C.40.CD:5.3 - A physical invariant replaces a timed operation

A device should dispense 40 cm³ from a reservoir. An earlier constant-flow model used V = q t; with q = 20 cm³/s, it prescribed two seconds. The receiving situation has unknown time-varying outflow, so that duration no longer determines the volume.

The reservoir has constant cross-sectional area A = 10 cm², starts at height h₀ = 15 cm and has no inflow or leakage. The supplied model treats the level reading and shutoff as ideal. Conservation and geometry give the dispensed volume:

V = A(h₀ − h).

The changed construction asks which level corresponds to the required volume. It gives h = 15 − 40/10 = 11 cm. The new operating rule is to stop outflow when that level is reached. At that event the model gives 40 cm³, independently of the outflow’s variation before it. If outflow stops earlier, the desired level is never reached; the volume relation alone supplies no completion guarantee.

This result connects physical conservation, a mathematical invariant and an event-based operating rule. Realizing it requires usable level observation and shutoff through C.16 and C.29.3. This supplies a design result under the model’s premises.

Suppose receiving work also requires completion within eight seconds. The level rule has left that condition open. A supplied bound q(t) ≥ 5 cm³/s until the target level makes the accumulated outflow at least 5t, so 40 cm³ is reached within eight seconds under the same ideal model. If no such flow support is available, keep the deadline unresolved or change the physical arrangement.