A cylindrical roll of cloth was soaked in a dye mixture to color it. After the dye binds to the cloth, the cloth is soaked in a well-mixed tank of solvent to remove the unbound dye. The concentration of dye in the cloth after soaking is \(1000 \mathrm{~mol} / \mathrm{m}^{3}\). This concentration is to be reduced to \(10 \%\) of its starting value. If the solvent initially contains no dye and the soaking process can take no more than \(20 \mathrm{~min}\)., what volume of solvent is required? The cloth roll is \(0.1 \mathrm{~m}\) long and \(0.04 \mathrm{~m}\) in diameter. The diffusivity of dye in the cloth is \(D_{d c}=2 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) and the mass transfer coefficient in the tank is \(k_{c}=2 \times 10^{-5} \mathrm{~m} / \mathrm{s}\). Assume the solvent is well mixed at all times.