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Re: [APD] dilution

After reading Liz's email: http://fins.actwin.com/aquatic-plants/month.200501/msg00416.html (or see below), I started to wipe off some dust off my calculus book and made some simple plots.

But it seems like I always get convergence for all %'s of water changes (90 % and 10% and even 1%). So maybe somebody can point out if there is anything wrong with the calculations below.

First thing is I used Liz's series and generelize it in:
   \            -n
   /__  a r              = a/(1-r), this will become divergent (has no maximum) if r>=1
limit n-> infinity

Which is called Geometric series. The "a" is the concentration (or X, the amount you dose your aquarium weekly as in Liz's mail, for now we go with Liz's definition, it doesn't matter what it is because it's only a constant) and "r" is "1-%waterchange" which is always <1 so that no matter how much you change the water, the buildup will not go to infinite. This idea seems rediculous, if I were to change only 1% weekly, the total amount after lots of water changes will not be infinite but just 100 times the amount I dose weekly.

Where did this generalization comes from? I'll start with the example Liz made with 10% water changes:

X+.9X+.81X+ .729X = X*(.9)^0+X*(.9)^1+X*(.9)^2+X*(.9)^3+.....+X*(.9)^n

Which is the same as:

   ___limit n-> infinity
   /__  X*(.9)^n              = X/(1-.9)=10*X

So after Infinite times (takes about 50) water changes of 10%, the amount in the tank will not build up any more then 10 times the amount you add every week. You can see the convergence limits in the plot below for all % of water changes:

It doesn't take infite times to "see" the convergence limit. When you do a 10% water changes, you can "see" the limit already within 50 times waterchanges (about a year when you do weekly water changes). And when you do larger weekly waterchanges, you'll see this limit even earlier (see the plot). So don't get scared off from the infinite limit series, it's closer then you think :).

On the y-as of both graphs, I called it "concentration factor", and it's no other than the 1/(1-r) in the first graph and the second graph it's
   \         n
   /__  r     .

To get the amount in the tank after n times water changes, you simply multiply this concentration factor with the amount you dose every waterchange-period (the X in Liz's mail). Some interesting result I get is that when u use Tom Barr's Estimative Index method, you'll get with 50% or 70% water change, respectively 2 - 1.4 concentration factor. So when the plant's don't use anything at all (which is extremely unlikely) you won't get infinite buildup but rather a factor 2 or resp 1.4 of the amount you add every waterchange-period.


  I amde no claims it was. It's just to show how dilution works. 

If you assume that nothing is consumed, and you dose the same amount each week (X), and you change 50% of the water each week...

Week 1 = X
Week 2 = X + X/2
Week 3 = X + X/2 + X/4
Week 4 = X + x/2 + x/4 + x/8

This is a simple series:

   \       -n
   /__  2   where n goes from 0 to infinity

It converges at 2. To find the maximum concentration multiply X times 2. So at a large number of weeks then you can safely say that the concentration after 50% water change + X is equal to 2X. Doesn't matter if it's KNO3 or fish poop or some allelopathic substance excreted by plants. If you are doing a 50% water change every 4 weeks but are dosing X each week then at the end of a sufficient number of months the maximum concentration of X is 2(4X) or 8X. You can set up a series for any dosing and dilution scheme. Some of them "converge" or have a maximum value. Some don't.

A 10% water change weekly and dosing X gives:

X+.9X+.81X+ .729X  which builds up concentration pretty fast.


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