A tube of radius \(r_{0}=0.2 \mathrm{~m}\) is being dissolved by passing a reactive fluid through the interior. The reactant in the fluid is in great excess and so we can express the enhancement as a first-order chemical reaction with reaction rate constant, \(k^{\prime \prime}=1 \times 10^{-6} \mathrm{~s}^{-1}\). The liquid enters the tube with zero concentration of wall material and at a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\). Liquid density and viscosity barely change throughout the tube, being essentially that of water at \(25^{\circ} \mathrm{C}\). The solubility of the wall material in the liquid is \(c_{a w}=10 \mathrm{~kg}\)-mole \(/ \mathrm{m}^{3}\) and the diffusivity of solute in the liquid is \(D_{a w}=3.5 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s}\).

a. Derive a model describing the average concentration of solute within the liquid as a function of position along the tube.

b. Determine a value for the mass transfer coefficient.

c. What is the outlet concentration of the solute if the tube is \(3 \mathrm{~m}\) long?