Problem $3$: Fick's Law. A permanent marker works by providing a dye $($A$,40M$ from a

reservoir $(8cm$ long, $0\mathrm{.}7cm$ diameter$)$ to the top of a felt tip $(1cm$ long, $0\mathrm{.}7cm$ diameter$)$

that wicks the dye in$,$ diffuses through the felt, then instantly consumes it when it hits the

paper. The binary diffusion coefficient of the dye $(A)$ in the felt tip is $2{10}^{-6}c\frac{m^2}{s},$ but

much faster in air $(0\mathrm{.}069c\frac{m^2}{s}).$ You are being tasked with estimating how long the

marker will last under several conditions.

From equilibrium measurements, we know:

The dye is only partially soluble in the felt $($solubility $=3M).$

The c$\text{'}$

a$.$ Uncapped in Air. Assume that the felt tip presents with no diffusional resistance

relative to evaporation of the dye into the air $(B{i}_{m}1).$ This results in a $0\mathrm{.}27$kPa

partial pressure of dye in the air at a distance of $0\mathrm{.}23mm$ from the tip. How long

before the uncapped marker dries out? Hints: list all assumptions, start with a flux

expression $($symbolically$),$ then evaluate

b$.$ Ultrafine Tip in Air. The same marker was used, but the tip was replaced with an

ultrafine tip $(d_{\text{t}\text{i}\text{p}}=1(mm)).$ How long will it last?

c$.$ Felt Diffuser. There is a hypothesis that Markers will last longer if you place a small

diffuser that limits inadvertent dye evaporation. The felt is modified to slow the

diffusion of dye. The new material is a porous $1cm$ long felt tip $(B)$ and has very

slow diffusional properties $(D_{AB}=5{10}^{-5}c\frac{m^2}{s})$ relative to the external

convection $(B{i}_{m}1).$ How long will this marker with the fine ultrafine tip last?

d$.$ Any problems with the hypothesis at part c$?$ How would you fix it$?$

e$.$ Draw concentration profile for cases a$-$c$.$ Be sure to label: axes, domain$($s$)$ for the

whole marker and air, known concentrations, direction of flux, etc.