Chapter 5, Problem 18P

The saturation concentration of dissolved oxygen in freshwater can be calculated with the equation (APHA, 1992)

$\begin{array}{l}\text{In }{o}_{sf}=-139.34411+\frac{1.575701\times {10}^5}{T_a}-\frac{6.642308\times {10}^7}{T_a^2}\\ +\frac{1.243800\times {10}^{10}}{T_a^3}-\frac{8.621949\times {10}^{11}}{T_a^4}\end{array}$

where $o_{sf}=$ the saturation concentration of dissolved oxygen in freshwater at 1 atm $\left(\text{mg/L}\right)$ and $T_a=$ absolute temperature $\left(\text{K}\right)$.

Remember that $T_a=T+273.15,\text{ where }T=\text{ temperature }\left(°\text{C}\right)$.

According to this equation, saturation decreases with increasing temperature. For typical natural waters in temperate climates, the equation can be used to determine that oxygen concentration ranges from $14.621\text{mg/L at 0}°\text{C to 6}\text{.413}\text{mg/L at 40}°\text{C}$. Given a value of oxygen concentration, this formula and the bisection method can be used to solve for temperature in $°\text{C}$.

(a) If the initial guesses are set as 0 and $40°\text{C}$, how many bisection iterations would be required to determine temperature to an absolute error of $0.05°\text{C}$?

(b) Develop and test a bisection program to determine T as a function of a given oxygen concentration to a prespecified absolute error as in (a) Given initial guesses of 0 and $40°\text{C}$, test your program for an absolute error $=0.05°\text{C}$ and the following cases: $o_{sf}=8,10,$ and 12 mg/L. Check your results.