# Streeter-Phelps Equation / DO Sag Curve

Sewage - not a pleasant thought! In this section of Fate you use the Streeter-Phelps model to look at the dissolved oxygen (DO) of a stream. Sewage and DO levels in streams are strongly tied to together, and this program models the DO concentration downstream from any constant input of organic waste.

Step 1: Enter problem statement
Step 2: Enter or calculate the longitudinal dispersion coefficient, E
Saturation concentration: mg O2/L
Results in a DO of: mg O2/L
$DO of stream = \frac{\% Saturation}{100}Saturation\ Concentration$
Step 3: Determine the temperature, DO, and BOD of the stream/waste mixutre
Stream Waste
Temperature °C
Flow rate
(in identical units)
Initial DO concentration mg/L
BOD5 mg/L
Calculations
Data Point Value Unit Equation
Equilibrium temperature °C $temp = \frac{(WasteFlow * WasteTemp) +(StreamFlow * StreamTemp)}{WasteFlow + StreamFlow}$
Initial DO of mixture mg 02/L $DO of Mixture = \frac{(WasteFlow * WasteDO) +(StreamFlow * StreamDO)}{WasteFlow + StreamFlow}$
Initial BOD5 of mixture mg BOD/L $BOD_5 = \frac{(WasteFlow * WasteBOD_5) +(StreamFlow * StreamBOD_5)}{WasteFlow + StreamFlow}$
BODL of mixture mg BOD/L $BOD_L = \frac{BOD_5}{1-e^{-5*k_1}}$
Step 4: Correct the BOD-rate constant and reaeraation-rate constant to the temperature of the mixture
Mixture temperature °C °C /day /day
Step 5: Data for the determination of critical points in the stream
Stream Temperature °C mg BOD/L mg O2/L mg O2/L mg O2/L
Calculation of critical points
Data Point Value Unit Equation
Initial DO deficit of the system (D0) mg O2/L $D_0 = Saturation\ Concentration\ at\ mixture\ temp - DO\ of\ mixture$
Time to reach the critical point (tc) days $t_c = \frac{1}{k'_2 - k'} ln(\frac{k'_2}{k'})(1-\frac{D_0(k'_2 - k')}{k'BOD_L})$
Location of critical point (Xc) km $X_c = Stream\ Velocity * t_c$
DO deficit at the critical point (DOL) mg O2/L $DO_L = Saturation\ Concentration - (\frac{k'\ corrected}{k'_2\ corrected} * BOD_L\ of\ mixture * e^{-k'\ corrected * t_c})$
DO at the critical point (DOc) mg O2/L $DO_L = Saturation\ Concentration - (\frac{k'\ corrected}{k'_2\ corrected} * BOD_L * e^{-k'\ corrected * t_c})$
BOD5 of a sample at the critical point (BOD5) mg BOD/L $BOD_5 = BOD\ of\ mixture (e^{-(k'corrected * t_c})(1-e^{-k'initial * 5})$
Calculate DO at a given distance below the input

Result: mg O2/L

$DO = \frac{\%\ Saturation}{100} * Saturation\ Concentration - (\frac{k'BOD_L}{k'_2 - k'}(e^{k' ({x / v})}-e^{k'_2 ({x / v})}) + D_0 * e^{k'_2 ({x / v})})$

Graph Data
$DO = \frac{\%\ Saturation}{100} * Saturation\ Concentration - (\frac{k'BOD_L}{k'_2 - k'}(e^{k' ({x / v})}-e^{k'_2 ({x / v})}) + D_0 * e^{k'_2 ({x / v})})$