A particle of pure \(\mathrm{NaCl}\) is dissolving in an aqueous liquid solution at \(18^{\circ} \mathrm{C}\). The dissolution of the particle is controlled by mass transfer. The system is vigorously stirred, and the mass transfer coefficient \(\mathrm{k}=7.2 \times 10^{-5} \mathrm{~m} / \mathrm{s}\) is constant (see Section 19.4.4 for stirred-tank mass transfer correlations). The aqueous solution is at \(\mathrm{NaCl}\) mole fraction \(=0.0932\). Assume the system is ideal and the particle is spherical.

a. If the initial particle diameter is \(D_{\text {initial }}=2.05 \mathrm{~mm}\), find the time it takes to totally dissolve the particle.

b. If you assume system is dilute and ignore convection, \(\mathrm{N}_{\mathrm{A}, \mathrm{mol}}=-\mathrm{k}\left(\mathrm{x}_{\mathrm{A}, \text { bulk }}-\right.\) \(\left.\mathrm{x}_{\mathrm{A}, \mathrm{i}}\right) ho_{\text {liq,avg,mol }}\), what is dissolution time?

c. If convection is included, but assume stagnant fluid, \(\mathrm{N}_{\mathrm{B}}=0\), what is dissolution time?

Data: Solubility of \(\mathrm{NaCl}\) in water at \(18^{\circ} \mathrm{C}=9.99 \mathrm{~mol} \%\). \(\mathrm{MW} \mathrm{NaCl}=58.45\).

Density pure solid \(\mathrm{NaCl}=2.163 \mathrm{~g} / \mathrm{cm}^{3}\). Density of aqueous solution of \(\mathrm{NaCl}=1.20\) \(\mathrm{g} / \mathrm{cm}^{3}\) (assume constant). \(\mathrm{MW}_{\text {water }}=18.016\).