Predicting the rate of chlorine loss based on CYA levels

[TreeFiter]

A while back, chemgeek provided some information regarding the rate at which chlorine is lost due to sunlight exposure at different concentrations of CYA. I took that information, and I came up with a series of charts that roughly predict how much FC levels will drop after each additional day passes.

This season, I have been plagued with problem pools that need to be "shocked" every week. I understand why this is happening, but unfortunately, my options are somewhat limited, and I can't necessarily do what needs to be done to correct the situation.

So because of these situations, I would like to create a chart that can roughly predict the same type of chlorine loss, but in terms of hours or even minutes. The problem is that I don't know the math involved.

My goal is to be able to approximate how long it will take before a pool will be swimable after adding a hefty dose of chlorine. That way I can try to dose a given pool such that it will be swimable by Saturday when the customer will want to swim.

Can anyone help me figure this out?

 

[chem geek]

The chlorine loss is a proportionate loss so a certain percentage per unit of time. It's an exponential decay rate, so FC = FC₀ * e^-k*t where FC₀ is the FC at time 0 (your starting or reference time) and "k" is a rate of loss. You can calculate k if you know the FC at two points in time since k = -ln(FC/FC₀)/t or perhaps more simply k = ln(FC₀/FC)/t so take the ratio of the starting FC to ending FC, take the natural logarithm of that, and divide by the time. That gives you "k" you can use in the exponential decay formula but of course you have to use the same units for time "t" as you used when deriving k. That can be hours if you want or even minutes, but your accuracy isn't going to be that precise.

Note also that the decay rate will vary a lot depending on how much sun there is on the pool so that not only varies by pool, but by season (angle of sun in the sky), by weather (clouds, etc.), and you already know it varies by FC. It can also vary by bather load or by nascent algae growth, but if there is sufficient minimum chlorine (FC/CYA ratio) levels and if the bather load is typical for a residential pool, then most of your FC loss will be from sunlight. If the pool has a pool cover, then there may be less loss from sunlight, but there will be additional loss from oxidizing the pool cover and that will come at night as well.

So the above formula assumes a constant "k", but the reality is that "k" will vary mostly due to the sunlight intensity. Obviously accounting for that makes it a more complicated formula and you'd be better off just figuring a 24-hour "k" and then assume that all the loss occurs during the day (say, over 12-hours).

Another complicating factor for this is that the decay at shock (SLAM) levels is faster than at normal levels for two reasons. One is that the FC/CYA ratio is higher so there is more active chlorine that decays faster than chlorine bound to CYA. The other reason is that the pH is usually higher during a SLAM unless you significantly lowered it first and at higher pH the higher hypochlorite ion concentration decays a lot faster than hypochlorous acid. At a pH of 7.5 where there is a 50/50 mix, the half-life near the water's surface is 35 minutes, but at a high pH with only hypochlorite ion the half-life is only 20 minutes.

So you'll have to do some measurements under different conditions, but once you determine "k" for that set of conditions you should be able to roughly predict the FC over time at least broadly.

If you were just to look at this in terms of days, then this becomes easier since you just figure out the percentage drop per day and then to figure out subsequent days just raise to that power the number of days. So if the 24-hour drop is 20% of the FC level, then the final FC is 80% of the initial FC. After two days it's 80% * 80% = 64%. After three days it's 0.8³ = 0.512 or 51.2% and so on. This is the same as using the exponential formula, but simpler because it uses a fixed time increment. I derive this below:

k = -ln(FC/FC₀)/1 = -ln(1-day-ratio)
FC/FC₀ = e^-k*t = e^{ln(1-day-ratio)*t} = (1-day-ratio)^t

So if you know the ratio of final to starting FC for a given time interval, then you can easily calculate what would happen over multiples of that time interval by simply raising that ratio to a power that represents how many such time intervals (which can be a fraction like 3.5 to figure out mid-day between days 3 and 4). Using our example of a 20% loss over 1 day so that after one day you have 80% of what you started with, then after 3-1/2 days you would have 0.8^3.5 = 0.458 = 45.8% left so a little less than half. In practice, at SLAM levels your losses may be substantially higher but you'll just have to see what those are.

Now let's say you want to go backwards and have a final FC target and a starting SLAM target and want to know how far back in time you need to go to make that work. Solve for "t" in the above so

t = ln(FC/FC₀)/ln(1-day-ratio)

So using our example of a 1-day-ratio of 0.8, let's say we shock to 20 ppm and want to end up at 5 ppm. Then t = ln(5/20)/ln(0.8) = -1.386/-0.223 = 6.2 days so roughly 6 days.

Now let's say that "t" is fixed at a week and you want to know what level to raise the chlorine so that it will end up at your target. The formula for this solves for FC₀:

FC₀ = FC/(1-day-ratio)^t

So for a 0.8 1-day-ratio and one week, so 7 days, we have 5/0.8⁷ = 23.8 so you need to raise the initial FC to around 24 ppm to have it end up at 5 ppm 7 days later.

 

[crek31]

Chem geek and I were typing at the same time. I was going to say what he said.

 

[BuckeyeChris]

yeah, me too.

 

[TreeFiter]

I went back to the information chemgeek provided way back when, and I realized that he provided approximations for %loss/hr. I was able to make a set of tables from that.

Chemgeek, thank you for the explanation, both past and present.

I realized as I was working on this that since the amount of sun light is the critical factor here, at the very least, I would have to think about what happens at night. So I decided that for my purposes, I should only need about 8 hours of predictability, which should capture the peak sunshine hours. So the charts won't be very useful for predictions in the early morning or evening, but could get me close during business hours.

I guess now all I have to do is compare it to a few real life situations, and see if it works.

For anyone interested in seeing the charts, here they are:

The top row is the Starting concentration of Free Chlorine, and as you move down the chart it predicts how much FC will be left after each hour passes. For some reason the top row shifted when I copied and pasted it in, but I think you can see whats going on.

100 ppm CYA
FC 10 20 30 40 50 100
1 hour 10 20 29 39 49 98
2 hours 10 19 29 38 48 96
3 hours 9 19 28 37 47 93
4 hours 9 18 27 37 46 91
5 hours 9 18 27 36 45 89
6 hours 9 17 26 35 44 87
7 hours 9 17 26 34 43 85
8 hours 8 17 25 33 42 83

90 ppm CYA
FC 10 20 30 40 50 100
1 hour 10 19 29 39 49 97
2 hours 9 19 28 38 47 95
3 hours 9 18 28 37 46 92
4 hours 9 18 27 36 45 90
5 hours 9 18 26 35 44 88
6 hours 9 17 26 34 43 85
7 hours 8 17 25 33 42 83
8 hours 8 16 24 32 41 81

80 ppm CYA
FC 10 20 30 40 50 100
1 hour 10 19 29 39 48 97
2 hours 9 19 28 38 47 94
3 hours 9 18 27 36 46 91
4 hours 9 18 27 35 44 88
5 hours 9 17 26 34 43 86
6 hours 8 17 25 33 42 83
7 hours 8 16 24 32 40 81
8 hours 8 16 23 31 39 78

70 ppm CYA
FC 10 20 30 40 50 100
1 hour 10 19 29 39 48 96
2 hours 9 19 28 37 46 93
3 hours 9 18 27 36 45 89
4 hours 9 17 26 34 43 86
5 hours 8 17 25 33 42 83
6 hours 8 16 24 32 40 80
7 hours 8 15 23 31 39 77
8 hours 7 15 22 30 37 74

60 ppm CYA
FC 10 20 30 40 50 100
1 hour 10 19 29 38 48 95
2 hours 9 18 27 36 46 91
3 hours 9 17 26 35 44 87
4 hours 8 17 25 33 42 83
5 hours 8 16 24 32 40 79
6 hours 8 15 23 30 38 76
7 hours 7 14 22 29 36 72
8 hours 7 14 21 28 35 69

50 ppm CYA
FC 10 20 30 40 50 100
1 hour 9 19 28 38 47 94
2 hours 9 18 27 35 44 89
3 hours 8 17 25 33 42 84
4 hours 8 16 24 31 39 79
5 hours 7 15 22 30 37 74
6 hours 7 14 21 28 35 70
7 hours 7 13 20 26 33 66
8 hours 6 12 19 25 31 62

40 ppm CYA
FC 10 20 30 40 50 100
1 hour 9 18 28 37 46 92
2 hours 8 17 25 34 42 85
3 hours 8 16 23 31 39 78
4 hours 7 14 22 29 36 72
5 hours 7 13 20 27 33 66
6 hours 6 12 18 24 31 61
7 hours 6 11 17 22 28 56
8 hours 5 10 16 21 26 52

30 ppm CYA
FC 10 20 30 40 50 100
1 hour 9 18 26 35 44 88
2 hours 8 16 23 31 39 78
3 hours 7 14 21 27 34 69
4 hours 6 12 18 24 30 61
5 hours 5 11 16 21 27 53
6 hours 5 9 14 19 24 47
7 hours 4 8 12 17 21 42
8 hours 4 7 11 15 18 37

20 ppm CYA
FC 10 20 30 40 50 100
1 hour 8 16 24 32 40 79
2 hours 6 13 19 25 32 63
3 hours 5 10 15 20 25 50
4 hours 4 8 12 16 20 40
5 hours 3 6 9 13 16 32
6 hours 3 5 8 10 13 25
7 hours 2 4 6 8 10 20
8 hours 2 3 5 6 8 16

10 ppm CYA
FC 10 20 30 40 50 100
1 hour 5 9 14 19 23 46
2 hours 2 4 6 9 11 22
3 hours 1 2 3 4 5 10
4 hours 0 1 1 2 2 5
5 hours 0 0 1 1 1 2
6 hours 0 0 0 0 0 1
7 hours 0 0 0 0 0 0
8 hours 0 0 0 0 0 0