If you maintain the pH, then that equation is not correct. For a constant CYA level, the loss in sunlight is proportional to the FC level. So you see a constant percentage FC loss. What you are seeing that has you think there is a power of 2 on the FC is that if you SLAM the pool or otherwise elevate the FC using a hypochlorite source of chlorine, then the pH rises and that produces much more hypochlorite ion compared to hypochlorous acid in the pool water. The half-life of hypochlorite ion in noontime sunlight is only 20 minutes while for hypochlorous acid it is 2 hours and 10 minutes. At a pH of 7.5 with a 50/50 mix of the two, the half-life is around 35 minutes (this is at the surface of the water and ignoring CYA shielding effects).
I've measured the FC drop in my pool after raising the FC to around 20% of the CYA level and then seeing it drop day to day. With 50 ppm Borates in the water, my pH didn't change very much from the addition or subsequent drop and I have noticed a consistent percentage loss day to day. However, this is with the pool covered most of the day except for an hour or so each day. Others on this forum seem to also report FC percentage drops except, as I wrote, when they elevate the FC a lot as with a SLAM in which case the pH rises significantly so an increased loss rate would be expected.
So I think the equation that should be used should be of the form where the d(FC)/dt = FC * f(CYA, pH, temp) so that the absolute FC loss is proportional to the FC level when CYA and pH and temperature are held constant. Another way to put this is that the percentage FC loss is a function of those factors so %FCloss/dt = f(CYA, pH, temp).
We know that the loss rate due to temperature is roughly doubled for every 13ºF of temperature. I can also calculate the pH effect for the loss from hypochlorite ion and hypochlorous acid. What I don't have good data for is the loss rate from CYA and its CYA (or CYA-Cl) shielding effect. The closest I have to that is an estimate for SWG pools in the table in
this post. That table intentionally has the same FC/CYA ratio for most of its entries so the loss from unbound CYA should be constant for all of them. It shows the CYA shielding effect because any loss rate associated with chlorine bound to CYA (which is most of FC) should have loss rates increase but instead one sees them decrease at higher CYA levels even with proportionally higher FC levels. So the loss rate formula should have CYA in the denominator at some higher power.
At the 5% FC/CYA ratio in the table and at a pH of 7.5 the absolute FC loss rate of hypochlorous acid and hypochlorite ion with no CYA shielding effect would be 118% of the unbound FC per hour and the unbound FC is 0.041 so that is 0.048 ppm FC per hour or over 8 hours of noontime equivalent sun that would be 0.38 ppm FC per day. Clearly most of the FC loss is not coming from the unbound chlorine. The amount of bound chlorine is close to the FC level (around 99% of the FC level at FC 4 ppm with CYA 80 ppm). If I sum the 0.04 with a presumed bound chlorine loss (subtracting out 0.04 from the 0.15 total) and then divide by CYA
n, I get an "n" of 1.6 but remember that the table is a very rough approximation of what is seen and is not carefully measured from experiment.
So that table's FC loss/hr is very roughly close to (80*FC/CYA + 10*FC) / (CYA
1.6)
The formula is of the form (unbound chlorine loss + bound chlorine loss) / (CYA shielding effect). The "80" will be a function of pH and FC/CYA ratio (it can be calculated from my Pool Equations spreadsheet and knowing the relative loss rates in sunlight of hypochlorous acid and hypochlorite ion) and will be higher at higher pH while both the "80" and the "10" will depend on temperature and amount of sunlight. As described in
this post the CYA shielding effect is more likely to be an exponential function of water depth rather than the simple power function shown above but the power function approximates the behavior for average pool depth. Note that the formula only works in the normal ranges of FC/CYA ratios and would not be correct at high FC/CYA ratios approaching a SLAM or a pool with little or no CYA.
How does this fit with your data? If you have pH data then we can calculate the fixed unbound chlorine loss factor.