A short connecting pipe between two tanks is clogged with a plug of \(\mathrm{NaCl}\) crystals. The plug formed as a cylinder of circular cross-sectional area with a constant diameter \(D=2.0 \mathrm{~cm}\) and an initial length of \(1.0 \mathrm{~cm}\). The pipe is \(2.0 \mathrm{~cm}\) in diameter and prevents the plug from increasing its diameter. However, the plug can grow or shrink from the two ends (it gets longer or shorter but has no change in diameter). The pipe is \(10-\mathrm{cm}\) long. Initially, the crystal is in the middle of the pipe from \(z=4.5\) to \(z=5.5 \mathrm{~cm}\). Tank 1 on the \(\mathrm{z}=0\) side of the pipe contains pure water and is well mixed so that the bulk mass fraction of \(\mathrm{NaCl}, \mathrm{x}_{\mathrm{NaCl}, 1}=0\). Assume the solution density from \(\mathrm{z}=0\) to the crystal plug is the density of water. Tank 2 on the \(\mathrm{z}=10 \mathrm{~cm}\) side contains a wellmixed, super-saturated aqueous solution of \(\mathrm{NaCl}\) with a mass fraction \(\mathrm{NaCl} \mathrm{x}_{\mathrm{NaCl}, 2}=\) 0.37. The rate of growth is controlled by mass transfer.

a. Find the length of the crystal plug after \(10^{4}\) seconds.

b. How far is the center of the plug from tank 1 after \(10^{4}\) seconds?

c. How many seconds does it take for the plug to disappear?

d. If we close a valve on the pipe at the tank 2 side so that the plug is no longer connected to the concentrated \(\mathrm{NaCl}\) solution, how many seconds does it take for the plug to disappear?

Data: Solubility of \(\mathrm{NaCl}\) in water \(=0.2647\) mass fraction. \(\mathrm{MW} \mathrm{NaCl}=58.45\).

Density pure solid \(\mathrm{NaCl}=2.163 \mathrm{~g} / \mathrm{cm}^{3}\).

Density of aqueous solution of \(\mathrm{NaCl}\) with \(\mathrm{x}=0.37\) is \(1.22 \mathrm{~g} / \mathrm{cm}^{3}\) (assume constant).

Density of pure water \(=1.0 \mathrm{~g} / \mathrm{cm}^{3}\). MW pure water \(=18\).

Mass transfer coefficient on side 1 (pure water) \(\mathrm{k}_{1}=5.5 \times 10^{-6} \mathrm{~m} / \mathrm{s}\).

Mass transfer coefficient on side \(2\left(\mathrm{x}_{2}=0.37\right) \mathrm{k}_{1}=2.4 \times 10^{-6} \mathrm{~m} / \mathrm{s}\).

The plug is cylindrical. Diameter is constant, and L varies.