to be equal to the viscosity of water. Both hydrodynamics and masstransfer processes are considered to be stationary, corresponding to the experiment where the measurements of absorbance were made at a relatively long time after the start of the flows. With these assumptions made, the hydrodynamics of the system is described by the set of Navie-Stokes equations in a cylindrical coordinate system:
ur ur
∂uz ∂uz 1 ∂p + uz )+ ν∆uz ∂r ∂z p ∂z
(
)
ur ∂ur ∂ur 1 ∂p + uz )+ ν ∆ur - 2 ∂r ∂z p ∂r r ∂uz ∂(urr) +r )0 ∂r ∂z
(1)
Here ∆ ) ∂2/∂r2 + 1/r ∂/∂r + ∂2/∂z2, ur and uz are radial and longitudinal components of fluid velocity, respectively; F and ν are fluid density and kinematical viscosity, respectively, and p is Analytical Chemistry, Vol. 71, No. 11, June 1, 1999 2199
pressure. The following are conditions for velocity profiles at z ) -∞ for inner and outer tubes:
uz )
ur ) 0
(2)
( )
(3)
2Qin πc2
1-
r2 , 0
