Fluids in chemical reactors are typically stirred by means of mechanical devices. Impellers provide a fast mixing of reactants and products at the expense of high levels of shear close to the blades. In biological systems reactions are usually slow and less intrusive mixing methods may be adopted. Al-Shannag et al. [1] have recently shown that the lid-driven flow in a cavity provides efficient mixing in an immobilized-enzyme batch bioreactor. The aim of the current communication is to design reactors where good mixing is achieved without moving parts. Mixing in flows induced by Rayleigh-Bénard convection in a cubical cavity has been investigated for Ra ≤ 1.5x105, following the stability domain reported elsewhere [2]. Values of Ra close to 105 can easily be obtained even in a small cavity of side L=10-2 m, filled with water and with a temperature difference of 10 K.
Two different approaches were followed. The first assumed a continuum model and a species conservation differential equation was solved to calculate the mixing of a solute that was initially confined within a small region inside the cavity. A finite-volume discretization method was implemented on uniform grids NxNxN=101x101x101. Velocities at each cell were computed in a preliminary step using the coefficients of the velocity expansions that were previously calculated by means of a Galerkin method in [2]. The resulting set of ordinary differential equations, one for each of the N3 computational cells, was integrated in time using a highly accurate explicit seventh-eighth order Runge-Kutta scheme (RK78) with an adaptive time step. This approach is appropriate for conditions ranging from moderate to low diffusivities (Sc/Pr<50). Notwithstanding, aqueous solutions typically encountered in bioreactors yield much lower diffusivities, typically Sc/Pr ≈ 400. In these conditions, advection of solute particles dominates over molecular diffusion to the point that the latter is almost negligible. Thus, mixing occurs at very small scales and it would be difficult to capture it by the continuum approach unless huge grids were adopted.
In the second approach the velocity fields obtained by the Galerkin method were analyzed by using tools from the dynamical systems theory. The Lagrangian evolutions of individual solute particles were computed. The system of three differential equations of motion for each particle was integrated explicitly in time using also the RK78 scheme. Since the convective flows considered were steady the inert particle paths (stream traces) coincide with flow streamlines. The qualitative theory of differential equations was used to investigate the shape and stability of the streamlines. In particular, a singular point analysis was performed to obtain information about the local flow topology. Singular points on the walls of the cavity were also investigated because the zero shear stress occurring in these points indicates flow separation or reattachment. Poincaré maps were used to obtain the global structure of the flows and to characterize well-mixed regions. Regions with regular motion, where nested tori and periodic orbits develop, were also identified by the Poincaré maps. The size and shape of regular regions were numerically estimated by dividing the domain into 8x106 cells of equal size, integrating the trajectories of 49 fluid particles, and identifying the cells not visited by any fluid particle after a large integration time of 106. The quantitative analysis of the mixing efficiency was carried out by computing the maximal Lyapunov exponents of particle trajectories. Since Lyapunov exponents can be interpreted as the long time average of the specific rate of stretching of fluid elements, an increase in the level of Lagrangian chaos can be associated to an increase in the number of orbits with positive maximal Lyapunov orbits. It is commonly assumed that the level of chaos of particle orbits is a good measure of the mixing efficiency [3, 4].
The mixing properties of several natural convection flow patterns were investigated. The continuum approach shows that a good level of mixing was achieved with all convective flow patterns considered for moderate diffusivities (1< Sc/Pr < 50; Pr =0.71 and 130). In all cases, the solute concentration was well dispersed through most of the cavity volume at dimensionless times significantly below unity. Moreover, complete mixing times compare favorably with the time scale typical of bio-reactions (e.g., Michaelis-Menten kinetics [1] with a perfectly stirred batch bioreactor in the cubical cavity).
The dependence of mixing properties on the Rayleigh number was also analyzed. The location, stability character and bifurcations of critical points were determined for values of Ra up to 1.5x105. The bifurcations of critical points turned out to be relevant in understanding both the changes in the three-dimensional topology and the mixing properties of the confined flows considered. Current results reveal that the size of the chaotic region tends to increase rapidly with Ra. The percentage of cavity volume where chaotic motion occurs is above 95% even at moderate Rayleigh numbers (80000 ≤ Ra ≤ 150000).
References
[1] Al-Shannag M., Al-Qodah Z., Herrero J., Humphrey J. A. C., Giralt F. (2008) Using a wall-driven flow to reduce the external mass-transfer resistance of a bio-reaction system, Biochem. Eng. J. 39: 554-565.
[2] Puigjaner D., Herrero J., Simó C., Giralt F. (2008) Bifurcation analysis of steady Rayleigh-Bénard convection in a cubical cavity with conducting sidewalls, J. Fluid Mech. 598: 393-427.
[3] Ottino J. M. (1990) Mixing, chaotic advection and turbulence, Annu. Rev. Fluid Mech. 22: 207-253.
[4] Aref H. (2002) The development of chaotic advection, Phys. Fluids 14: 1315-1325.