First-Order Chlorine Decay

Free chlorine does not sit still. It reacts — with natural organic matter it did not finish with at the plant, with biofilm on the pipe wall, with corrosion products, with whatever else the distribution system offers — and the residual falls exponentially as it goes. $C = C_0 e^{-kt}$ is the standard first-order description, and what "first order" means in practice is that the residual loses the same fraction in every equal interval: if it halves in twelve hours, it halves again in the next twelve, whatever it started at.

Take a main carrying 1.4 mg/L out of the plant with a bulk decay coefficient of 0.15 per hour. Eight hours of travel gives $kt = 1.2$, and $e^{-1.2} = 0.301$, so 0.42 mg/L reaches the sample point. The implied half-life is $\ln 2 / k = 4.6$ hours — a useful figure to carry in your head, because it converts a decay coefficient into something you can reason about without a calculator.

The coefficient belongs to your water and should be measured on it. The standard method is a bottle test: fill headspace-free bottles at the plant, hold them at distribution temperature in the dark, and read the residual at intervals over a day or two. Plot the natural log of the residual against time and $k$ is the negative of the slope. Three cautions about that fit. Decay is strongly temperature dependent, so a coefficient measured in February badly understates August loss. The first hour or so is usually much faster than everything after it, because the fast-reacting organics go first, and a $k$ fitted through that initial plunge will overpredict decay at the ages your system actually runs — fit the long tail. And a bottle test measures bulk decay only.

That last point matters more than it sounds. A glass bottle has no biofilm, no tuberculation and no corroding iron. A real main has all three, and wall demand can dominate bulk demand entirely in old unlined cast iron. So expect the field residual to come in below what this equation predicts, and treat the difference as information rather than as error: a large gap between bottle and field is a measurement of your pipe's condition.

Travel time is not a single number either. A distribution system is not one pipe. The trunk main might deliver in three hours while a dead-end leg holds water for four days and an oversized storage tank stratifies and turns over a fraction of its volume a day. It is the old water that loses its residual, and losing the residual is where nitrification starts in a chloraminated system and where a coliform positive eventually appears. Running the equation backwards to estimate water age from a residual drop is a legitimate blunt instrument for finding those places, but it assumes bulk decay and a single path, and a sample point fed by a blend of fresh trunk water and old tank water returns a meaningless average of the two. Use it to decide where to point a hydraulic model or a fluoride tracer, not as a substitute for either.

The design direction is the honest one to end on. Solve for $C_0$ and you get the plant setpoint needed to hold a target residual at the far end — and the exponential is unforgiving, because every additional half-life of water age doubles what you must feed. Which is why the answer to a distant dead-end is almost never "raise the plant residual". That dulls the whole system with taste and odour complaints and drives disinfection by-products up everywhere, not just where the problem is. Flushing the dead end, or re-operating the tank so it actually turns over, treats the cause.