# Step 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 a step input; for example, the continuous release of industrial waste and mixing with river water. 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

Calculated value for E: m2/s

$E = 0.011 \frac{v^2*w^2}{d\sqrt{gds}}$
Step 4: Calculate first order rate constant
$ln(\frac{C}{C_o}) = -kt_\frac{1}{2}$

k =

Step 5: Verify data
Data Point Value Unit
Stream depth (d) m
Stream width (w) m
Mass input (W) Kg/s
Water velocity (v) m/s
Stream flow rate (Q) m3/s
Longitudinal dispersion coefficient (E) m2/s
First order rate constant (k) /second
Concentration calculations

Result:

Graph varying distance
$C_{(x,t)} = \frac{W}{Q\sqrt{1+\frac{4kE}{v^2}}}e^{\frac{vx}{2E}\bigl(1±\sqrt{1+\frac{4kE}{v^2}}\bigr)}$