Sponges are pumps. Each one drives water through thousands of tiny chambers, keeps the bacteria, and expels clean water through its osculum. How many does a lagoon need to stay clear?
Integrates the lagoon balance numerically for different sponge counts and compares it with the closed-form prediction.
The science
Continuity at the osculum. A sponge of tissue volume Vs pumps Q = p·Vs, where p is the specific pumping rate (mL of water per mL of sponge per second; typical values are of order 0.01 to 0.5 s-1, so this is a dial, not a measurement). All that water leaves through the osculum of area Ao, so by conservation of mass vo = Q / Ao. The jet Reynolds number is Re = vo do / ν with ν ≈ 1.0×10-6 m²/s.
Retention. Collar cells (choanocytes) trap particles; the fraction of a particle type removed from the water that passes through, η, is high (typically 0.8 to 0.99) for bacteria and small algae. The values in the particle menu are rounded typical figures, not measurements of one species. The clearance rate is F = η·Q.
Lagoon balance. A well-mixed lagoon of volume V is flushed by the tide with residence time τf: dC/dt = (Cin − C)/τf − (η Qtot/V)·C. With k = ηQtot/V the solution is C(t) = C* + (C0 − C*)·e−t/τ, 1/τ = 1/τf + k, C* = Cin / (1 + k τf). The page integrates the equation with a 4th-order Runge-Kutta stepper and compares it with this closed form.
Temperature and silt. Pumping follows a Q10 law, p(T) = p20·2(T−20)/10 (Q10 = 2 is a common rule of thumb). Suspended sediment makes sponges close their pores and stop pumping; here 1/(1 + (S/30 mg/L)²), an illustrative curve of the right shape, not a calibrated one.
Simplifications. The lagoon is one well-mixed box, so there are no concentration gradients between sponges, no re-filtration of the exhaled jet, no growth or death of sponges, no bacterial growth, and a constant inflow concentration. Sponge tissue volumes come from simple geometric estimates of the three modelled shapes (tube 0.04, vase 0.07, barrel 0.12 L per (height/10 cm)³). The 3D particle motion is schematic and exaggerated; only the numbers on this page are physical.
Please spot-check against primary literature before citing any constant: p, η, Q10, the carbon per cell and the silt curve are typical or illustrative values.