Gas-liquid reactors

micro reactor
size: millimeter
source: SPP 1740

prediction of

• mass transfer
• enhancement
• mixing
• conversion
• selectivity
• yield
• ...

bubble column reactor
size: meter
source: R. M. Raimundo, ENI

High Péclet number problem

Specimen calculation

$d_b=1~mm$ water/oxygen at room temperature

• $Pe = Sc\ Re = \nu_l / D_{O_2} \cdot U_b d_b/\nu_l \approx 10^5 $
• $$ Re\approx 250;\quad \delta_h/d_b \propto Re^{-1/2};\quad\delta_h\approx 45~\mu m $$
• $$ Sc\approx 500;\quad \delta_c/\delta_h \propto Sc^{-1/2};\quad\delta_c\approx 2.5~\mu m $$

$\delta_h/\delta_c$ typically 10 ... 100

feasible simulations up to $Pe\approx 1000$ (3D, HPC)

Underresolved meshes

$$ \partial_t c_A + \nabla \cdot (c_A\mathbf{u} - D_A \nabla c) = 0 \quad \text{in}\quad \Omega^+ $$

• $c_A|_\Sigma = const.$
• $\mathbf{u}$ - given velocity vector field
• finite volume discretization

$\Omega^\pm$ - liquid/gas domain, $\Sigma$ - interface, $f_i$ - cell faces

Transfer species - A

Interpolation errors

Subgrid-scale modeling

Idea: Leverage coherent structures in convection-dominated species boundary layers.

1. Substitute problem
   Find a representative substitute problem to obtain a parameterized profile function.
2. Reconstruction
   Adjust the profile function based on local simulation data (cell-width, average concentration, ...).
3. Correction
   Correct convective fluxes, diffusive fluxes, and reaction source terms based on the adjusted profile function.

(1) Substitute problem

dimensionless form
$$ \partial_\tilde{y} \tilde{c} = \frac{1}{Pe}\partial_{\tilde{x}\tilde{x}} \tilde{c}$$

• $\tilde{c}(\tilde{x}=0,\tilde{y}) = 1$
• $\tilde{c}(\tilde{x}>0,\tilde{y}=0) = 0$
• $\tilde{c}(\tilde{x}\rightarrow\infty,\tilde{y}) = 0$

(2.1) Reconstruction

(3.1) Correction

1. compute interface normal derivative:
   $\partial_\tilde{x} \tilde{c}|_{\tilde{x}=0} = 2/(\sqrt{\pi}\delta_{num})$
2. correct numerical diffusive flux at $\tilde{x}=0$
3. solve transport equation for $\tilde{c}$

→ saves one refinement level (half cell-width)

(3.2) Correction

Idea:
Correct convective and diffusive fluxes over all cell faces of an interface cell.

Control volume (cube) with embedded interface (plane).

(2.2) Reconstruction

Results

Results

$Sh = \frac{k_L d_b}{D}$   with   $k_L = \frac{\dot{N}}{A_{eff}\Delta c}$

Data-driven modeling

Idea: Replace analytical solution with machine learning (ML) model.

1. Substitute problem
   Find a representative substitute problem and solve it numerically to obtain boundary layer data.
2. Model creation
   Select meaningful features (variables) and train a ML model.
3. Correction
   Correct convective fluxes, diffusive fluxes, and reaction source terms based on the ML model.

ML terminology

$s$ instances of feature - label pairs

(1) Substitute problem

numerical solution

• OpenFOAM®
• steady-state flow dynamics
• parameters: $Re$, $Sc_i$, $Da_i$, shape, ...
• finished in minutes

data extraction

• features (input variables) should be available in target simulation
• features motivated by boundary layer theory and previous modeling
• down-sampling to coarse meshes
• stored as comma separated values (csv)

(2) Model creation

Multilayer Perceptron

• 6 layers with 40 neurons per layer
• SELU activation function

model training

• mean squared error loss
• one model per label
• ≈ 5 min training time per model
• training on CPU with double precision

implementation

• PyTorch®
• export to TorchScript
• dynamically loaded at runtime

SGS Model assessment

Aim:
Create reference data for complex shapes and flow scenarios to assess model generalization.

Idea:
Decoupling of two-phase flow and species transport.

Mesh dependency

Prismatic cell layers around a spherical-cap bubble with increasing mesh resolution.

Model assessment

Comparison with experiments

Mathematical model

mass balance and jump

$\frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho\mathbf{u} \right) = 0\quad\text{in}\quad \Omega^\pm (t)$

$[\![ \mathbf{u}] \!] \cdot \mathbf{n}_\Sigma = 0\quad\text{on}\quad \Sigma (t)$

Interface Tracking

• numerical equivalent of sharp interface model
• surface tension isotherms
  $\sigma = \sigma (c^\Sigma)$
• sorption library $ s^\Sigma + [\![ -D\nabla c ]\!]\cdot \mathbf{n}_\Sigma = 0$
• here: Langmuir fast sorption model

C. Pesci, A. Weiner, H. Marschall, D. Bothe:
Computational analysis of single rising bubbles influenced by soluble surfactant (2018)

Surfactant - surface active agent

Velocity

Local velocity

Oxygen transport

Experiment vs. Simulation

Local Sherwood number for clean (left) and contaminated (right) interfaces.

Summary

Outlook

Complementary material

Publications

Reproducibility

ML Challenges

Feature density for the source terms of a parallel-consecutive reaction of type $A+B\rightarrow P$ and $A+P\rightarrow S$.

Video by courtesy of David Merker.

Pendant bubble method

Image by courtesy of David Merker. Measurement apparatus by dataphysics-instruments.

Bulk species - B

Interpolation error B

Product species - P

Interpolation error P

Source term

Feature selection

Idea: ranking of feature importance.

algorithms

• regression forest
• sequential backward selection
• sequential forward selection

example (left)

$f_m = f_m(\bar{c},Da,\tilde{r})$

ML model assessment

Histogram (left) and heatmap (right) of the label error for the transfer species (A). Reminder: the height of a bar in a histogram depicts the number of instances in the range the bar covers. The index $01$ indicates a min-max-scaling.