Microreactors in Organic Chemistry and Catalysis, Second Edition (2013)
8. Liquid–Liquid Biphasic Reactions
8.4. Biphasic Flow in Microchannels
For a biphasic reaction in a flask, the liquid phase with greater density is found at the bottom, while the lighter liquid is on top forming an undisturbed flat interface. When the system is agitated, drops of various sizes of one phase form within the other phase. In a microchannel, the two immiscible liquids create various flow patterns, from segregation of phases by each other to stratified parallel flow. These flow patterns are characterized by a number of dimensionless parameters, which depend on the channel dimensions, channel surface, and the liquid properties. Variation of these parameters affects the stability of the flow pattern and can lead to transition from one flow pattern to another. The characteristics of liquid flow in a channel were first studied in 1883 by Osborne Reynolds by pumping a liquid continuously into a glass tube while introducing a fine strand of colored water to the flow. Reynolds observed that at low flow rate, laminar flow behavior was dominated in which the colored strand flew in straight parallel streams along the flow direction. As the flow rate was increased, the colored strands were broken into vortices until a point where turbulent flow behavior was dominant across the tube [11, 12] (Figure 8.2).
Figure 8.2 Schematic illustration of the liquid flow in a channel shown by arrows: (a) laminar flow at low flow rate and (b) turbulent flow at high flow rate.
Transformation from laminar to turbulent flow is characterized by a significant value of a dimensionless quantity known as Reynolds number (Re). Reynolds number relates inertial and viscous forces as represented in Equation 8.2; where ρ is the fluid density, vs is the fluid velocity, L is the length of the channel, and μ is the fluid viscosity.
Reynolds Number (Re)
At low values of the Reynolds number (Re < 2000), viscous force dominates resulting in laminar flow. At a high Reynolds number (Re > 3000), inertial forces are dominant resulting in turbulent flow. However, within a certain range of Reynolds number (Re: 2000–3000) the flow is neither laminar nor turbulent since the transformation occurs gradually.
The velocity profile of the flow in a channel varies across the diameter of the channel regardless of the flow rate. It has a minimum value (~0) near the channel walls, and a maximum value at the center of the flow. This variation in velocities arises from the adhesive forces between the channel walls and the liquid, causing the liquid layers nearest to the walls to be slower than that in the center. Consequently, the layer region nearest to the wall, always exhibit a laminar flow even at high Reynolds numbers [12, 13] (Figure 8.3).
In microscale channels, the viscous forces dominate the inertial effect resulting in a low Reynolds number. Hence, laminar flow behavior is dominant and mixing occurs via diffusion. However, in a liquid–liquid system the interfacial forces acting on the interface add complexity to the laminar flow as the relationship between interfacial forces and other forces of inertia and viscous results in variety of interface and flow patterns. Günter et al.  illustrated this relationship as a function of the channel dimension and velocity as shown in Figure 8.4. The most regularly shaped flow pattern is achieved when interfacial forces dominate over inertia and viscous forces at low Reynolds numbers, which is represented in Figure 8.4 by the area below the yellow plane [14, 15].
Viscous and inertial forces are related to the interfacial forces by the dimensionless capillary and Weber numbers. Capillary number (Ca) (Equation 8.3) describes the relative importance of viscosity and surface tension, where μ represents the viscosity, υ is the velocity, and σ is the surface tension.
Capillary Number (Ca)
The Weber number (We) is shown in Equation 8.4 and relates the inertial forces to the surface tension, where ρ is the density of the fluid, l represents the characteristic channel length, υ is the velocity of the fluid, and σ is the surface tension.
Weber Number (We)
In a macroscale channel, gravity force has an effect on the flow pattern of a biphasic system; consequently the flow pattern varies between vertical and horizontal channels. However in a microchannel, the gravity effect is dominated by the viscous and interfacial force, which is expressed by the ratio of gravity force to surface tension using the Bond number (Bo) as expressed in Equation 8.5; where Δρ is the density difference between two immiscible liquids, g is acceleration due to gravity, dh is the channel dimension, and σ the surface tension.
Bond Number (Bo)
Low Bond numbers indicate the dominance of interfacial forces over the gravitational force in a system. Applying factors that can affect the strength of the interfacial forces, using surfactants or applying high temperatures for example, the gravity force can then dominate resulting in a high Bond number.
Figure 8.3 Velocity profile of single laminar flow in a microchannel.
Figure 8.4 Effect of interfacial forces on inertia, viscosity, gravity forces with velocity, and channel diameter: the balance between these forces where all have a value of 1 is represented by the plane with dotted boundaries. Source: Reproduced with permission from The Royal Society of Chemistry .