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Date: Mon, 18 Nov 96

I hear over and over again this "aquarium rule of thumb" in directions for adding X mg/liter of PPDD, magnesium, or whatever to add to your aquarium that "you can't overdose as long as you do 25% water changes every two weeks". (PPDD == Poor Persons Dupla Drops) At first, it seemed to me, by adding some material every day/week and then only taking out 25% of it leaving 3/4 of it (with a 25% water change) every two weeks, that one would build up stuff until one exceeds some limit and things in your aquarium would start dying. [Wow - say that last sentence ten times without breathing.] Anyway, for some people this "aquarium rule of thumb" may seem obvious, but for me it wasn't. So after a little digging around, I can now try to explain why this rule seems to be a "fact" and not a "myth". First, let's say we add 50mg/day of material (NaCl) to a 700 liters aquarium for two weeks, then do a 25% water change. After we continue doing this for a number of weeks, what would be the eventual concentration of this material (NaCl) in the water? Of course, we need to assume that the living inhabitants or nonliving whatevers are _not_ consuming/storing this material so that it can removed by a water change. On the other hand, if we know that we add 60mg of nitrogen a day and somehow we figure out that bacteria are consuming 10mg of this nitrogen each day, then we use the net 50mg/day for our calculations. Table 1. Let's See What's Happening So Far... Before Water Change After Water Change Time Total Amount - Concentration Total Amount - Concentration - ---- ------------ ------------- ------------ ------------- 2nd Week 700mg 1mg/L 525mg 0.75 mg/L [50mg*14days] [700mg/700L] [700*3/4] [525/700] 4th Week 1225mg 1.75mg/L 919mg 1.31mg/L [525 + 700] [1225mg/700L] [1225*3/4] [919/700] 6th Week 1619mg 2.31mg/L 1214mg 1.73mg/L [919 + 700] [1619mg/700L] [1619*3/4] [1214/700] 8th Week 1914mg 2.73mg/L 1435mg 2.05mg/L [1214 + 700] [1914mg/700L] [1914*3/4] [1435/700] At this point, it's not clear what the final amounts are going to be. One may observe that in the Before Water Change that the _difference_ in Concentration from one water change to the previous water change seems to be going down: (Likewise for the After Water Change Concentration.) 4th Week - 2nd Week => 1.75 mg/L - 1mg/L = 0.75 mg/L 6th Week - 4nd Week => 2.31 mg/L - 1.75mg/L = 0.56 mg/L 8th Week - 6nd Week => 2.73 mg/L - 2.31mg/L = 0.42 mg/L However, a pattern (series) is starting to form: The total amount this week total amount last week + 700 before water change = after water change and The total amount this week total amount this week * 3/4 after water change = before water change If we let X be the net amount of material we add before every water change, and (1-r)*100 be the percent water change over this same time period. (e.g. For 25% water change r=3/4, or for 20% water change r=4/5) We have: Before Water Change After Water Change Time Total Amount Total Amount - ---- ------------ ------------ 1 period X units X(r) units 2 period X(1 + r) units X(r + r*r) units [X*r + X] [(X*r + X)*r] 3 period X(1 + r + r*r) X(r + r^2 + r^3) 4 period X(1 + r + r^2 + r^3) X(r + r^2 + r^3 + r^4) 5 period X(1 + r + r^2 + r^3 + r^4) X(r + r^2 + r^3 + r^4 + r^5) etc..... where: r^2 = r*r r^3 = r*r*r r^4 = r*r*r*r etc r^n = r*r*r* etc... *r } n terms of r Some people may recognize (1 + r + r^2 + *** + r^(n-1) + r^n) where 0 < r < 1, as the well known, at least to Math majors, Geometric Series. - ---------------------------------------------------------------- Geometric Series: For n time periods, (1 + r + r^2 + *** + r^(n-1)) = (1 - r^n) --------- 1 - r which, after a long time, n get real big, (i.e., n approaches infinity) (1 - r^n) 1 ---------- becomes ----- where 0 < r < 1 1 - r 1 - r - ---------------------------------------------------------------- So for the nth water change: Before Water Change After Water Change Time Total Amount Total Amount - ---- ------------ ------------ nth period (1 - r^n) (1 - r^n)) X * --------- X * --------- * r 1 - r 1 - r When n gets big, the r^n terms get very small, so after a long time (steady state) we have: Before Water Change After Water Change Time Total Amount Total Amount - ---- ------------ ------------ long time 1 r X * ----- X * ----- 1 - r 1 - r >From this, we can construct the following table. With X net units of material being added before every water change the ultimate (steady state) amount of material in the aquarium can be found. Table 2. The Ultimate (Steady State) Amount of Material In the Aquarium by Adding X amount Before/After Each % Water Change Before Water Change After Water Change Ultimate Ultimate % Water Change Total Amount Total Amount - -------------- ------------------- ------------------ 10% (r=9/10) 10 * X 9 * X 20% (r=4/5) 5 * X 4 * X 25% (r=3/4) 4 * X 3 * X 33% (r=2/3) 3 * X 2 * X 50% (r=1/2) 2 * X X 66% (r=1/3) 3/2 * X 1/2 * X 75% (r=1/4) 4/3 * X 1/3 * X Smoke and mirrors one may say! Basically what's happening is that material builds up in the aquarium until the amount we put in before each water change is equal to the amount one takes out during the water change. So we know what the ultimate (steady state) amount of material will be in the aquarium after "a long enough" period of time (water changes). But, one may ask, how long is "a long enough" period of time? Let's compute the number of water changes it takes so that the total amount of material in the aquarium reaches within 20%, 10%, or 5% of the steady state value. All we got do is, for a given r, solve for n when: (1 - r^n) --------- = 0.8 or 0.9 or 0.95 1 - r Table 3. For a given Percent water Change, the Number of Water changes to 80%, 90%, or 95% Of the Steady State Amount Number of Water Number of Water Number of Water Changes to 80% Changes to 90% Changes to 95% % water change Total Amount Total Amount Total Amount - -------------- --------------- --------------- --------------- 10% (r=9/10) 16 22 29 20% (r=4/5) 8 11 14 25% (r=3/4) 6 9 11 33% (r=2/3) 4 6 8 50% (r=1/2) 3 4 5 66% (r=1/3) 2 3 3 75% (r=1/4) 2 2 3 It's interesting to note that for a 10% water change, the steady state variation in the amount of material before and after water change is only 10%, but it takes about a year (for water changes every two weeks) to get there. For a 50% water change, it only takes about two months (4 water changes) to get close to the steady state, but one gets a 50% variation in the amount of material that being added between every water change. The 25% water change seems to be a pretty good compromise between time to steady state and concentration variation between water changes. Of course, if one knew how much to add, one could just add the amount needed to reach the desired steady state, and then add just the right amount at water change to keep the total concentration constant. (i.e. It's not too difficult to measure the KH of the water before the water change, and using George Booth's recipe of "one teaspoon (about 6 grams) of sodium bicarbonate (NaHCO3) per 50 liters will increase KH by 4 degrees and will not increase general hardness," one could adjust the water to the desired level.) On the other hand, it's difficult to determine which micronutrient may be overdosing the plants before it's too late. So, at least with frequent water changes, and knowing what you're adding, one can at least keep the micronutrients below a certain amount. Sorry for the long post, but I hope some people find this information useful. Ron Wozniak Allentown PA, USA rjwozniak-at-lucent.com

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