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)

Basic idea

mitigate complexity by decoupling two-phase flow and mass transfer

Hybrid approach overview

Hybrid approach overview

Volume-of-Fluid simulations

Characteristic bubble shapes

Velocity at the interface

• mathematical framework: sharp interface model
• mass conservation at the interface: $\quad u_{\bot}^{gas} = u_{\bot}^{liq} $
• steady shape: $\quad u_{\bot}^{gas} = u_{\bot}^{liq} = 0$
• assumption: $\quad u_{\parallel}^{gas} = u_{\parallel}^{liq}$

How to model $u_{\parallel}^{liq}$ in single-phase simulations?

Parametrization and machine learning

1. extract velocity vector from two-phase simulations
2. create ML model based on extracted data (here: multilayer perceptron, PyTorch)
3. export model to torch script (.pt)
4. • star-shaped: $u_\parallel = u_\parallel (\varphi)$ with polar angle $\varphi$
   • complex shapes: $u_\parallel = u_\parallel (s)$ with arc length $s$

Final model

Learn more: publication, Jupyter notebooks

Single-phase flow dynamics

Implementation in OpenFOAM

bubbleSurfaceVelocitySimpleFvPatchVectorField.H

class bubbleSurfaceVelocitySimpleFvPatchVectorField
:
    public fixedValueFvPatchVectorField
{
    // Private data
        ...
        //- name of the PyTorch model
        word model_name_;

        //- PyTorch model
        torch::jit::script::Module pyTorch_model_;
        ...
						

Implementation in OpenFOAM

bubbleSurfaceVelocitySimpleFvPatchVectorField.C

Foam::bubbleSurfaceVelocitySimpleFvPatchVectorField::
bubbleSurfaceVelocitySimpleFvPatchVectorField
(
    const fvPatch& p,
    const DimensionedField<vector, volMesh>& iF,
    const dictionary& dict
)
:
    fixedValueFvPatchField<vector>(p, iF, dict, false),
    ...
    model_name_(dict.lookupOrDefault<word>("model", "velocity_model.pt"))
{
    pyTorch_model_ = torch::jit::load(model_name_);
}
						

Implementation in OpenFOAM

bubbleSurfaceVelocitySimpleFvPatchVectorField.C

void Foam::bubbleSurfaceVelocitySimpleFvPatchVectorField::updateCoeffs()
{
    ...
    // run forward pass to compute tangential velocity
    std::vector<torch::jit::IValue> modelFeatures{phiTensor};
    torch::Tensor uTensor = pyTorch_model_.forward(modelFeatures).toTensor();
    auto uAccessor = uTensor.accessor<double,1>();

    vectorField surfaceVelocity(Cf.size(), Zero);
    forAll(surfaceVelocity, faceI)
    {
        surfaceVelocity[faceI] = tau[faceI] * uAccessor[faceI];
    }
	...
						

Boundary condition

bubble patch in 0/U

bubble
{
    type            bubbleSurfaceVelocitySimple;
    origin          (0 0 0);
    axis            (0 1 0);
    normal          (0 0 -1);
    model           "bhaga_02_u_tau.pt";
}
						

Meshes

Wedge mesh creation: blockMesh, snappyHexMesh, extrudeMesh, collapseEdges

Comparison to free slip

Single-phase mass transfer

Physisorption $ A^{gas} \rightarrow A^{liq}$

Global Sherwood number

Comparison of global Sherwood number computed with Dirichlet and slip boundary condition.

Summary

• hybrid approach: decoupling of two-phase flow and species transfer problem
• computationally efficient: minutes to hours on a single core (end to end)
• data-driven: machine learning maps information from two-phase to single-phase simulations

Outlook

assessment for dynamic interfaces