Computational Modeling of Biological Filtration Systems

Abstract

A biofilter removes a dissolved contaminant by letting it diffuse into a microbial film and be consumed there. Performance is set at two scales at once: a millimeter-scale competition between diffusion and reaction inside the biofilm, and a meter-scale advection of fluid through the packed column. This work models both and couples them. The biofilm scale is a reaction–diffusion boundary value problem whose solution is condensed into an effectiveness factor; the column scale is an advection–dispersion–reaction equation that consumes that factor as its local sink. The result predicts removal efficiency and the design parameters (bed depth, loading, film thickness) that govern it.

keywords: biofilm · reaction–diffusion · Thiele modulus · effectiveness factor · packed bed

## Biofilm scale

Inside the film of thickness L_f, the substrate concentration S(z) obeys a balance between diffusion and Monod uptake. At steady state:

D_f d²S/dz² = μ_max X_f S / Y(K_s + S) (1)

with no-flux at the support, S′(0) = 0, and the bulk concentration imposed at the liquid interface, S(L_f) = S_b. The dimensionless group that governs (1) is the Thiele modulus, the ratio of reaction rate to diffusion rate:

φ = L_f √( μ_max X_f / Y K_s D_f ) (2)

The two limits are the whole design story. When φ ≪ 1 the film is reaction-limited: substrate penetrates fully and the entire biomass works. When φ ≫ 1 the film is diffusion-limited: substrate is consumed in a thin outer shell and the interior is starved, dead weight. The effectiveness factor η, the fraction of the film actually contributing, collapses the boundary value problem into one number:

η = (observed rate) / (rate with no diffusion limit) ∈ (0, 1] (3)

  S(z)                          phi << 1 : full penetration
  Sb |======= = = = .                       eta -> 1
     |              ' .
     |                 ' .   phi ~ 1
     |                    ' .
     |                       ' . . _      phi >> 1 : outer shell only
   0 |____________________________ ' . . _____ eta -> small
     0                                       L_f  (depth into film)
   support                              liquid interface

Fig. 1 Substrate profile into the biofilm as the Thiele modulus grows. High φ starves the interior, dropping the effectiveness factor.

## Column scale

Zoom out to the packed bed. Bulk concentration S_b varies along the column coordinate ξ under advection, axial dispersion, and the sink supplied by the biofilm model. At steady state with superficial velocity u and specific surface area a:

u dS_b/dξ = D_ax d²S_b/dξ² − a η r(S_b) (4)

The coupling is the point: η from (3) makes the local removal rate honest about diffusion limitation, so a column model that would otherwise overpredict removal (by assuming every cell in every film is fed) is corrected at each depth. Removal efficiency is the inlet-to-outlet drop:

E = 1 − S_b(H) / S_b(0) (5)

for a bed of depth H. Equation (5) is what a designer actually buys, and (1)–(4) are what determine it.

# two-scale solve: biofilm BVP feeds the column ODE
def removal(Sb):
    eta = solve_biofilm(Sb,                 # Eq (1) BVP, no-flux + bulk BC
                        phi=thiele(Sb))     # Eq (2)
    return a * eta * rate(Sb)               # effectiveness-corrected sink

Sb = solve_bvp(column_adr,                  # Eq (4): advection-dispersion-reaction
               sink=removal,
               span=(0, H), bc=(Sb_in, danckwerts))

E  = 1 - Sb[-1] / Sb[0]                      # Eq (5): removal efficiency

## What the model is for

The model converts operating and geometry choices into a single predicted efficiency, and it exposes which regime a given design sits in. That distinction drives the decision:

Diffusion-limited (high φ): thicker film does nothing: the interior is already starved. The lever is surface area a and loading, not biomass.
Reaction-limited (low φ): removal scales with biomass and residence time; bed depth H and velocity u are the levers.
Sizing: sweeping H and u against a target E traces the feasible operating envelope without a pilot column for every case.

A model that resolves both scales answers a question a lumped first-order rate constant cannot: why a column underperforms, and which knob recovers it.

Environmental bioprocess modeling. Reaction–diffusion at film scale, advection–dispersion at column scale.
Liam Kozma · liam.kozma@protonmail.com