You do some $($cursory$)$ research into decaffeinating coffee beans. While there is apparently a market for coffee with no caffeine, there seems to be a bigger market for products that have caffeine without all that coffee nonsense. Caffeinated water is a thing $-$ water, a pure beverage that has been a human staple for millennia now needs a little bit of a pop so we add caffeine. What's next? Caffeinated condiments? Caffeinated salads!

Regardless of the eventual destination of the caffeine, we still need to get it from the coffee bean. In this problem, you will look into the process of caffeine removal where coffee beans are contacted with a solvent used to extract the caffeine. For this process, the following information is available:

the average diameter of a coffee bean is $0\mathrm{.}6cm$

The diffusivity of caffeine through the bean at process temperature is $1\mathrm{.}8E-10\frac{m^2}{s}$

a$)$ Assuming that there is no convective resistance in the solvent $(Bi),$ find the time required to reduce the amount of caffeine in the bean to $3\%$ of its original value. $($Use the Gurney$-$Lurie charts$)$

b$)$ If the size of the coffee bean is reduced by $50\%,$ what is the time required to remove the caffeine to $3\%$ of original content? Why did a smaller size impact the time?

c$)$ In parts a$)$ and b$),$ we assumed there is now convective resistance, however, you know the old saying about when you "assume"... In this scenario, convective resistance would be affected by the convective coefficient $k$ and amount of caffeine in the solvent. For this problem, recalculate the time required to remove the caffeine but account for convective resistance. Note the following:

Convective coefficient: $k=1\mathrm{.}25E-05c\frac{m}{s}$

Initial concentration of caffeine in bean : $20m\frac{g}{g}$

Bulk caffeine concentration ins solvent: $0\mathrm{.}21m\frac{g}{g}$