Quantifying mixing processes

Rob Sturman

Department of MathematicsUniversity of Leeds

Graduate course, Spring 2007Leeds

Rob Sturman Quantifying mixing processes

Aim of a mixing process

To mix:1. trans. a. To put together or combine (two or moresubstances or things) so that the constituents orparticles of each are interspersed or diffused more orless evenly among those of the rest; to unite (one ormore substances or things) in this manner withanother or others; to make a mixture of, to mingle,blend. (OED)

To produce a mixture:

Substances that are mixed, but not chemicallycombined. Mixtures are nonhomogeneous, and maybe separated mechanically. (Hackh’s ChemicalDictionary)

Rob Sturman Quantifying mixing processes

Mixing is intuitive

Rob Sturman Quantifying mixing processes

Mixing via stretching and folding

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

Rob Sturman Quantifying mixing processes

Mixing via chaotic advection

Mixing of two highly viscous fluids between eccentric cylinders[Ottino, 1989]

Rob Sturman Quantifying mixing processes

An mixer based on a chaotic dynamical system

The Kenics R©Mixer

(a)

(b)

(c)

Figure 1: Examples of different Kenics designs: (a): a “standard” right-left layout with 180◦ twist of the blades (RL-180); (b): right-right layout with blades of the same direction of twist (RR-180); (c): (RL-120) right-leftlayout with 120◦ blade twist.

degrees) to specify a particular mixer geometry. Thus, for example RL-180 stands for the mixer, combiningthe blades twisted 180◦ in both directions, figure 1a, as analyzed in [3]. Figure 1b shows the RR-180configuration as was considered by Hobbs and Muzzio [8], while figure 1c illustrates the RL-120 geometry,which was suggested as more energy efficient in [9].

1.2 Principle of Kenics operationThe Kenics mixer in general is intended to mimic to a possible extent the “bakers transformation” (see [1]):repetitive stretching, cutting and stacking. To illustrate the principles of the Kenics static mixer a series ofthe concentration profiles inside the first elements of the “standard” ( RL-180) are presented in figure 2.All these concentration distributions are obtained using the mapping approach. The first image shows theinitial pattern at the beginning of the first element: each channel is filled partly by black (c = 1) and partlyby white (c = 0) fluid, with the interface perpendicular to the blade. The flux of both components is equal.

The images in figure 2b-e show the evolution of the concentration distribution along the first blade, thethin dashed line in figure 2e denotes the leading edge of the next blade. From the point of view of mimickingthe bakers transformation it seems that the RL-180 mixer has a too large blade twist: the created layers donot have (even roughly) equal thickness. The configuration achieved 1/4 blade twist earlier (figure 2d)seems to be much more preferable. The next frame, figure 2f, shows the mixture patterns just 10 ◦ into thesecond, oppositely twisted, blade. The striations, created by the preceding blade are cut and dislocated atthe blade. As a result, at the end of the second blade (figure 2g) the number of striations is doubled. Afterfour mixing elements, figure 2h, sixteen striations are found in each channel. The Kenics mixer roughlydoubles the number of striations with each blade, although some striations may not stretch across the wholechannel width. Note, that the images in figure 2 show the actual spatial orientation of the striations andmixer blades. In all further figures the patterns are transformed to the same orientation: the (trailing edgeof the) blade is positioned horizontally. This simplifies the comparison of self-similar distributions.

1.3 Existing approaches to Kenics mixer characterizationThe widespread use of the Kenics mixer prompted the attention to the kinematics of its operation and at-tempts to find ways to improve its performance. Khakhar et al. [10] considered the so-called partitionedpipe mixer, designed to mimic the operation of Kenics. The analogy is incomplete, since the partitionedpipe mixer is actually a dynamic device, consisting of rotating pipe around a number of straight, fixed,

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(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 2: How the Kenics mixer works: the frames show the evolution of concentration patterns within the first fourblades of the RL-180 mixer.

perpendicular placed, rectangular plates1. This device, however, gives the possibility to control its effi-ciency by changing the rotation speed of the pipe (which may be considered to be analogous to the twistof the blades in a Kenics mixer) and allowed relatively simple mathematical modeling using an approxi-mate analytical expression for the velocity field. The expression for the velocity field (and, consequently,the numerical simulations) was improved by Meleshko et al. [11], achieving even better agreement withexperimental results of [10]. However, these studies were dealing with a simplified model, which fails tocatch the details of the real flow in a Kenics static mixer.

The increasing computational power allowed different researchers to perform direct simulations of thethree-dimensional flow in Kenics mixers [3, 8, 12–16]. The last paper considers even flows with higherReynolds numbers up to Re = 100. These studies analyzed only certain particular flows and, unlike [10],did not allow for the optimization of the mixer geometry, due to high cost of 3D simulations.

More systematic efforts on exploring the efficiency of the Kenics mixer were made by [9], who sug-gested a more energy efficient design with a total blade twist of 120 ◦. They explored different mixerconfigurations, but, since the velocity field had to be re-computed every time, the scope was limited: onlyseven values of the blade twist angle were analyzed. The aim of the current work is to study numerically thedependence of the mixer performance on the geometrical parameter (blade twist angle) and to determinethe optimal configuration within the imposed limitations. Since it was shown in [9] that the blade pitch hasrather minor effect on mixer performance, it is fixed in the current work.

The Kenics static mixer was also considered as a tool to enhance the heat exchange through the pipewalls [17]. They found that the Kenics mixer may offer a moderate improvement in heat transfer, but itsapplicability in this function is limited by difficulty of i.e. wall cleaning. However, only mixers with the“standard” 180◦ blade twist were considered. In the current work we also analyse the influence of theblade twist angle on refreshing of material on the tube surface. Recently, Fourcade et al. [16] addressedthe efficiency of striation thinning by the Kenics mixer both numerically and experimentally, using theso-called “striation thinning parameter” that describes the exponential thinning rate of material striations.This was done by inserting a large number of “feed circles” and numerically tracking markers along themixer. Their method allows to characterize the efficiency of the static mixer. However, adjusting thegeometry would necessitate repetition of all particle tracking computations. Optimization of the mixergeometry calls for a special tool that allows to re-use the results of tedious, extensive computations in orderto compare different mixer layouts. A good candidate for such a tool is the mapping technique.

1Note that the partitioned pipe mixer is actually a simplified model of a RR type of Kenics mixer.

3

from [Galaktionov et al., Int. Polymer Proc. 18 138–50 (2003)]

Rob Sturman Quantifying mixing processes

Mixing via MHD

Amagnetohydrodynamic chaotic stirrer, [Qian & Bau, 2002]

Rob Sturman Quantifying mixing processes

Oceanographic Mixing

Plankton bloom at the Shetland islands. [NASA]

Rob Sturman Quantifying mixing processes

Duct flows

Schematic view of a ductflow with concatenatedmixing elementsRed and blue blobs of fluidmix well under a smallnumber of applicationsChanging only the positionof the centres of rotationcan have a marked effecton the quality of mixing

Rob Sturman Quantifying mixing processes

Mixing through turbulence

Scalar concentration distribution from a high resolutionnumerical simulation of a turbulent flow in a two-dimensionalplane for a Schmidt number of 144 and a Reynolds number of22. (Courtesy of G. Brethouwer and F. Nieuwstadt)

Rob Sturman Quantifying mixing processes

Mixing via dynamos

magnetic field generated by inductive processes by the motionsof a highly conducting fluid. The prescribed velocity is of a typeknown to be a fast dynamo, i.e., capable of field amplification inthe limit of infinite conductivity (Cattaneo et al. 1995).

Rob Sturman Quantifying mixing processes

A good definition...

THOMASINA:

When you stir you rice pudding, Septimus, the spoonful of jam spreadsitself round making red trails like the picture of a meteor in myastronomical atlas. But if you stir backward, the jam will not come togetheragain. Indeed, the pudding does not notice and continues to turn pink justas before. Do you think this odd?

SEPTIMUS:

No.

THOMASINA:

Well, I do. You cannot stir things apart.

SEPTIMUS:

No more you can, time must needs run backward, and since it will not, wemust stir our way onward mixing as we go, disorder out of disorder intodisorder until pink is complete, unchanging and unchangeable, and we aredone with it for ever.

Arcadia, Tom Stoppard

Rob Sturman Quantifying mixing processes

How to quantify all this...

Danckwerts (1952): "...two distinct parameters are required tocharacterize the ’goodness of mixing’...the scale ofsegregation...and the intensity of segregation" [Denbigh, 1986]

How big?

How wide?

How does this compare with this?

Rob Sturman Quantifying mixing processes