The numbers below are taken from the dry lab's reactor calculation sheet of 23 September 2026. They are computed from the module geometry and were not measured, so an error in the assumed geometry propagates into all of them.
Wall shear and the operating envelope
Wall shear rate sweeps the membrane surface, sets the critical flux, and is the quantity held constant to make the scale-up linear. For laminar flow in a round lumen it is determined by the flow per fibre alone:
\[ Q_1 = \frac{Q_{xf}}{N} \qquad \dot\gamma_w = \frac{32\,Q_1}{\pi d_i^{3}} \qquad \tau_w = \mu\,\dot\gamma_w \qquad \mathrm{Re} = \frac{4\rho Q_1}{\pi \mu d_i} \]
The cross-flow has to be divided by the fibre count before it enters the single-fibre expression; substituting the pump total overstates the wall shear by a factor of N, and we found that error in more than one published reactor write-up. Taking the module as eight fibres of 1.0 mm bore, the operating envelope is as follows.
| Cross-flow | Lumen velocity | Wall shear | Shear stress | Re | Axial drop | Single pass |
|---|
| 230 mL/min (vendor floor) | 61 cm/s | 4,881 s−1 | 4.9 Pa | 610 | 11.6 kPa | 0.98 s |
| 377 mL/min (band closes) | 100 cm/s | 8,000 s−1 | 8.0 Pa | 1,000 | 19.0 kPa | 0.60 s |
| 792 mL/min (Re 2,100) | 210 cm/s | 16,806 s−1 | 16.8 Pa | 2,100 | 40.0 kPa | 0.28 s |
| 2,600 mL/min (wide open) | 690 cm/s | 55,174 s−1 | 55.2 Pa | 6,897 | 131.5 kPa | 0.09 s |
Conventional hollow-fibre microfiltration of bacterial cultures operates at 2,000 to 8,000 s−1, and Maiorella and colleagues place the balance between sweeping the membrane and damaging the cells nearer 1,500 to 3,000 s−1 [8]. The pump's vendor floor of 230 mL/min [9] already produces 4,881 s−1, in the upper half of the conventional band, and the band closes at 377 mL/min, which is 6% of a dial that reads to 2,600 mL/min. The lowest available pump setting is therefore the only one that falls inside the band.
B. subtilis is a walled organism, so the 3 to 5 Pa wall-stress ceiling quoted for mammalian perfusion does not apply, and single-pass exposure carries a wide margin at every setting in the table. Cumulative exposure we have not established: at 230 mL/min a 100 mL loop passes the fibres about 138 times an hour. Van Reis and colleagues attribute protein aggregation in such loops to pump passages and air entrainment rather than to shear itself [3].
Critical flux and the fouling margin
Below a critical permeate flux, particle deposition is negligible and transmembrane pressure does not drift. Above it, cells accumulate on the membrane surface as a cake layer and the membrane fouls [5]. For a particulate suspension in laminar cross-flow, the shear-induced diffusion model estimates that threshold [6][7]:
\[ J_{crit} = 0.072 \left( \frac{a^{4}}{L} \right)^{1/3} \dot\gamma_w \ln\!\left( \frac{\phi_w}{\phi_b} \right) \]
With a = 0.71 µm as the equivalent-sphere radius of a 0.8 × 3 µm rod, the wall fraction pinned at random close packing (0.60) and the bulk fraction computed from cell density (OD600 1.454 at 109 cells per mL per OD unit), the threshold at the 263 mL/min operating point with eleven fibres, wall shear 4064 s−1, is 46 LMH against a design flux of 2. The ratio J/Jcrit is 0.04.
The membrane is therefore oversized for the duty by more than an order of magnitude, which follows from selecting the module for its fibre geometry and accepting the area supplied with it. Flux could be raised about 23-fold before fouling becomes limiting. That headroom is still short of what cell-specific perfusion would require, which begins at about 60 LMH for a litre of culture (below), so on this module the membrane and the medium supply become limiting together. Two qualifications apply to the estimate. Local flux is not uniform across the surface, so low-level fouling can occur below the nominal threshold and no operating point should be quoted as sitting exactly at Jcrit [4]. And the model treats the feed as cells in water, whereas secreted protein, media components and cell debris each add resistance that the expression does not account for.
Cell-specific perfusion rate
The governing quantity for this reactor is the cell-specific perfusion rate, the volume of medium delivered per cell per day. The chemostat washout inequality D < µmax does not apply, because a 0.2 µm membrane retains rods of 1 to 2 µm: permeate leaves the system and biomass does not, so the dilution rate can exceed the growth rate without washing the culture out. This retention of biomass at high dilution is the defining property of perfusion culture.
Modern perfusion practice sits between 0.02 and 0.5 nL; at 2 LMH, with OD600 1.454 in 1 L and 109 cells per mL per OD unit, ours is 0.0007, about thirty times under the band, so the culture is being fed very lean per cell. Reaching the bottom of the band, 0.02, needs about 60 LMH, above the 46 LMH fouling threshold, and 0.05 needs about 150. The medium arithmetic points the same way: at 2 LMH the module draws 0.67 mL/min, which is 0.96 L of medium a day and about ten turnovers of the 100 mL loop; at 60 LMH it is about 29 L a day. At the design flux, 0.96 L a day is unremarkable for a bench run over a shift. For a device meant to sit unattended in a field for a week, the medium it would take to feed the cells conventionally is the binding constraint, and more membrane alone does not remove it.
Oxygen, energy and dead time
Oxygen supply limits cell density before any other quantity does. At the carrying capacity of OD600 1.454, taking 0.4 g dry cell weight per litre per OD unit and a conservative uptake of 3 mmol g−1h−1, the culture consumes 1.74 mmol L−1h−1 and requires a kLa of 12.5 h−1. That figure is modest for a sparged fermenter and demanding for a closed loop with no gas port. The dissolved inventory shows the constraint more clearly than the rate does: a 100 mL loop held at 30% of saturation contains 0.006 mmol of dissolved oxygen against a consumption of 0.174 mmol per hour, so an interruption in supply exhausts it in about two minutes, and in just under seven minutes from full saturation. Oxygen behaves here as a throughput quantity, and cell density has to be capped at what the loop can transfer.
Energy determines whether a unit can be deployed in a field at all. We sized the supply against Taiwan's worst-case winter peak-sun-hours of about three per day, and not against the annual average. At a 75% harvest efficiency from panel through battery to load, a 45 W panel returns 101 Wh per day against a draw of 72 to 192 Wh; at a mid-point draw of 5 W the balance supplies 0.84 of the requirement, and a 53 W panel would close it. The whole of that spread comes from one unmeasured quantity, the continuous draw of the recirculation pump, which a clamp-meter reading would resolve.
Loop dead time θ is the sum of sensor sampling, controller cycle, LED ramp, the CcaSR activation half-time, secretion lag and pump transit, of which the activation half-time of 105.1 ± 1.5 min [2] exceeds the others by an order of magnitude. The system time constant τ is the shell exchange time, 150 minutes at 2 LMH and 15 minutes at 20 LMH, so θ/τ falls between 0.7 and 7.0: comparable to the time constant at the design flux and dead-time dominated above it. Such a loop remains controllable but responds slowly. Raising the flux shortens τ and increases the ratio, so the delivery rate cannot be selected on plant-response grounds alone.
The transmembrane pressure the scale-up method is supposed to hold constant is read at ±10 kPa against an axial drop of 11.6 kPa at the pump floor [10], so fouling has to be read from a trend over time and not from a single reading. A 0 to 100 kPa sensor resolves ±1 kPa and is the least expensive upgrade available to this reactor; the sensor specification is on Measurement.
Where the geometry comes from
Our own records give the module as both 150 cm² with eight fibres and 200 cm² with eleven. The one place the arithmetic has been checked against the bench agrees with eleven: on 16 August 2026, water at 22 °C, eight pump settings with the module fitted and with a jumper in its place, both fitting ΔP = aQ + bQ² through the origin at module R² 0.9954; at 263 mL/min the net module drop is 0.096 bar, 1.03 times the Hagen–Poiseuille prediction for eleven open fibres. A lumen weighed on 13 August gave 7.0 to 7.2 mL of holdup against a computed 3.74 mL. The shear table above uses eight fibres, so the shear values it reports are the conservative, higher ones. The membrane area, permeate and medium figures use 200 cm² and eleven fibres, which is what the bench check and the rig software's constants agree on. Bench calibration of the pump puts setting 10 at about 100 mL/min against the vendor floor of 230, and three calibrations across three days span 100 to 140, which widens the usable shear window rather than closing it. The hydraulic fit is R1 on Digital Twin.
