Single laminar flow, velocity profile

Figure 4.11 Velocity profile of single laminar flow in a microchannel.

Examples of values of Pe are provided in Fig. 19-8. When Pe is large, n =k Pe/2 and the dispersion model reduces to the PFR model. For small values of Pe, the above equation breaks down since the lower limit on n is n = 1 for a single CSTR. To better represent dispersion behavior, a series of CSTRs with backmixing may be used e.g., see Froment and Rischoff (Chemical Reactor Analysis and Design, Wiley, 1990). A model analogous to the dispersion model may be used when there are velocity profiles across the reactor cross-section (eg., for laminar flow). In this case, the equation above will contain terms associated with the radial position in the reactor. 

Laminar flow is the usual flow regime met in monolith reactors, given that the typical Reynolds number has values below SOO. The radial velocity profile in a single channel develops from the entrance of the monolith onward and up to the position where a complete Poiseuille profile has been established. The length of the entrance zone may be evaluated from the following relation [3] ... 

In conventional single phase microfluidic systems, flow in the microchaimel is laminar a parabolic velocity profile is established with fluid velocity zero at the channel walls and maximum at the channel center [18, 21] (Fig. 9a). There are two implications to this behavior (1) a reagent sample plug will constantly dissipate along the microchannel, and (2) the mixing of samples could be very slow in co-flow streams. 

Fig. 3.9 a Velocity vector field for single phase run at mixture velocity of 0.015 m s . b Velocity profile along Y direction averaged across the channel length, compared with the analytical laminar flow profile (solid line)... 

The system was validated by performing single aqueous phase experiments and comparing velocity profiles with the analytical solutions for laminar circular channel flow. In Fig. 3.9 results for single phase experiments are presented (average velocity equal to 0.015 m s ). The average velocity profile in Fig. 3.9a matches well the analytical solution. The uncertainty of velocity measurements was estimated based on the standard deviation of velocity data (order of 10 %) and plotted in the form of error bars on Fig. 3.9b. 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

Figure 3 shows the measurement result of an application of the profile sensor at a microchannel flow obtained without traversing the sensor. Each point corresponds to a single tracer whose position and velocity were determined. A good agreement with the expected parabolic velocity profile of the laminar channel flow was achieved. The spatial resolution was approximately 3.5 xm (which is the standard deviation of the tracer position). It was limited by sensor properties and the size of the tracers, which was about 2 xra Using submicrometer tracers as well as improved sensor properties a spatial resolution in the nanometer range will be possible. 

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