Reynolds and Péclet Number Effects inFolding Flow Mixers
Mark Bown • Jordan MacInnes • Alexander Vikhansky† • Ray Allen • Marko Blom‡
Department of Chemical and Process Engineering, University of Sheffield, www.shef.ac.uk/cpe† Department of Engineering, Queen Mary University of London, www.eng.qmul.ac.uk‡ Micronit Microfluidics BV, www.micronit.com

Folding Flow Mixers
Folding flow mixers consist of a chain of elements. In each element theflow is folded upon itself, decreasing the length scale for mixing and thus
reducing the mixing time. Each element has a pressure drop associatedwith it and therefore it is important to optimise the efficiency of theelements, the number of elements and the flowrate to achieve the desiredmixing in the desired time.
The topology of a folding flow mixer varies according to whether it isdesigned for low or high Reynolds number (Re) flow. At low Re number,the flow is folded by the splitting and recombining of the channels. Manymixer geometries along these lines have been studied over the years, herea tear-drop shaped geometry and an F-shaped geometry (Chen andMeiners, 2004, Appl. Phys. Lett. 84, 2193-2195) are investigated. At higher
Re numbers, the topology of the mixer is different. In these flows, inertia
can be used to fold the flow. In this work, a double swirl chamber geometry
is used. The fluid spirals down through the swirl chambers, rapidly
reducing the mixing length.

F-shaped mixer for low
Re number mixing

Swirl mixer for highRe number mixing

The mixers are all fabricated by powder blasting and thermal bonding inglass by Micronit Microfluidics (Enschede, The Netherlands).
Tear-drop mixer for low
Re number mixingComputations at Pe = 1000
Re = 1

σ rms

Pe Number Scaling

Re = 100σ rms

σ rmskkk

Computational Fluid Dynamics (CFD) computations can produce accurate simulations for each mixing geometry up to Pe numbers around1000. The two plots show FLUENT computations of the number of mixer elements (k) required to produce a given rms deviation incomposition (σ rms ). At low Re number the split and recombine topology performs better, since there is insufficient inertia in the flow for the swirl
mixer to operate correctly. At higher Re number, as the inertia increases, the swirl mixer performance improves dramatically. The split andrecombine topologies show small improvement at the higher Re number, but are now less efficient than the swirl mixer. The Re number will
depend on the flowrate, fluid properties and channel dimensions. If higher Re number flows can be achieved, the use of a swirl mixer will
dramatically reduce the number of elements required and the mixing time. However, if the inertial flow regime cannot be accessed the split
and recombine topologies will provide the best mixing solution.

Flow Visualisation at Pe ~ 105Flow visualisation experiments can be used to obtain a qualitative insight into micro-scale mixing. The technique isbased on the combination of two streams of the same fluid, one containing a fluorescent dye. Fluorescencemicroscopy images can then be taken to visualise the mixing of the two streams. However, the results must beused with caution. The images naturally provide depth averaged information and thus horizontal layers of unmixed
fluid will appear identical to mixed fluid. This is clearly seen in every other element of the low Re number flow
through the F-shaped mixer. Additionally, images taken through a curved surface may have apparent dark bands
at the edges of the channel, even if the fluid there is actually dyed. This can be due to the difference in refractive
index between the fluid and the glass. This effect is particularly evident here in the high Re number flows, forexample through the second element of the swirl mixer. In general, visualisation experiments can provide usefulsupporting evidence for the computational studies above, but on their own are insufficient for a complete study ofmixing effects.

Currently, accurate computations are possible up to a Pe number of around 1000. Above this, MonteCarlo simulations can determine mixer performance. Most micro-scale mixing problems occur above
a Pe number of 104. For folding flow mixers, a scaling law can be derived for a particular geometry(MacInnes et al, 2006, submitted to Chem. Engng. Sci.). The law falls into two regimes at low andhigh Pe number, as seen in the plot above. The dots represent CFD computations and Monte Carlosimulations in a rectilinear version of the F-shaped mixer geometry, whilst the solid lines are thescaling law derived for the same geometry. The scaling law should allow performance for thisparticular mixing geometry to be predicted at useful Pe numbers for mixer design.

High Re (~102) Flow
At higher Re number the flow pattern in the split and recombine topology ismore complicated, with evidence of the flow twisting as it passes through themixer. This effect is particularly striking in the first element shown here. The
computations suggest that this only results in a small improvement in mixer
efficiency and that the swirl mixer will be more efficient still at these Renumbers.

The Swirl Mixer

Low Re (~10-1) Flow

The images show the first few elements of the swirl mixer at Re = 0.25 (left) and Re = 250 (right). The effect of inertia is clearly
evident with the two bands of fluid passing unaltered through the mixer at low Re number compared to a significant reduction in
mixing length due to the swirl at high Re number. The computations suggest that the swirl mixer operating in the inertial regime
will be the most efficient mixing solution and the evidence from the flow visualisation supports this.

Conclusions for Mixer Design
At low Re number there is no significant inertia in the flow and thus the swirl mixer is not expected to perform well.
This is clearly seen in the flow visualisation experiments, with two distinct bands of dark and light fluid evident
throughout the mixer. There appears to be a slow rotation of the interface along the mixer, which reduces thewidth of the dark band, but mixing is clearly inefficient. The split and recombine topology does effectively reducethe mixing length from element to element even at low Re number. The effect is particularly striking in the images
from the tear-drop geometry where the bands are seen getting thinner and thinner from element to element.
However, computations suggest that the F-shaped mixer will be slightly more efficient and there is no evidence in
the images to disprove this.

Chemical &ProcessEngineering.

The principal constraints when designing a mixer are the flowrate required through that particular section of the micro-device, thefluid properties of the streams to be mixed and the desired level of mixing. The design variables are the channel dimensions, thenumber and geometry of the mixing elements and the pressure drop across the mixer. There may be additional special
constraints, for example a maximum pressure or a target mixing time. It is likely that the mixing problem can be broken down into
one of two forms:
1. What is the minimum pressure drop required to produce a given rms deviation in a target time?
2. For given flow conditions, what is the quickest time in which a desired rms deviation can be achieved?
The computational and experimental work here shows that the most efficient mixing solution is a swirl mixer operating in theinertial flow regime. However, if the liquid is highly viscous or the flowrate in that portion of the micro-device is low, the inertial
flow regime may be inaccessible and a split and recombine topology should be used instead.