A static dose calculation assumes constant flow and constant influent — real plants have neither. This page covers the maths that keeps a dose on target as conditions change: the feed-forward pacing equation, the PID feedback equation, the compound (cascade) loop, dead-time and why it limits feedback, and how to size a usable control range.
Proactive control — act on the disturbance before it reaches the process.
Feed-forward sets pump output directly from measured flow using a fixed pacing constant. It is fast and stable but has no self-correction; calibration drift shows up as slow process drift.
k = pacing constant (L solution per m³ flow)C = target dose (mg/L), S = solution strength (g/L)k by bench/jar testing across the seasonal range.Reactive control — measure the result, correct the error.
A downstream analyser measures the controlled variable; the PID controller drives the error to zero. Robust and self-correcting, but limited by loop dead-time.
e = setpoint − measurement (the error)Kp proportional, Ki integral (removes offset), Kd derivative (rarely used on noisy water signals)Fast bulk dose from flow, accurate trim from the analyser.
The workhorse of modern dosing. Feed-forward delivers the bulk dose instantly; a bounded PID trims for accuracy and drift. Tolerant of calibration error and a wide operating range.
The transport + reaction delay that destabilises pure feedback.
Dead-time is the lag between changing the dose and seeing the effect at the analyser — transport time plus reaction time. Large dead-time relative to the controller's response makes feedback oscillate or go sluggish; the fix is feed-forward-dominant control or a closer analyser.
Vpipe = volume between injection and analyser; Q = flow; treact = reaction time30 m of DN100 pipe (V ≈ 0.236 m³) between dosing point and pH probe at Q = 30 m³/h, near-instant reaction.
The pump must cover the full flow × dose envelope without losing accuracy.
The required turndown is the ratio of maximum to minimum dose duty. The pump's accurate turndown must exceed it — dilution (Equation 3 on the equations page) is the usual way to extend range.
Send us trend data and the P&ID. We will calculate dead-time, recommend the right control mode, retune the PID and size the turndown so the loop holds setpoint across the full flow range.
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