Biofilm Diffusion-Reaction Kinetics: Penetration Depth, Flux and the Design of Attached-Growth Reactors

Why biofilms behave differently from suspended growth

In suspended growth, every organism sees essentially the bulk liquid composition, so a single Monod expression on bulk concentration describes the whole reactor. In a biofilm, substrate must diffuse from the bulk through a stagnant liquid layer, then through the film itself, while being consumed en route. Concentration therefore falls with depth, and organisms deep in the film may be starved of a substrate that is abundant in the bulk.

The consequences are structural, not incidental:

• Stratification. Aerobes occupy the outer layer, anoxic and anaerobic populations the interior — so a single film can nitrify and denitrify simultaneously.
• Sludge age decoupling. Slow-growing organisms are retained by attachment regardless of hydraulic retention time, which is why biofilm reactors nitrify reliably at short HRT.
• Diminishing returns on thickness. Beyond the penetration depth, additional biofilm contributes nothing to flux while adding diffusion resistance and detachment risk.
• Area, not volume, is the design currency. Doubling reactor volume without adding surface area does very little.

The governing diffusion-reaction equation

Consider a one-dimensional biofilm of thickness L on an impermeable surface, with substrate diffusing in from the bulk at z = L and no flux at the substratum z = 0. At steady state, diffusion balances reaction:

The reaction term is Monod, r(S) = μ_maxX_fS/[Y(K_S+S)], but the two asymptotes are what make the problem tractable and are usually sufficient:

• Zero-order (S >> K_S): r = k₀, constant. The equation integrates to a parabolic profile.
• First-order (S << K_S): r = k₁S. The solution is hyperbolic-cosine and gives the classical effectiveness factor.

For the zero-order case the profile and the penetration depth follow directly:

Worked example: how far does oxygen actually get?

Take a dense nitrifying biofilm at 20 °C. Free-water oxygen diffusivity is 2.1×10⁻⁹ m²/s; take D_e = 0.8 × 2.1×10⁻⁹ = 1.7×10⁻⁹ m²/s. Volumetric oxygen uptake in an active nitrifying film is of the order of 20 kg O₂/m³·d = 20,000 g/m³·d = 0.232 g/m³·s. Bulk dissolved oxygen 6 mg/L, and assume negligible external film resistance so S_s ≈ 6 g/m³.

• L_p = √(2 × 1.7×10⁻⁹ × 6 / 0.232) = √(8.8×10⁻⁸) = 2.97×10⁻⁴ m = ≈300 µm.
• Now repeat for ammonia at a bulk concentration of 6 mg N/L. Nitrification consumes about 4.57 g O₂ per g N, so the volumetric ammonia uptake is 20,000/4.57 = 4,376 g N/m³·d = 0.0507 g/m³·s, with D_e for ammonium of roughly 1.4×10⁻⁹ m²/s.
• L_p,NH4 = √(2 × 1.4×10⁻⁹ × 6 / 0.0507) = √(3.31×10⁻⁷) = ≈575 µm.

This single ratio explains why MBBR nitrification stalls in tanks with adequate carrier area, and why the fix is nearly always more air rather than more media. The equipment side of that is covered in our guide to aeration and oxygen transfer.

Surface flux: the number that sizes the reactor

For a deep, zero-order, fully oxygen-limited biofilm, the flux into the film has a compact closed form obtained by integrating the profile:

Continuing the example: J_O2 = √(2 × 1.7×10⁻⁹ × 0.232 × 6) = 6.87×10⁻⁵ g/m²·s = 5.9 g O₂/m²·d. Dividing by 4.57 gives a nitrification flux of 1.30 g N/m²·d — squarely within the 0.8–1.5 g N/m²·d range quoted for well-aerated MBBR nitrification at moderate temperature. The theory reproduces the field data because it captures the actual mechanism.

Sizing follows immediately. To nitrify 300 kg N/d:

• Required biofilm area = 300,000 g/d ÷ 1.30 g/m²·d = 230,800 m².
• With carriers of protected surface area 500 m²/m³ at 50% fill, effective specific area = 250 m²/m³.
• Reactor volume = 230,800 ÷ 250 = 923 m³.
• Oxygen demand for the nitrification alone = 300 × 4.57 = 1,371 kg O₂/d, before any carbonaceous demand — which is what actually sizes the blowers.

Note the structure of the answer: area came from flux, and volume came from area divided by carrier packing. Hydraulic retention time never entered the calculation. That is the defining difference from suspended growth design, and it is developed further in our MBBR design guide and the MBR versus MBBR comparison.

The Thiele modulus and the effectiveness factor

For first-order kinetics the classical chemical-engineering formulation applies directly, and it is the cleanest way to express how much of a biofilm is doing useful work.

Worked values. With k₁ = 0.05 s⁻¹ and D_e = 1.7×10⁻⁹ m²/s, √(k₁/D_e) = √(2.94×10⁷) = 5,424 m⁻¹. Then:

The conclusion is uncomfortable for anyone who equates thick biofilm with a healthy reactor: a thick film is a liability. It adds diffusion resistance, increases the risk of sloughing in slugs, promotes anaerobic activity and odour at the base, and can bridge and clog carriers. Thin, well-sheared films are the objective.

External mass transfer and the role of shear

Before substrate reaches the film it must cross a hydrodynamic boundary layer, adding a resistance in series with the internal one:

Mixing intensity therefore does double duty. It thins the boundary layer, raising the surface concentration and hence flux; and it applies the shear that controls film thickness through detachment. The steady-state thickness is where growth balances detachment:

This is why an MBBR carrier is designed with protected internal surface: enough shielding that carriers colliding at speed do not scour the film away entirely, and enough exposure that the film is kept thin. It is also why an under-mixed MBBR fails in two ways at once — starved surfaces and overthick films — and why fixing the mixing usually fixes both.

The same balance governs trickling filters, where hydraulic loading provides the shear, and rotating biological contactors, where rotational speed does.

Stratification: simultaneous nitrification and denitrification

Because oxygen penetrates only a few hundred micrometres, a film thicker than that has an anoxic interior supplied with nitrate from the aerobic exterior. If biodegradable carbon also reaches that depth, denitrification proceeds inside the same film that is nitrifying on its surface.

This simultaneous nitrification–denitrification (SND) is genuinely useful — it recovers alkalinity and reduces the aeration demand — but it is difficult to control, because the two requirements conflict. Thicker films give more anoxic volume and better denitrification but lower nitrification flux; thinner films do the reverse. Reported SND performance therefore varies widely between plants, and it should be treated as a bonus rather than as a design basis unless the reactor has been specifically configured and proven for it.

The membrane-aerated biofilm reactor inverts the geometry deliberately: oxygen is supplied through a gas-permeable membrane at the base of the film, so the aerobic layer is innermost and the anoxic layer faces the bulk liquid. Nitrate produced at the membrane must diffuse outward through carbon-rich biomass, which makes SND the natural rather than the accidental outcome, and allows oxygen transfer efficiencies far above bubble aeration because no bubble ever leaves the water. That reversal is explored in our MABR guide.

Temperature, and why biofilms are more robust in winter

Reaction rates fall with temperature according to the usual Arrhenius correction, k_T = k₂₀θ^(T−20) with θ ≈ 1.06–1.10 for nitrification. Diffusivity, however, falls much more slowly — roughly with the Stokes–Einstein dependence D ∝ T/µ(T), about 2–3% per degree.

Since deep-biofilm flux goes as J = √(2D_ek₀S), it depends on the square root of the reaction rate. A rate that halves reduces flux only by a factor of √2 = 1.41, not by 2. Diffusion-limited biofilms are therefore intrinsically less temperature-sensitive than suspended growth — a real and often-overlooked advantage in UK winter conditions.

Note the caveat: this holds only while the film remains deep and oxygen-limited. If the film is fully penetrated, flux is k₀L and the full Arrhenius penalty applies. Knowing which regime a design sits in is the whole point of computing the penetration depth.

Applying the theory: a checklist for attached-growth design

1. Establish which substrate limits. Compute penetration depth for oxygen and for the target substrate at design bulk concentrations. Design against whichever is shorter.
2. Check the bulk DO to ammonia ratio. Below roughly 3–4 g/g, expect oxygen limitation and size aeration accordingly.
3. Derive area from flux, not volume from HRT. Use a flux appropriate to the temperature, the bulk concentration and the loading regime.
4. Use the winter flux. With the square-root dependence, the penalty is milder than for suspended growth, but it is not zero.
5. Design mixing for both duties. Boundary layer thinning and detachment shear are the same variable; specify carrier circulation velocity, not just air flow.
6. Provide for sloughing. Attached growth releases solids episodically; downstream clarification or flotation must handle the peak, not the average.
7. Validate with a pilot where the feed is unusual. Flux correlations derived from municipal sewage transfer poorly to inhibitory or high-salinity industrial effluents.

Getting the limiting-substrate question right at concept stage is the difference between a reactor that meets its consent in February and one that needs a retrofit — which is why treatability data precedes equipment selection in any competent biological process design.