The residual that leaves the plant is not the residual that arrives. Free chlorine keeps reacting — with organics still in the water, with biofilm, with corrosion products on the wall — and it does so exponentially: $C = C_0\,e^{-k t}$, read aloud C equals C-nought e to the minus k t.

The letters: $C$ is the residual remaining in mg/L at the sample point; $C_0$ — C-nought — is the starting residual leaving the plant, also mg/L; $t$ is the travel time — the water age from plant to sample point; and $k$ is the bulk decay coefficient, a fraction lost per unit time, measured on your own water in a bottle test. $k$ and $t$ must agree on their unit of time, and in distribution work that unit is almost always the hour.

Read the shape before the number. Decay is proportional, not fixed: the water does not take 0.2 mg/L per hour, it takes the same FRACTION every hour. So the natural measure is the half-life, $t_{1/2} = \dfrac{\ln 2}{k}$. A $k$ of 0.0693 per hour is a ten-hour half-life; twenty hours of travel is two half-lives and a quarter of the residual survives; thirty hours is three and an eighth survives. Counting half-lives is the sanity check that catches a $k$ off by a factor of ten before the arithmetic does.

Run backwards, $C_0 = C\,e^{k t}$ sets the plant's setpoint from a far-end target, and it is unforgiving: each extra half-life doubles what the plant must feed. Run backwards the other way, $k = \dfrac{\ln(C_0/C)}{t}$ is the fit itself — natural log, not base 10, because the model is written on e.

Two things the model does not know about, and both make the field residual lower than the page. First, a bottle test measures bulk decay only; a real main adds wall demand from biofilm and tuberculation, which can dominate in old unlined cast iron and is entirely absent from your bottle. Second, travel time in a distribution system is not one number: a dead-end leg or an oversized tank can hold water for days while the trunk main delivers in hours. It is the old water that loses its residual, grows nitrifiers and returns the coliform positive.

Which is why chasing a distant dead end by raising the plant residual is the wrong move. It dulls the taste everywhere, drives trihalomethanes up across the whole system, and treats a hydraulic problem with chemical. Flushing the dead end, or re-operating the tank so it actually turns over, is almost always the better answer.