Initial DO deficit of the system (D_{0}) 

mg O_{2}/L 
\[D_0 = Saturation\ Concentration\ at\ mixture\ temp  DO\ of\ mixture\] 
Time to reach the critical point (t_{c}) 

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 (X_{c}) 

km 
\[X_c = Stream\ Velocity * t_c\] 
DO deficit at the critical point (DO_{L}) 

mg O_{2}/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 (DO_{c}) 

mg O_{2}/L 
\[DO_L = Saturation\ Concentration  (\frac{k'\ corrected}{k'_2\ corrected} * BOD_L * e^{k'\ corrected * t_c})\] 
BOD_{5} of a sample at the critical point (BOD_{5}) 

mg BOD/L 
\[BOD_5 = BOD\ of\ mixture (e^{(k'corrected * t_c})(1e^{k'initial * 5})\] 