[Paper Review] On the wall boundary conditions for species-specific mass conservation equations in mathematical modelling of direct precipitation fouling from supersaturated, multi-component fluid mixtures

Sverre Gullikstad Johnsen, Tiina Pääkkönen+3 more|arXiv (Cornell University)|Mar 4, 2017

Particle Dynamics in Fluid Flows7 references3 citations

TL;DR

This paper proposes a framework for accurately modeling species-specific mass conservation in turbulent boundary layers during direct precipitation fouling from supersaturated, multi-component fluids. By deriving physically consistent wall boundary conditions based on nano-scale phase-change kinetics and interfacial mass transfer, it enables predictive coupling of molecular-scale phenomena with macro-scale CFD models, significantly improving fouling rate prediction in industrial heat exchangers.

ABSTRACT

The mathematical modelling of species transport in the turbulent boundary layer of fluids that precipitate on the wall, is an important topic at the heart of one of the biggest challenges in efficient energy utilization in all process industries; namely the fouling of heat exchangers. A major contributor to the complexity of the problem is the multi-length-scale nature of the governing phenomena. That is, transport mechanisms dominating at the nano-scale may be responsible for the macroscopic performance of the industrial process. This paper addresses some of the challenges that need to be met in modelling the boundary conditions, i.e. the atomic/molecular-scale conditions, for the species-specific mass conservation equations, at the wall, for single-phase, multi-component fluids that precipitate at the wall.

Motivation & Objective

• To address the critical challenge of defining accurate wall boundary conditions for species-specific mass conservation equations in multi-component, supersaturated fluid flows.
• To bridge the gap between nano-scale precipitation kinetics at the solid-fluid interface and macro-scale CFD simulations of turbulent boundary layers.
• To establish a physically consistent framework for modeling deposition rates by linking interfacial mass flux to thermodynamic saturation and kinetic parameters.
• To highlight the importance of interface reaction coefficients and surface mass fractions in determining fouling behavior.
• To enable predictive modeling of direct precipitation fouling by incorporating atomic/molecular-scale phenomena into continuum-scale transport equations.

Proposed method

• Derives wall boundary conditions for species-specific mass conservation equations using a combination of Maxwell-Stefan diffusion and turbulent single-phase Navier-Stokes equations.
• Introduces a formulation for interfacial mass flux based on the interface reaction rate coefficient and deposition flux: $ j_{ ext{IR}, ho} = k_{ ext{IR}} imes (X_{ ho, ext{Sat},w} - X_{ ho,w})^{n_{ ext{IR}}} $.
• Uses the Arrhenius equation to model the temperature dependence of the interface reaction coefficient: $ k_{ ext{IR}} = k_{ ext{IR},0} \exp(-E_a / RT_w) $.
• Proposes that interface mass fractions $ X_{ ho,w} $ are determined iteratively from the saturation state and deposition flux: $ X_{ ho,w} = X_{ ho,\text{Sat},w} + (j_{ ho,\bot,w} / k_{\text{IR}})^{1/n_{\text{IR}}} $.
• Emphasizes the need for thermodynamic equilibrium calculations or experimental data to determine $ X_{\rho,\text{Sat},w} $ at wall conditions.
• Advocates for molecular dynamics and atomic-scale simulations to inform kinetic parameters, surface structures, and Soret coefficients for predictive modeling.

Experimental results

Research questions

• RQ1 How can physically consistent wall boundary conditions be defined for species-specific mass conservation equations in multi-component, supersaturated fluid mixtures?
• RQ2 What are the key differences in boundary condition formulation between depositing and non-depositing species in turbulent boundary layers?
• RQ3 How do nano-scale phase-change kinetics and interfacial reaction rates influence macro-scale fouling rates in industrial heat exchangers?
• RQ4 What role do thermodynamic saturation states and kinetic parameters (e.g., Arrhenius pre-exponential factor and activation energy) play in determining interfacial mass flux?
• RQ5 To what extent can molecular dynamics simulations improve the predictive capability of CFD-based fouling models?

Key findings

• The interface mass fraction $ X_{ ho,w} $ for a depositing species is determined by the balance between saturation composition at the wall and the interfacial deposition flux, via $ X_{ ho,w} = X_{ ho,\text{Sat},w} + (j_{ ho,\bot,w} / k_{\text{IR}})^{1/n_{\text{IR}}} $.
• The interface reaction coefficient $ k_{\text{IR}} $ is temperature-dependent and can be modeled using the Arrhenius equation, requiring experimental or simulation-derived parameters $ k_{\text{IR},0} $ and $ E_a $.
• Accurate determination of $ X_{\rho,\text{Sat},w} $ requires thermodynamic or experimental data, as it depends on local wall temperature and fluid composition.
• The sign and magnitude of the Soret coefficient—critical for thermophoretic transport—can be predicted using advanced molecular dynamics simulations, which may explain reversible deposition behavior near critical wall temperatures.
• Molecular-scale simulations can provide essential data on surface structure, reaction mechanisms, and free energy changes, enabling more accurate and predictive boundary conditions.
• The framework enables coupling of nano-scale interfacial kinetics with meso- and micro-scale transport models, forming a foundation for predictive CFD modeling of direct precipitation fouling.