Water · The arithmetic

Aquarium top-offs and water changes: the calculations

This is the part of the water change argument that is not a study and does not need to be. Evaporation concentrates, topping up adds more, and a repeated water change does not reset a tank but gives it a ceiling. Both lines are written out with their inputs named, worked against the measured Yerevan tap water figures, and printed so they can be re-run rather than trusted.

A canister filter with a drain hose
A canister filter connected to a drain hose and bucket.Fish Tank Society · CC BY-SA 4.0 · Resized; uncropped.

Original work · Mechanism 01

Evaporation and top-off. The strongest item on this page, because it is not a study.

Every week, evaporation takes away some pure water and you put back the same amount of tap water. The tank keeps everything the vapour left behind and gains everything the tap water brought. Write that down and it turns into one line.

V tank volume, held constant by topping up e fraction of V lost to evaporation in one week C_tap concentration of anything non-volatile in your tap water C(n) = C_tap x (1 + n e) after n weeks, if the tank was filled from the tap1

There is no ceiling in that line. It rises for ever. And it gives one number worth memorising: the tank doubles its dissolved load after 1 divided by e weeks1, whatever your tap water is and whatever the substance is. At 5 percent a week that is 20 weeks. At 2 percent it is 50 weeks. At 8 percent it is 12 weeks.1 The doubling time does not depend on how hard your water is. It depends only on how fast your tank evaporates.

Total dissolved solids in a Davtashen tank over two years, at three evaporation rates, with no water changes Three straight rising lines starting together at 305 milligrams per litre. At 2 percent evaporation per week the line reaches 939 after two years. At 5 percent it reaches 1891. At 8 percent it reaches 2843. The 1000 milligram per litre Armenian drinking water mineralisation limit is crossed by the 5 percent line at about one year and by the 8 percent line at about eight months. Every figure in this chart is calculated by this site from the model in source 25, using the Yerevan tap water values in source 1; the limit line is source 4. 1000 mg/L, the Armenian total mineralisation limit 0 500 1000 1500 2000 2500 3000 week 0 26 52 78 104 Weeks with no water change, topping up with Davtashen tap water at 305 mg/L 2 percent evaporation per week, a lidded tank. 305 at week 0, 464 at 6 months, 622 at 1 year, 939 at 2 years, all in mg/L. 5 percent evaporation per week, a typical open tank. 305 at week 0, 702 at 6 months, 1098 at 1 year, 1891 at 2 years, all in mg/L. 8 percent evaporation per week, an open rimless tank in a dry summer. 305 at week 0, 939 at 6 months, 1574 at 1 year, 2843 at 2 years, all in mg/L. 2 percent per week, 6 months: 464 mg/L 2 percent per week, 1 year: 622 mg/L 5 percent per week, 6 months: 702 mg/L 5 percent per week, 1 year: 1098 mg/L 8 percent per week, 6 months: 939 mg/L 8 percent per week, 1 year: 1574 mg/L 2% a week 5% a week 8% a week
  • 2 percent a week, lidded1
  • 5 percent a week, typical open tank1
  • 8 percent a week, open rimless in a dry summer1

The three lines are three levels of one variable, so they are three steps of one color, light to dark, in the same order as the variable. Every line is labeled at its end, so nothing here depends on telling two blues apart. The numbers are in the table below.

Calculated by us1 from Yerevan tap water measurements3. No water changes, topping up only. General hardness in dGH4, dissolved solids in mg/L
District and evaporation rateStart6 months1 year2 years
Avan, Kanaker, Nork, at 2 % a week13.4 / 2305.2 / 3506.9 / 46910.5 / 708
Central districts, at 2 % a week16.4 / 2909.7 / 44113.1 / 59219.7 / 893
Davtashen, at 2 % a week17.7 / 30511.7 / 46415.7 / 62223.7 / 939
Avan, Kanaker, Nork, at 5 % a week13.4 / 2307.8 / 52912.2 / 82821.1 / 1426
Central districts, at 5 % a week16.4 / 29014.7 / 66723.0 / 104439.7 / 1798
Davtashen, at 5 % a week17.7 / 30517.7 / 70227.7 / 109847.7 / 1891
Avan, Kanaker, Nork, at 8 % a week13.4 / 23010.5 / 70817.5 / 118731.7 / 2144
Central districts, at 8 % a week16.4 / 29019.7 / 89333.0 / 149659.6 / 2703
Davtashen, at 8 % a week17.7 / 30523.7 / 93939.7 / 157471.8 / 2843

The headline. A tank in Davtashen, filled from the tap at about 7.7 dGH and 305 mg/L3, evaporating 5 percent a week, topped up with the same tap water and never water changed, is at roughly 28 dGH and 1,100 mg/L after one year1. That is harder than any tap water ever recorded in Yerevan3, and the dissolved solids figure is above the 1,000 mg/L total mineralisation limit in the Armenian drinking water standard2. Nothing went wrong. Nobody added anything. Water evaporated and was replaced.

A Caridina shrimp keeper aiming at 4 to 6 dGH left the target range in about three months1.

The one measurement this page asks you to make

Tape on the glass, seven days, one number.

Nobody knows their own evaporation rate, and it is the number that decides everything above. Put a strip of tape at the water line, do not top up for seven days, and measure how far the level fell. If it fell h centimeters and the water is d centimeters deep, your rate is h divided by d1.

Calculated by us1. Turning a weekly drop into a weekly percentage. Every cell is the drop divided by the depth
Water depth0.5 cm drop11 cm12 cm13 cm1
30 cm11.7 %3.3 %6.7 %10.0 %
35 cm11.4 %2.9 %5.7 %8.6 %
40 cm11.2 %2.5 %5.0 %7.5 %
45 cm11.1 %2.2 %4.4 %6.7 %

Correction 01

"Topping up is maintenance, so a careful keeper is safe."

The opposite. This is the one mechanism on the page that gets worse the more careful you are. Topping up promptly and keeping the level exactly right is precisely the behavior that draws the lines above. Someone who lets the tank drop and then refills with a big change is accidentally doing the right thing.

Correction 02

"Top up the tank."

Top up with what. If the top-up water is distilled or reverse osmosis water, it carries nothing, and evaporation plus top-off becomes a null operation. The line goes flat. This is the cheapest intervention available and almost nobody does it, because the advice is never phrased to make you ask the question.

Honest limit

The line is exact for some things and an overestimate for others.

Sodium and chloride have no sink at all, so the line is exact. Calcium bends below it, because snail shells, shrimp moults, pruned plant tissue and carbonate scale on the heater all remove some. So hardness may plateau while total dissolved solids keep climbing. A general hardness reading that has stopped rising has not stopped accumulating. It has run out of the one thing you were measuring.

The other half of the arithmetic

A water change does not reset the tank. It gives the tank a ceiling.

Put a water change back into the model. Change a fraction f of the volume each week, after topping up. Solve for the value that stops moving.

C_next = (C + e C_tap)(1 - f) + f C_tap C* = C_tap x ( 1 + e (1 - f) / f ) the value just after a change peak = C* + e C_tap the value just before the next one1

That equation is the whole argument in one line. With f at zero it has no solution, which is the mathematical way of saying there is no ceiling. With any f above zero there is a ceiling, and the ceiling is a fixed multiple of your tap water Change more often and the multiple gets smaller. It never reaches one, and it never goes below one.

Calculated by us1. Where a tank settles, at 5 percent evaporation a week, on Davtashen tap water3
Water changeSettles atHardnessDissolved solids
50 % weekly11.05x tap8.1 dGH1320 mg/L1
30 % weekly11.12x tap8.6 dGH1341 mg/L1
20 % weekly11.20x tap9.2 dGH1366 mg/L1
10 % weekly11.45x tap11.2 dGH1442 mg/L1
5 % weekly11.95x tap15.0 dGH1595 mg/L1
2 % weekly13.45x tap26.6 dGH11052 mg/L1
50 % monthly11.40x tap at the peak10.8 dGH1427 mg/L1
30 % monthly11.67x tap at the peak12.8 dGH1508 mg/L1
20 % monthly12.00x tap at the peak15.4 dGH1610 mg/L1
10 % monthly13.00x tap at the peak23.1 dGH1915 mg/L1
None at allno ceilingrises for everrises for ever

Turn it round and it becomes a decision you can actually make. Choose how far above your tap water you are willing to let the tank sit, call it M, and the weekly change fraction you need is f = e divided by (M minus 1 plus e)1.

Calculated by us1. Weekly change fraction needed to hold the tank at a chosen multiple of tap
Your evaporation rateWithin 10 % of tap1Within 20 %1Within 50 %1Within twice tap1
2 % a week, lidded117 % weekly19 % weekly14 % weekly12 % weekly1
5 % a week, typical133 % weekly120 % weekly19 % weekly15 % weekly1
8 % a week, dry and open144 % weekly129 % weekly114 % weekly17 % weekly1

Read that honestly, because it is smaller than the hobby's default advice. On evaporation grounds alone, a tank losing 5 percent a week needs a 20 percent weekly change to stay within a fifth of its tap water1. It does not need 50 percent. The 50 percent weekly rule comes from somewhere else5, and the section on how often says where.

  1. Our own calculation, not a published finding. The evaporation, top-off and ceiling model. Two lines: C(n) = C_tap (1 + n e) for topping up only, and C* = C_tap (1 + e (1 - f) / f) for the ceiling under a repeated change, with f = e / (M - 1 + e) inverted from it. Named inputs: Yerevan district tap water from 3; the hardness unit conversion from 4; and evaporation rates of 2, 5 and 8 percent a week, which are assumed parameters and not measurements, because no measured evaporation rate for an aquarium in Yerevan exists. Nobody has peer reviewed this. Re-run it rather than trust it. How we handle our own arithmetic
  2. Republic of Armenia sanitary standards. Norms N 2-III-A2-1 for the 1,000 mg/L total mineralisation limit, and Rules 2.1.4.001-23 of 2023, which drops the hardness limit but obliges the operator to publish hardness and dissolved mineral data for consumers from 2027. arlis.am
  3. Shahinyan, H. V. (2005). On the hydrogeochemical characteristics of some drinking waters supplying the city of Yerevan. Proceedings of the National Academy of Sciences of Armenia, Earth Sciences LVIII(1): 41 to 45. Sampling 2002 to 2004. Open access. The source for Yerevan tap water hardness and dissolved solids by district, the bicarbonate and sulfate fractions, and the finding that copper, zinc, lead, cadmium, iron and arsenic were at or below detection. PDF at arar.sci.am
  4. Hem, J.D. (1985). Study and Interpretation of the Chemical Characteristics of Natural Water, 3rd edition. US Geological Survey Water-Supply Paper 2254. Public domain. The hardness unit conversions used throughout, including 1 German degree equals 17.8 mg/L as calcium carbonate. Full text PDF
  5. Barr, T. The Estimative Index of Dosing. Barr Report. The 50 percent weekly change, and the statement that with it the maximum build-up is about twice the weekly dose. The origin of the hobby's default number. barrreport.com

Continue

Where to go next.

Water changes

The rest of the argument: zero on the kit, the rescue change, and how often.

Fifteen mechanisms, rated

What the accumulating material actually does, rated mechanism by mechanism.

The Armenian scene

The Yerevan tap water figures the model runs on, district by district.

Chemistry

Units done properly, because half the disagreements here are two people using different ones.

Old tanks and low-change tanks

Apply the arithmetic to an old, neglected or deliberately low-change aquarium.