Waste gases at a flowrate of \(15 \mathrm{~kg} / \mathrm{min}\) leave a plant through a chimney stack \(120 \mathrm{~m}\) high. The outer diameter of the chimney is \(1.5 \mathrm{~m}\) and the walls are made of brick, \(0.15 \mathrm{~m}\) thick. The gases also contain particulates that deposit on the chimney walls and must be removed periodically. One approach is to use a hot gas steam of mostly oxygen \(\left(180^{\circ} \mathrm{C}\right)\) to burn it off. The rate of reaction is first order in oxygen concentration.

\[-r_{p}=k^{\prime \prime} c_{O} \quad k^{\prime \prime}=5.5 \times 10^{-3} \mathrm{~s}^{-1}\]

The flow of gas is high enough that the heat generated by the reaction is easily dissipated. The oxygen concentration within the flow during the burnout period is also effectively constant at \(25 \mathrm{~mol} / \mathrm{m}^{3}\).

a. Calculate the temperature of the waste gases leaving the chimney when there are not deposits on the walls.

b. If the outlet temperature is determined to be \(145^{\circ} \mathrm{C}\), how thick are the particulate deposits on the wall?

c. For the conditions in part (b), determine the time required to remove \(90 \%\) of the deposit. You may neglect any variation in diameter when calculating the values of the transport coefficients.

Properties:

Gases: \(k=0.0312 \mathrm{~W} / \mathrm{m} \mathrm{K} \quad v=2.6 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\)

\[\operatorname{Pr}=0.68 \quad ho=0.73 \mathrm{~kg} / \mathrm{m}^{3} \quad C_{p}=1007 \mathrm{~J} / \mathrm{kg} \mathrm{K}\]

Particulate Deposit:

\[k=0.047 \mathrm{~W} / \mathrm{m} \mathrm{K} \quad ho=3.33 \times 10^{4} \mathrm{~mol} / \mathrm{m}^{3}\]

Atmospheric Conditions:

\[T_{\infty}=20^{\circ} \mathrm{C} \quad v_{\infty}=25 \mathrm{~km} / \mathrm{hr}\]