Page 204 - Soil and water contamination, 2nd edition

11

Substance transport

11.1 MASS BALANCE

The amounts of chemical constituents are measured in mass quantities (e.g. mg, kg, or
tonnes) or related units (moles, or becquerels (Bq) in the case of radioactive elements). These
amounts are expressed as such, or as concentration per unit mass of water, rock, soil, or
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sediment (mass concentration, e.g. mg kg , % (percent), ppm (parts per million), ppb (parts
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-1
per billion)), or per unit volume (volume concentration, e.g. mg l , M (= moles l )). Since
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fresh water has a specific density of 1.00 kg l , the mass concentrations in fresh water are
approximately equal to volume concentrations. Furthermore, the amount of chemicals can
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also be expressed per unit area (e.g. Bq m , tonnes ha ) or per unit time (e.g. kg s , tonnes
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y ). The amount of chemicals or sediment transported per unit time is also referred to as
load .
The variety of options available for expressing the amount of substances stresses the need
for a consistent use of physical dimensions and units. A careful check for consistency of
units is a simple yet powerful manner to intercept errors in mathematical equations. To test
whether a calculation result has the correct dimensions, you should check that the left- and
right-hand sides of the equation resolve to the same units of measure. Besides giving insight
into the mathematical expression, a check like this will also reveal missing or redundant
terms that cause spurious units and erroneous results. For example, the chemical load of a
river can be calculated by:
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2
Load = Flow velocity (m s ) × Cross-sectional area (m ) × Concentration (g m ) (11.1)
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The units of the load would be (g s ) with dimensions [M T ]. Note that the unit for
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concentration (g m ) is equal to (mg l ). If we want to express the load in (tonnes y ),
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the load in (g s ) should be multiplied by (31 536 000 s y × 10 tonnes g = 31.536). If
the cross-sectional area of the river channel had been omitted from the equation, the load
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would have had the units of (g m s ) with dimensions [M L T ], which are clearly not the
dimensions for chemical load. On the other hand, it is also worth noting that consistency
of units and dimensions does not guarantee that the result is correct. For example, in
Equation (11.1) if we use the surface area of a part of the river instead of the cross-sectional
area, though the units are consistent the result is meaningless. If we calculate the inorganic
nitrogen load in a river but in Equation (11.1) we fill in the sum of the nitrate and
ammonium concentrations in the river water, the result is also obviously wrong even though
the units are again consistent. In this case, we have to convert the nitrate and ammonium
concentrations to nitrogen concentrations using stoichiometric constants:
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2
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Nitrogen load (g N s ) = Flow velocity (m s ) × Cross-sectional area (m ) ×
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(14/62 × Nitrate Concentration (g NO m ) +
3
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+
14/18 × Ammonium Concentration (g NH m )) (11.2)
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