Temperature effect on BOD; 5-day BOD

Effect of temperature on the rate of decomposition

Decomposition of organic matter is done by bacteria, and their activity increases as temperature increases (up to about 35-40°C). The rate constant follows the van't Hoff-Arrhenius relation:

$$K_T = K_{20}\,\theta^{\,(T-20)}$$

where $\theta$ is the temperature coefficient, commonly taken as 1.047 (some books use 1.056 for 20-30°C and 1.135 for 4-20°C). With $K_{20}=0.1$ per day and $\theta=1.047$:

• Higher temperature: the reaction is faster, the BOD is satisfied in a shorter time, and also less oxygen dissolves in water. The ultimate BOD $L$ remains the same.
• Lower temperature: bacterial activity slows, the BOD exerted in 5 days is less, and the sewage decomposes more slowly.

 BOD exerted
   |        ______________ 30 C
   |     __/  _____________ 20 C
   |   _/  __/  ___________ 10 C
   |  / __/ __/
   +--------------------------> time

All curves reach the same ultimate BOD $L$, but at different times. For this reason the standard BOD test is done at 20°C.

Numerical

Given $y_1=100$ mg/L at 20°C, $K_{20}=0.1$ day⁻¹ (base 10).

$$\begin{aligned} L&=\frac{y_1}{1-10^{-K t}}=\frac{100}{1-10^{-0.1}}=\frac{100}{1-0.7943}=486.2\ \text{mg/L}\\ y_5&=L\left(1-10^{-0.1\times5}\right)=486.2\left(1-0.3162\right)=332.5\ \text{mg/L} \end{aligned}$$

Answer: Five-day BOD at 20°C = 332.5 mg/L (ultimate BOD = 486.2 mg/L).