Quantifying mixing processes

Transcription

1 Department of Mathematics University of Leeds Graduate course, Spring 2007 Leeds

2 Aim of a mixing process To mix: 1. trans. a. To put together or combine (two or more substances or things) so that the constituents or particles of each are interspersed or diffused more or less evenly among those of the rest; to unite (one or more substances or things) in this manner with another or others; to make a mixture of, to mingle, blend. (OED) To produce a mixture: Substances that are mixed, but not chemically combined. Mixtures are nonhomogeneous, and may be separated mechanically. (Hackh s Chemical Dictionary)

3 Mixing is intuitive

4 Mixing via stretching and folding "None other than Osborne Reynolds advocated in a 1894 lecture demonstration that, when stripped of details, mixing was essentially stretching and folding and went on to propose experiments to visualize internal motions of flows." [Ottino, Jana, Chakravarthy, 1994]

5 Mixing via chaotic advection Mixing of two highly viscous fluids between eccentric cylinders [Ottino, 1989]

6 An mixer based on a chaotic dynamical system The Kenics R Mixer (b) (a) (b) (c) (d) (c) (e) (f) (g) (h) Figure 2: How the Kenics mixer works: the frames show the evolution of concentration patterns within the first four from blades [Galaktionov of the RL-180 mixer. et al., Int. Polymer Proc (2003)] 1: Examples of different Kenics designs: (a): a standard right-left layout with 180 twist of the blades (RLperpendicular placed, rectangular plates 180); (b): right-right layout with 1. This device, however, gives the possibility to control its efficiency by changing the rotation speed of the pipe blades (which may of be the considered sametodirection be analogous toof the twist (RR-180); (c): (RL-120) right-left of the blades in layout with 120 a Kenics mixer) and allowed relatively simple mathematical modeling using an approxi- blade twist. mate analytical expression for the velocity field. The expression for the velocity field (and, consequently, the numerical simulations) was improved by Meleshko Rob et al. Sturman [11], achieving even Quantifying better agreement mixing with processes

7 Mixing via MHD A magnetohydrodynamic chaotic stirrer, [Qian & Bau, 2002]

8 Oceanographic Mixing Plankton bloom at the Shetland islands. [NASA]

9 Duct flows Schematic view of a duct flow with concatenated mixing elements Red and blue blobs of fluid mix well under a small number of applications Changing only the position of the centres of rotation can have a marked effect on the quality of mixing

10 Mixing through turbulence Scalar concentration distribution from a high resolution numerical simulation of a turbulent flow in a two-dimensional plane for a Schmidt number of 144 and a Reynolds number of 22. (Courtesy of G. Brethouwer and F. Nieuwstadt)

11 Mixing via dynamos magnetic field generated by inductive processes by the motions of a highly conducting fluid. The prescribed velocity is of a type known to be a fast dynamo, i.e., capable of field amplification in the limit of infinite conductivity (Cattaneo et al. 1995).

12 A good definition... THOMASINA: When you stir you rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backward, the jam will not come together again. Indeed, the pudding does not notice and continues to turn pink just as before. Do you think this odd? SEPTIMUS: No. THOMASINA: SEPTIMUS: Well, I do. You cannot stir things apart. No more you can, time must needs run backward, and since it will not, we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it for ever. Arcadia, Tom Stoppard

13 How to quantify all this... Danckwerts (1952): "...two distinct parameters are required to characterize the goodness of mixing...the scale of segregation...and the intensity of segregation" [Denbigh, 1986] How big? How does this compare with this? How wide?