Dynamical compartments in stirred tank reactors and Markov state modeling for mixing quantification: a transfer operator approach

Abstract

Identifying coherent flow structures in chemical reactors is crucial for understanding the mixing dynamics, which is essential for optimizing reactor performance. We demonstrate the use of a transfer operator method to find coherent flow structures such as almost-invariant sets and coherent sets, which are characterized by minimal mixing with the surrounding fluid, in a lab-scaled stirred tank reactor using both simulated and experimental Lagrangian trajectory data. The proposed method further enables a detailed analysis of the mixing behavior by computing expected residence times and mixing times. Additionally, a Markov-state-model describes the macroscopic transport dynamics between compartments in the reactor.

Figures (9)

• Figure 1: Schematic representation of the transition matrix construction according to equation \ref{['eq:pij_approx_traj']}: Each entry $P_{ij}$ is estimated as the fraction of particles from $B_i$ that end in $B_j$ (circled in orange).
• Figure 2: Schematic representation of the stirred tank reactor. Dimensions are: length (x): $127.94 \,\text{mm}$, width (z): $127.94 \,\text{mm}$, height (y): $223.64 \,\text{mm}$.
• Figure 3: Leading 12 eigenvalues for ten symmetrized transition matrices for the time intervals $[0,1], [1,2], \ldots [9,10]$ (time expressed in number of rotations) for the simulated data (▲, … ,▲). Combined transition information for one stirrer rotation into a single transition matrix: Eigenvalues of the symmetrized transition matrix for the simulated (■) and the experimental (●) data.
• Figure 4: Extracted five almost-invariant sets on time intervals $[0,1], [1,2], \ldots [9,10]$ with time expressed in number of rotations.
• Figure 5: Eigenvectors to the second (a) and fourth (b) eigenvalue of the symmetrized transition matrix (see appendix \ref{['appendix:AIS']}) computed from the combined transition information for one stirrer rotation and $\bm{s}_{max}$ (c) serving as a cluster indicator; all shown in the middle plane $z=\qty{0.0635}{\m}$.