Fouling - Fouling Modelling

Fouling Modelling

Fouling of a system can be modelled as consisting of several steps:

• Generation or ingress of the species that causes fouling ("foulant sourcing");
• Foulant transport with the stream of the process fluid (most often by advection);
• Foulant transport from the bulk of the process fluid to the fouling surface. This transport is often by molecular or turbulent-eddy diffusion, but may also occur by inertial coasting/impaction, particle interception by the surface (for particles with finite sizes), electrophoresis, thermophoresis, diffusiophoresis, Stefan flow (in condensation and evaporation), sedimentation, Magnus force (acting on rotating particles), thermoelectric effect, and other mechanisms.
• Induction period, i.e., a near-nil fouling rate at the initial period of fouling (observed only for some fouling mechanisms);
• Foulant crystallisation on the surface (or attachment of the colloidal particle, or chemical reaction, or bacterial growth);
• Sometimes fouling autoretardation, i.e., reduction (or potentially enhancement) of crystallisation/attachment rate due to changes in the surface conditions caused by the fouling deposit;
• Deposit dissolution (or re-entrainment of loosely attached particles);
• Deposit consolidation on the surface (e.g., through Oswald ripening or differential solubility in temperature gradient) or cementation, which account for deposit losing its porosity and becoming more tenacious with time;
• Deposit spalling, erosion wear, or exfoiliation.

Deposition consists of transport to the surface and subsequent attachment. Deposit removal is either through deposit dissolution, particle re-entrainment, or deposit spalling, erosive wear, or exfoliation. Fouling results from foulant generation, foulant deposition, deposit removal, and deposit consolidation.

For the modern model of fouling involving deposition with simultaneous deposit re-entrainment and consolidation, the fouling process can be represented by the following scheme:

Following the above scheme, the basic fouling equations can be written as follows (for steady-state conditions with flow, when concentration remains constant with time):

where:

• m is the mass loading of the deposit (consolidated and unconsolidated) on the surface (kg/m2);
• t is time (s);
• k_d is the deposition rate constant (m/s);
• ρ is the fluid density (kg/m3);
• C_m - mass fraction of foulant in the fluid (kg/kg);
• λ_r is the re-entrainment rate constant (1/s);
• m_r is the mass loading of the removable (i.e., unconsolidated) fraction of the surface deposit (kg/m2); and
• λ_c is the consolidation rate constant (1/s).

This system of equations can be integrated (taking that m = 0 and m_r = 0 at t = 0) to the form:

where λ = λ_r + λ_c.

This model reproduces either linear, falling, or asymptotic fouling, depending on the relative values of k, λ_r, and λ_c. The underlying physical picture for this model is that of a two-layer deposit consisting of consolidated inner layer and loose unconsolidated outer layer. Such a bi-layer deposit is often observed in practice. The above model simplifies readily to the older model of simultaneous deposition and re-entrainment (which neglects consolidation) when λ_c=0. In the absence of consolidation, the asymptotic fouling is always anticipated by this older model and the fouling progress can be described as:

where m* is the maximum (asymptotic) mass loading of the deposit on the surface (kg/m2).