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Instantaneous Release into a Stream

In this section of Fate you can enter data to predict the concentration of a pollutant in a stream. The input of the pollutant is treated as an instantaneous input; for example, the immediate release and mixing of 50 gallons of acetone into a river. The model used here is a one-dimensional advection-dispersion equation that can also account for the first-order degradation or removal. A sample problem has been loaded with values similar to those you might need.

Step 1 Manually convert input data to metric units: meters, kilograms, or curies.
Step 2: Data entry
Step 3: Enter or calculate the longitudinal dispersion coefficient, E
Step 4: Calculate first order rate constant
\[ln(\frac{C}{C_o}) = -kt_\frac{1}{2} \]
Step 5: Verify data
Data Point Value Unit
Stream depth (d) m
Stream width (w) m
Total mass (M0) Kg
Water velocity (v) m/s
Longitudinal dispersion coefficient (E) m2/s
First order rate constant (k) /minute
Concentration calculations

Result:

Graph varying time
\[C_{(x,t)} = \frac{M_o}{wd\sqrt{4\pi Et}}e^{-\frac{(x-vt)^2}{4Et}-kt}\]
Graph varying distance
\[C_{(x,t)} = \frac{M_o}{wd\sqrt{4\pi Et}}e^{-\frac{(x-vt)^2}{4Et}-kt}\]
Graph varying distance and time
\[C_{(x,t)} = \frac{M_o}{wd\sqrt{4\pi Et}}e^{-\frac{(x-vt)^2}{4Et}-kt}\]
Distance:
Concentration 1:
Concentration 2:
Concentration 3:
Concentration 4: