Identifiability limits of a two-scale biofilter model

## The bench column and the two-scale model

The object is a laboratory biofilter: a 0.5 m packed bed, 0.1 m diameter, fed 4 L/min of a synthetic waste-gas stream at an empty-bed contact time near 59 s, sampled at three to five axial ports over several runs in 2006. The target compounds are two branched pentanols (3- and 2-methyl-1-butanol) and the aldehyde hexanal. Two length scales govern removal. Inside the biofilm, substrate S(z) diffuses and is consumed by immobilized biomass:

with no-flux at the support and bulk concentration at the interface. Here D_f is the effective diffusivity in the wet film, R_max the maximum volumetric reaction rate, and K_s the half-saturation constant. Non-dimensionalizing (1) gives the Thiele modulus φ = L_f√(R_max/(D_fK_s)) and an effectiveness factor η ∈ (0,1], the fraction of biomass actually fed. At column scale the gas concentration C_g obeys an advection–dispersion–reaction BVP with Danckwerts boundary conditions, the biofilm entering only through η. One forward solve is the η-table BVP family plus the column BVP.

The effectiveness factor is the numerical bottleneck, and solving the stiff biofilm BVP everywhere is both slow and unreliable. effectiveness_factor is a three-regime closure: the first-order form η = tanh(φ)/φ at small surface load; the exact deep-film closed form for φ ≥ 12, validated below 0.1% against the solved BVP; the solved BVP in between, with the reaction term clamped to s ≥ 0 so the integrator cannot reach the 1 + s pole. This is roughly 15× faster on the η-table and repairs an earlier fallback that returned an order-of-magnitude-low η in the diffusion-limited corner where the solver failed to converge.

## Calibration, sensitivity, and the identifiability wall

Reaction order is chosen before calibration by a hierarchical Bayesian fit of the three integrated rate laws to each compound, compared by PSIS-LOO rather than by R² on a linearized transform that is undefined once a concentration hits zero. Hexanal is rate-limiting with posterior probability 0.977, so it sets the industrial duty. The two-scale model is then calibrated with emcee, embedding one coupled BVP solve in every likelihood evaluation: 40000 steps, effective sample size ≈ 3397, run on UGA's Sapelo2 cluster under SLURM with MPI-parallel walkers and HDF5 checkpoint/resume across a 14-hour wall time. A Sobol analysis over 147,456 model evaluations apportions variance in removal across the seven log-parameters. Simulation-based calibration over 400 replicated datasets closes the loop.

SBC is unambiguous: every one of the twelve calibrated parameters is rank-uniform (chi-square p between 0.13 and 0.96). The sampler has no detectable bias. The design stage nonetheless fails. Propagating the calibrated posterior through the industrial scale-up gives a median required volume of 2579 m³ but a 95% credible interval of [1393, 698131] m³, and only 53.4% of posterior draws yield a bed that meets the removal target under the 3:1 length-to-diameter rule. A calibrated inference producing a design interval three orders of magnitude wide is not a bug to be tuned away: it is the correct report of what the data contain.

The mechanism is structural non-identifiability. The bench inlet concentrations sit far below K_s, so S / (K_s + S) → S/K_s and the reaction is pseudo-first-order everywhere in the column. Only the lumped ratio R_max/K_s is identifiable; R_max and K_s separately are not. The free Henry constant and the L_f–a product add further confounding, and near-complete removal in the first third of the bed leaves the slow-kinetics tail of the posterior unbounded. SBC passes precisely because the posterior faithfully represents that unbounded tail.

## The identifiable first-order reanalysis

The second model infers only the identifiable quantity: one macroscopic first-order rate k₁ (1/s) per compound and one observation-noise scale, with D_f and the Henry constants fixed to literature values and axial dispersion dropped (bed Péclet number >> 100, so plug flow). The forward column model is then analytic,

so the likelihood is milliseconds per evaluation instead of a nested BVP solve and the whole pipeline runs on a laptop. Calibration is 64 walkers × 8000 steps, acceptance 0.51, effective sample size ≈ 3972, elapsed 164 s. The rate constants are tight, and hexanal — the design-critical compound — is well constrained.

SBC over 400 datasets, chains thinned by 12, confirms the reanalysis is calibrated where it matters: all three k₁ are rank-uniform (p = 0.81 / 0.11 / 0.55), including hexanal at 0.55. The nuisance noise scales for the two alcohols are mildly non-uniform (p ≈ 0.003 / 0.006), a small real effect from under-exploring the more correlated rate–noise posterior of the steeper decays. It survives 400 datasets and thinning, and it touches no reported quantity.

## Design under uncertainty

Sizing the industrial bed for 95% hexanal removal at the report duty (Q = 2831.68 m³/min, the 3:1 length-to-diameter rule) propagates the k₁ posterior through the analytic model. The required volume has median 2421 m³ and 95% credible interval [1527, 3493] m³, a factor of 2.3 rather than 500, and every posterior draw is feasible. Sizing on the conservative 95th percentile of k₁ gives 3277 m³ (L = 33.5 m, D = 11.2 m), which reproduces the original report's point design of ≈ 3500 m³ (L = 34.23 m, D = 11.41 m) — now carrying an interval instead of a bare number.

The mechanistic model is not wrong and its HPC machinery is not wasted; it is the correct tool for data rich enough to resolve R_max and K_s separately. These bench data are not that. Separating the two constants requires experiments that leave the starvation regime: higher inlet load approaching K_s, shorter contact time, or transient step-loading. Until such data exist, the identifiable first-order rate is the whole of what the biofilter tells you, and the honest design is an interval.