(102b) Evaluation of Numerical Methods for Describing Mass Transfer in Microstructured Devices Using Experimentally Determined Residence Time Distributions

The benefits of miniaturisation in chemical reactors for the intensification of chemical processes have been demonstrated by numerous examples over the last few years. Enhanced mass and heat transfer are the most important phenomena contributing to improved reactor performance. Despite experimental proofs-of?concepts for various microreactors, there remains a considerable need for accurate reactor models describing the actual reactor behaviour. In traditional chemical engineering three main modelling levels are distinguished: (i) ideal reactors, (ii) globally characterised real reactors and (iii) locally resolved rigorous reactor models.

The work presented here is focused on an evaluation of the precision and reliability of different numerical approaches to describe the liquid phase mass transfer within microreactors (i.e. level (iii)). Due to the enormous aspect ratios in microreactors, i.e. the ratio of reactor length to the microscaled dimension, an accurate treatment of the mass transfer equations is a major challenge for reliable simulations using computational fluid dynamics (CFD) of an acceptable complexity. In particular, the propagation of steep fronts can be distorted by virtual numerical diffusion effects caused by the discretisation of the second order derivatives in the mathematical model.

An experimental technique generating such steep concentration gradients is the determination of the residence time distribution by measuring the transient response to a Dirac- or an ideal step stimulus signal applied to the reactor inlet at the outlet of the reactor. For this reason data obtained using a novel non-intrusive method for the determination of residence time distributions in microreactors [1] are used to evaluate the simulation results at the global modelling level (ii). An estimate of the error introduced by numerically superimposed diffusion can thus be obtained.

Numerical concepts for simulating residence time distribution in microreactors

Two alternative approaches are conceivable. On the one hand, an Euler-Lagrange approach using virtual (mass-free) particles can be applied to a steady-state solved flow field (Navier-Stokes equation together with continuum condition) as a post-processing tool [2]. In this case, the movement of the individual particles is along the streamlines of the flow field, i.e. no diffusive terms are included in the transport equations for the tracer phase. For the comparison between this kind of simulation and experimental data, a given number of particles is introduced into the inlet cross-section of the microreactor at time zero, weighted according to the actual local flow rate, and the exit time for the individual particles is subsequently recorded at the outlet. Summing the particles obtained over a particular time interval in a histogram then yields the residence time distribution.

In the second technique, the tracer phase for the determination of the residence time distribution is treated as a second continuous phase according to an Euler-Euler approach. Unlike the virtual particles, the continuous tracer phase is affected by two transport mechanisms, namely both convection and diffusion. The resulting system of partial differential equations considers the physically diffusion actually occurring, together with a certain amount of additional numerical diffusion introduced when solving with standard CFD-methods. For comparison of the simulated concentration field, the exit concentration is monitored and integrated over the cross-section, weighted according to the actual flow velocity in each section of the exit plane.

Finally, the Euler-Euler approach was refined by implementing advanced mathematical methods within the solving algorithm for the partial differential equations to diminish the contribution of numerical noise to the diffusive mass transfer (Total variance diminishing (TVD) method) [3].

Experimental technique for determining residence time distribution in microreactors

The experimental determination of residence time distribution is conducted using a novel non-intrusive technique developed by the authors [1]. Basically, a caged fluorescent dye dissolved in a carrier fluid is used and an ideal stimulus step signal is generated by exposing the reactor inlet under steady-state flow conditions to UV-light emitted by a laser diode with a Gaussian-shaped intensity profile, to ensure compatibility of the optical tracer ?injection' process with the prevailing laminar flow conditions. The determination of the residence time distribution is then easily conducted by detecting the two dimensionally resolved fluorescence signal at the reactor outlet and integrating the detected concentrations, weighted according to the laminar flow profile at each position, over the whole cross-section.

Results and discussion

Two different geometries were analysed using the different concepts described: a simple straight rectangular channel (150 µm x 70 µm) with a length of 20 mm was investigated for five different mean residence times and the behaviour of an intricately structured reactor with several parallel channels, as an example of a device designed for internal parallelisation, was examined at the same mean residence times.

In the first case, the analytical solution is well-known for both the Euler-Lagrange as well as the Euler-Euler approach. By virtue of the purely laminar flow behaviour, the residence time distribution obtained using the virtual tracer particles matches the theoretically calculated analytical solution for laminar flow in pipes, first derived by Levenspiel [4], exactly. As would be expected from the enhanced radial mass transfer in microstructured reactors, both the Euler-Euler approach and the experimental results show a significantly stronger influence of diffusive mass transfer and thus deviations from the ideal solution form. Nevertheless, the standard Euler-Euler approach exhibits an even stronger deviation on the same grid, probably caused by additional numerical diffusion effects. First results demonstrate that this effect occurring in particular at high flow rates can be suppressed successfully by applying TVD-methods, for example, when simulating the tracer as a continuous second phase. These results can also be used to estimate the amount of numerical diffusion taking place in microstructured devices. Using the second intricately structured device, the consequences of such effects are elucidated for a system without an analytical solution.

Conclusion

The comparison between various numerical approaches and experimental results for microreactor modelling, using the example of residence time distribution determination is an appropriate test system for the evaluation of mass transfer in a system with sharp concentration gradients, providing a useful indicator of the potential reliability of simulation results using the currently available methods for describing heat and mass transfer with chemical reaction. An accurate determination of diffusive mass transfer is of particular interest in such instances e.g. for the simulation of fast reactions in catalytic wall reactors, where steep gradients over the miniaturised dimension might arise.

[1] S. Lohse, I. Gerlach, D. Janasek, P.S. Dittrich and D.W. Agar, IMRET 9 (2006)

[2] J. F. Acker and S. Turek, Ergebnisbericht 193, Universität Dortmund (2000)

[3] D. Kuzmin and S. Turek, J. Comput. Phys., 198, p. 131 (2004)

[4] O. Levenspiel and W.K. Smith, Chem. Eng. Sci., 6, p. 227 (1957)