Calculation of the Content of a Substance in a Lake

Used are hypsography and profile of the chemical

Assumptions: Variation of concentration is linear with depth between measured values and
fall line is straight (stump of a cone).

Black lines are measured, coloured lines are calculated with linear interpolation. If this is not done, the calculated volume of lake is not constant even if there is no variation in the water level.

The calculation uses: the upper area (A1), the lower area (A2), the upper concentration (C1), the lower concentration (C2) and the thickness of the layer (D z).

Content of one layer according to Kepler's barrel rule:

Content = D z/6*(A1*C1 + A2*C2 + 4* (C1+C2)/2 * ((√A1+ √A2)/2)2 )

i.e.
(content at upper boundary + content at upper boundary + 4* content in the middle) times thickness/6
where as the middle is the arithmetic average of the concentrations times the arithmetic mean of the square roots of areas.

For the first figure Kepler's rule has to be applied eight times for the total content.

Literature:
Bührer, H.:
Die Berechnung der totalen Menge gelöster Stoffe in Seen. Schweiz. Z. Hydrol. 41/2, 418-420 (1979).

 

P.S.
It's really the same Kepler with the planet orbits. As young assessor he invented this trick for exact integrals of a polynominal equation of third degree (applied on the volume of wine barrels). (published in Linz 1615)


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