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 Stream depth
d = m Stream width
w = m Mass input
W = Water velocity
v = m/s Stream flow rate
Q = m³/s

Step 3: Enter or calculate the longitudinal dispersion coefficient, E Channel slope s = (unitless and not required if E is entered) Gravity constant
g = m/s²

Calculated value for E: m²/s

\[E = 0.011 \frac{v^2*w^2}{d\sqrt{gds}}\]

Entered value for E
E = m²/s

Step 4: Calculate first order rate constant

Half-life

\[ln(\frac{C}{C_o}) = -kt_\frac{1}{2} \]

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 Distance
Km

Result:

Graph varying distance

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